#  Constrained Systems, Generalized Hamilton–Jacobi Actions, and Quantization

Alberto S. Cattaneo  Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>,
Pavel Mnev  University of Notre Dame  St. Petersburg Department of V. A.
Steklov Institute of Mathematics of the Russian Academy of Sciences
<EMAIL_ADDRESS>and  Konstantin Wernli  University of Notre Dame<EMAIL_ADDRESS>In memory of Kirill Mackenzie

###### Abstract.

Mechanical systems (i.e., one-dimensional field theories) with constraints are
the focus of this paper. In the classical theory, systems with infinite-
dimensional targets are considered as well (this then encompasses also higher-
dimensional field theories in the hamiltonian formalism). The properties of
the Hamilton–Jacobi (HJ) action are described in details and several examples
are explicitly computed (including nonabelian Chern–Simons theory, where the
HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative
quantization, limited in this note to finite-dimensional targets, is performed
in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of
the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity
check of the method, it is proved that the semiclassical contribution of the
physical part of the evolution operator is still given by the HJ action.
Several examples are computed explicitly. In particular, it is shown that the
toy model for nonabelian Chern–Simons theory and the toy model for 7D
Chern–Simons theory with nonlinear Hitchin polarization do not have quantum
corrections in the physical part (the extension of these results to the actual
cases is discussed in the companion paper [21]). Background material for both
the classical part (symplectic geometry, generalized generating functions, HJ
actions, and the extension of these concepts to infinite-dimensional
manifolds) and the quantum part (BV-BFV formalism) is provided.

###### Key words and phrases:

Hamilton–Jacobi, (generalized) generating functions, Chern–Simons,
Wess–Zumino–Witten, nonlinear (Hitchin) phase space polarization,
Kodaira–Spencer (BCOV) action, Batalin–Vilkovisky, Batalin–Fradkin–Vilkovisky

###### 2020 Mathematics Subject Classification:

81T70, 53D22, 70H20, 53D55, 53D50 (Primary); 81T13, 81S10, 70H15, 57R56, 81T45
(Secondary).

This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss
National Science Foundation. A.S.C. and K.W. acknowledge partial support of
SNF Grant No. 200020_192080. K. W. also acknowledges support from a BMS
Dirichlet postdoctoral fellowship and the SNF Postdoc.Mobility grant
P2ZHP2_184083, and would like to thank the Humboldt-Universität Berlin, in
particular the group of Dirk Kreimer, and the university of Notre Dame for
their hospitality.

###### Contents

  1. 1 Introduction
    1. 1.1 Structure of the paper
    2. 1.2 How to read this paper
    3. 1.3 Notations
    4. 1.4 Teaser
  2. 2 Hamilton–Jacobi for nondegenerate actions
  3. 3 Systems with one constraint
    1. 3.1 Nondegenerate Lagrange multiplier
    2. 3.2 The general case
  4. 4 Systems with several constraints
    1. 4.1 The strictly involutive case
    2. 4.2 The Lie algebra case
    3. 4.3 The general case
  5. 5 Systems with nontrivial evolution and constraints
    1. 5.1 Classical mechanics
    2. 5.2 The free relativistic particle
  6. 6 Generalized generating functions for “bad” endpoint conditions
    1. 6.1 No evolution and no constraints
    2. 6.2 The partial Legendre transform
  7. 7 Infinite-dimensional targets
    1. 7.1 Three-dimensional abelian Chern–Simons theory
    2. 7.2 Nonabelian Chern–Simons theory
    3. 7.3 Nonabelian $BF$ theory
    4. 7.4 More examples: 2D Yang–Mills theory and electrodynamics in general dimension
    5. 7.5 Higher-dimensional Chern-Simons theory
    6. 7.6 A nonlinear polarization in 7D Chern–Simons theory and the Kodaira–Spencer action
  8. 8 BFV, AKSZ and BV
  9. 9 An outline of elements of BV-BFV quantization
    1. 9.1 The classical BV-BFV setting
    2. 9.2 The quantum BV-BFV setting
  10. 10 BV-BFV boundary structures for linear polarizations
    1. 10.1 Three cases
    2. 10.2 The boundary structure in Case I
    3. 10.3 The boundary structure in Case II
    4. 10.4 The boundary structure in Case III
  11. 11 BV-BFV quantization with linear polarizations
    1. 11.1 Quantization in Case II
    2. 11.2 Quantization in Case III
    3. 11.3 Quantization in Case I
    4. 11.4 Gluing
    5. 11.5 Quantum mechanics
    6. 11.6 The quantum relativistic particle
  12. 12 BV-BFV quantization with nonlinear polarizations
    1. 12.1 Boundary structure
    2. 12.2 Gluing
    3. 12.3 The Case IIINL
    4. 12.4 The computation of $Z_{\text{INL}}$
  13. A Symplectic geometry and generating functions
    1. A.1 Symplectic spaces
    2. A.2 Symplectic manifolds
    3. A.3 Generalized Hamilton–Jacobi actions with infinite-dimensional targets
  14. B The modified differential quantum master equation
    1. B.1 Assumptions
    2. B.2 The propagator and the mdQME
    3. B.3 Proof of the mdQME
    4. B.4 Historical remarks

##  1\. Introduction

The Hamilton–Jacobi (HJ) action for a nondegenerate system is a generating
function for the graph of the hamiltonian flow of the system and is obtained
by substituting a solution of the Euler–Lagrange (EL) equations into the
action functional. We extend this result to degenerate systems (i.e., systems
with constraints). More precisely, we consider one-dimensional field theories
with constraints in involution (a.k.a. first-class constraints), i.e.,
theories constrained to live in a coisotropic submanifold of the (possibly
infinite-dimensional) target. In the finite-dimensional case, we also study
their perturbative quantization.

This paper is mainly based on a discussion of examples. The two main general
results, on the role of the HJ action, are presented in Theorem 4.2 on page
4.2 at the classical level and in Theorem 11.4 on page 11.4 at the quantum
level.

Among the many examples, we discuss in particular Chern–Simons theory and its
finite-dimensional toy model versions, get the gauged WZW action as the
generalized generating function of the evolution relation via the HJ action,
and, in the finite-dimensional version, show that there are no quantum
corrections in the physical sector. We will prove the latter result in the
infinite-dimensional case in the companion paper [21].

We start with a recollection of the known results for a regular mechanical
system (i.e., a one-dimensional field theory with regular lagrangian). In this
case, the EL equations define a hamiltonian flow and therefore its graph,
which we call the evolution relation, is a lagrangian submanifold of the
product of the symplectic manifolds associated to the two endpoints (in the
framework of this paper the symplectic manifold associated to an endpoint is
the target symplectic manifold, up to the sign of the symplectic form which
depends on the orientation of the endpoint). The HJ action, defined as the
evaluation of the classical action on solutions of the EL equation for a given
choice of endpoint polarizations111By polarization we mean, in this paper, an
explicit realization of a symplectic manifold as a cotangent bundle.
Generating functions, wave functions and integral kernels of operators will
always be understood as functions on the base of the cotangent bundle.
(boundary conditions), is then a generating function222A generating function
for a lagrangian submanifold $L$ of a cotangent bundle is a function on the
base that yields $L$ as the graph of its differential. for the evolution
relation. At the quantum level, the HJ action appears as the dominant
contribution in the semiclassical expansion of Feynman’s path integral (simply
as the evaluation of the action on a background classical solution).

In a constrained system, the following differences occur:

  1. (1)

In the framework of this paper,333with the exception of the first part of
Section 7.4 which portrays a more general situation the EL equations split
into two classes which we call the evolution equations (those that involve a
time derivative) and the constraints (those that do not involve time
derivatives). The fields actually split into two classes: maps to the target
symplectic manifolds and Lagrange multipliers; the evolution equations arise
from variations with respect to the former and the constraints from variations
with respect to the latter.

  2. (2)

Solutions of the EL equations still define a lagrangian submanifold—which we
keep calling the evolution relation—of the product of the symplectic manifolds
associated to the two endpoints. Yet, this submanifold is no longer a
graph.444Geometrically, the constraints define a coisotropic submanifold $C$,
and the evolution equations describe a motion along a leaf ot the
characteristic distribution. The evolution relation then consists of pairs of
points of $C$ that lie on the same leaf.

  3. (3)

The HJ action, defined as the evaluation of the classical action on solutions
of the evolution equation for a given choice of endpoint polarizations, is
then a generalized generating function555A generalized generating function is
a generating function, see footnote 2, that depends on additional parameters.
It defines the lagrangian submanifold of the given cotangent bundle as the
intersection of the graph of its differential with respect to the base
variables and the vanishing locus of the differential with respect to the
additional parameters. for the evolution relation. Note that the constraints
are not used in the definition of the HJ action but are recovered from it (as
they appear in the definition of the evolution relation).

  4. (4)

The HJ action depends parametrically on the Lagrange multipliers but is
invariant under a (Lie algebra or, more generally, Lie algebroid) gauge
transformation thereof that is trivial at the endpoints.

  5. (5)

At the quantum level, the path integral requires a gauge-fixing procedure one
has to keep track of. The formalism we use in this paper is the one due to
Batalin and Vilkovisky, in particular in its version coupled to the boundary,
referred to as the BV-BFV formalism. The HJ action then appears as the
physical (i.e., ghost independent) part of the dominant contribution of the
perturbative expansion of the path integral for certain choices of endpoint
polarizations and of residual fields (the latter being, in physical parlance,
a choice of infrared components, not to be integrated out, of the fields). The
HJ action now arises from the resummation of tree diagrams and not from the
evaluation of the action on a background classical solution, which is not at
hand in the BV-BFV formalism.

Points (3) and (4) are the content of Theorem 4.2 and point (5) of Theorem
11.4—see also equations (101) and (117).

One topic we are interested in is the change of (say, final) endpoint
polarization. This is realized by a generating function of the corresponding
symplectomorphism. At the classical level, the HJ action for the new
polarization is simply the composition of the HJ action for the old
polarization with the generating function for the change of polarization. At
the quantum level, one has to find an appropriate quantization of the latter
generating function. There are two, in general irreconcilable, wishes:
unitarity and compatibility with the BV-BFV formalism. In this paper we focus
on the latter and show that a BV-BFV quantization is possible if the
constraints are linear in the momenta of both endpoint polarizations. This is
a serious limitation, yet it applies to some very interesting examples.

###  1.1. Structure of the paper

This paper consists of three parts.

In the first part (Sections 2 to 6), we discuss the HJ action for constrained
systems (after briefly reviewing the regular, unconstrained case). We discuss
several examples (in particular constraints that are linear or affine in both
the positions and the momenta), prove Theorem 4.2, i.e., point (2) of the
above list, describe how to incorporate a nontrivial evolution in addition to
the constraints, and extend the discussion of generating functions and HJ
actions to some bad endpoint conditions.

In the second part (Section 7), we compute the HJ action for several examples
with infinite-dimensional target (Chern–Simons theory, $BF$ theory, 2D
Yang–Mills theory, abelian Yang–Mills theory in any dimension).

In the third part (Sections 8 to 12), we discuss the BV-BFV quantization. We
start with a review of BV and BFV, tailored to the theories we consider in
this paper (which are ultimately one-dimensional AKSZ models). Next we discuss
the quantization in the case of finite-dimensional targets, both with linear
and nonlinear endpoint polarizations, proving in particular Theorem 11.4,
i.e., point (5) of the above list, and show that in the linear and affine
examples (the toy models for Chern–Simons theory) no quantum corrections arise
in the physical sector: see equations (108), (109) and (118).

In Appendix A we review several results on symplectic geometry and on
(generalized) generating functions, both in the finite- and in the infinite-
dimensional case.

In Appendix B we prove a technical result that ensures that the BV-BFV
quantization works in the case at hand.

We defer the BV-BFV quantization with infinite-dimensional target (mainly,
Chern–Simons theories) to the companion paper [21].

###  1.2. How to read this paper

The first part is necessary for the whole paper, with the exception of Section
5 which is relevant only for Sections 7.4, 11.5 and 11.6.

The second and the third part can be read independently from each other, as
they focus on different developments of the first part.

Appendix A contains additional material that completes the discussion in the
first part and provides a rigorous foundation for the second part. Appendix B
contains results for the foundation of the third part.

###  1.3. Notations

To simplify the reading of the paper, we collect here some of the main
notations and some terminology.

Arguments (e.g., of an action, a HJ action, a partition function) are written
in square brackets if they are fields (i.e., if they depend on the variable on
the source interval) and in round brackets otherwise.

We will use the subscript HJ to denote a HJ action. We will have several
versions thereof, with notations like $S_{\text{HJ}}$, $S^{f}_{\text{HJ}}$ or
$\hat{S}^{f}_{\text{HJ}}$ referring to different incarnations (change of
polarization by $f$; dependency on gauge classes of the parameters). This
pedantic notation is important to avoid confusion, but the only important
point to focus on is that these are all HJ actions in their own way. In the
arguments of HJ actions, or more generaly of generalized generating functions,
we will consistenly separate the generalized position variables from the
parameters by a semicolon.

In the BFV formalism, the classical master equation (CME) is the equation
$\\{S,S\\}=0$, where $\\{\ ,\ \\}$ is the given even Poisson bracket and $S$
is an odd functional. By $\Omega$ we will denote a coboundary operator that
quantizes $S$.

In the BV formalism, the CME is the equation $(S,S)=0$, where $(\ ,\ )$ is the
given odd Poisson bracket and $S$ is an even functional. By $\Delta$ we will
denote the BV Laplacian (on functions or on half-densities, depending on the
context). The quantum master equation (QME) is the equation $\Delta\psi=0$,
where $\psi$ is an even function or half-density. For $\psi$ of the form
$\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S}$, under the condition $\Delta S=0$
(which we usually assume), the QME for $\psi$ is equivalent to the CME for
$S$.

In the BV-BFV formalism, where we have both $\Delta$ and $\Omega$, the
modified quantum master equation (mQME) is the equation
$(\Omega+\hbar^{2}\Delta)\psi=0$. We often use the abbreviation
$\Delta^{\Omega}\mathrel{\mathop{\mathchar
58\relax}}=\Delta+\frac{1}{\hbar^{2}}\Omega$.

For a partition function (or a state), we will use the notation $Z$, possibly
with some label denoting the case we are considering. If we define it as a
half-density, we will use the notation $\hat{Z}$.

###  1.4. Teaser

We illustrate one of the results of the paper (summarizing the content of
Sections 4.2, 11.3, and 12, with simplified, adapted notations for this
introduction).

The data of the system are a quadratic Lie algebra $\mathfrak{g}$, a vector
space $Z$ and two derivations: $v\colon\mathfrak{g}\to\mathfrak{g}\otimes Z$
and $\bar{v}\colon\mathfrak{g}\to\mathfrak{g}\otimes Z^{*}$. The target
symplectic manifold is $T^{*}(\mathfrak{g}\otimes Z)$ which we identify with
$\mathfrak{g}\otimes Z^{*}\oplus\mathfrak{g}\otimes Z\ni(\bar{q},q)$ (using
the nondegenerate pairing $\langle\ ,\rangle$ on $\mathfrak{g}$).666We write
$\bar{q}$ for the image of the momentum $p$ under the isomorphism
$\mathfrak{g}^{*}\otimes Z^{*}\to\mathfrak{g}\otimes Z^{*}$. The fields are a
path $(\bar{q},q)\colon[0,1]\to T^{*}(\mathfrak{g}\otimes Z)$ and a
$\mathfrak{g}$-valued one-form $e$ (i.e., a connection one-form) on $[0,1]$.
The action functional, a toy model for nonabelian Chern–Simons theory (see
Section 7.2),777In this case, we have
$\mathfrak{g}=\Omega^{0}(\Sigma,\mathfrak{g}^{\prime})$ and
$Z=\Omega^{0,1}(\Sigma)$, where $\mathfrak{g}^{\prime}$ is a finite-
dimensional quadratic Lie algebra, and $\Sigma$ is a closed, oriented surface
with a choice of complex structure. The derivations $v,\bar{v}$ are now the
Dolbeault operators $\partial,\bar{\partial}$. is

$S[\bar{q},q,e]=\int_{0}^{1}(\langle\bar{q},\mathrm{d}_{e}q\rangle-\langle\bar{q},v(e)\rangle-\langle\bar{v}(e),q\rangle),$

where $\mathrm{d}_{e}$ denotes the covariant derivative.

Under the assumption that the exponential map to the simply connected Lie
group $G$ integrating $\mathfrak{g}$ is surjective, we compute the HJ
action—for initial value $q_{a}$ of $q$, final value $\bar{q}^{b}$ of
$\bar{q}$, and a parameter $g=\mathrm{e}^{\xi}\in G$—as

$\hat{S}_{\text{HJ}}(q_{a},\bar{q}^{b};g)=-\langle\bar{q}^{b},{g}^{-1}q_{a}g\rangle-\langle\bar{q}^{b},{g}^{-1}v(g)\rangle-\langle\bar{v}(g)g^{-1},q_{a}\rangle-\text{WZW}(g)$

with the Wess–Zumino–Witten term

$\text{WZW}(g)=\frac{1}{2}\langle\bar{v}(g)g^{-1},v(g)g^{-1}\rangle+\frac{1}{2}\int_{0}^{1}\langle\bar{v}(h)h^{-1},[v(h)h^{-1},\mathrm{d}hh^{-1}]\rangle,$

where $h=\mathrm{e}^{t\xi}$.

The BV-BFV quantization then provides an integral kernel which, in adddition
to the classical coordinates $(q_{a},\bar{q}^{b})$, also depends on “ghost
variables” $c_{a},c_{b}\in\mathfrak{g}[1]$ and is a half-density on
$T^{*}[-1]G\ni(g,g^{+})$:

$\hat{Z}(q_{a},c_{a},\bar{q}^{b},c_{b};g,g^{+})=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\hat{S}_{\text{HJ}}(q_{a},\bar{q}^{b};g)}\;\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\langle
g^{+},gc_{a}-c_{b}g\rangle}\,\mu_{G},$

where $\mu_{G}$ is the Haar measure.

This integral kernel satisfies the modified quantum master equation (mQME)
$(\Omega+\hbar^{2}\Delta)\hat{Z}=0$—and is defined up to the image of
$\Omega+\hbar^{2}\Delta$—where $\Delta$ is the canonical BV Laplacian on half-
densities on $T^{*}[-1]G$ and

$\begin{split}\Omega&=\left\langle\mathrm{i}\hbar\frac{\partial}{\partial
q_{a}},v(c_{a})\right\rangle+\left\langle\bar{v}(c_{a}),q_{a}\right\rangle-\frac{\mathrm{i}\hbar}{2}\left\langle[c_{a},c_{a}],\frac{\partial}{\partial
c_{a}}\right\rangle\\\
&\phantom{=}-\left\langle\bar{q}^{b},v(c_{b})\right\rangle-\left\langle\bar{v}(c_{b}),\mathrm{i}\hbar\frac{\partial}{\partial\bar{q}^{b}}\right\rangle-\frac{\mathrm{i}\hbar}{2}\left\langle[c_{b},c_{b}],\frac{\partial}{\partial
c_{b}}\right\rangle\end{split}$

is a quantization of the endpoint BFV action describing the coisotropic
reduction by the constraints.

In the abelian case, where we use the multiplicative convention for the group
$G$, we write $g=\mathrm{e}^{T}$, and the HJ action becomes

$\hat{S}_{\text{HJ}}(q_{a},\bar{q}^{b};T)=-\langle\bar{q}^{b},q_{a}\rangle-\langle\bar{q}^{b},v(T)\rangle-\langle\bar{v}(T),q_{a}\rangle-\frac{1}{2}\langle\bar{v}(T),v(T)\rangle.$

In this case it is also interesting, as a toy model for 7D Chern–Simons theory
with endpoint Hitchin polarization (see Section 7.6), to change the final
Darboux coordinates by some generating function $f(q,Q)$ with
$\langle\bar{q},\mathrm{d}q\rangle=\langle\bar{Q},\mathrm{d}Q\rangle+\mathrm{d}f$.
The HJ action for initial value $q_{a}$ of $q$ and final value $Q_{b}$ of $Q$
is then

$\hat{S}_{\text{HJ}}^{f}(q_{a},Q_{b};T)=-f(q_{a}+v(T),Q_{b})-\langle\bar{v}(T),q_{a}\rangle-\frac{1}{2}\langle\bar{v}(T),v(T)\rangle.$

The BV-BFV quantization turns out to be possible when the transformed
$\mathfrak{g}^{*}$-valued hamiltonian
$\tilde{H}(\bar{Q},Q)=v^{*}(\bar{q}(\bar{Q},Q))+\bar{v}^{*}(q(\bar{Q},Q))$,
with $*$ denoting transposition, is linear in $\bar{Q}$. In this case, the
integral kernel, with $(T,T^{+})\in T^{*}[-1]\mathfrak{g}$, is

$\hat{Z}^{f}(q_{a},c_{a},Q_{b},c_{b};T,T^{+})=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};T)}\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}T^{+}_{\alpha}\,(c_{a}^{\alpha}-c_{b}^{\alpha})}\sqrt{\mathrm{d}^{k}T\mathrm{d}^{k}T^{+}},$

with $k=\dim\mathfrak{g}$. This integral kernel also satisfies the modified
quantum master equation (mQME), with $\Delta$ the canonical BV Laplacian on
half-densities on $T^{*}[-1]\mathfrak{g}$ and

$\Omega=\left\langle\mathrm{i}\hbar\frac{\partial}{\partial
q_{a}},v(c_{a})\right\rangle+\left\langle\bar{v}(c_{a}),q_{a}\right\rangle-\left\langle
c_{b},\tilde{H}\left(-\mathrm{i}\hbar\frac{\partial}{\partial
Q_{b}},Q_{b}\right)\right\rangle.$

######  Remark 1.1 (Parameters and quantum corrections).

In the examples just presented, there are no quantum corrections in the
physical part of the partition function $\hat{Z}$. This statement if of course
correct only if we compare the classical and quantum theory in the presence of
parameters. The point is that usually one can get rid of (some of) the
parameters. Let us discuss for simplicity the case when all parameters can be
eliminated. In the classical theory, the constraints are recovered by setting
to zero the differential of the HJ action as a function of the parameters $g$
or $T$ (parametrized by the generalized position variables). If the Hessian is
nondegenerate, we can (locally) solve these equations and write the parameters
as functions of the generalized position variables. Inserting this back into
the HJ action yields a new, equivalent generating function, now without
parameters. At the quantum level, we can instead integrate out the parameters
$g$ or $T$ in the partition function, after setting their odd momenta $g^{+}$
or $T^{+}$ to zero. In the abelian case with linear polarization (generalized
position variables $q_{a}$ and $\bar{q}_{b}$), this is equivalent to the
classical procedure (apart from an irrelevant constant depending on the
determinant of the quadratic form $\langle\bar{v}(T),v(T)\rangle$). In the
other cases (abelian case with nonlinear polarization and nonabelian case),
the quantum procedure yields nontrivial quantum corrections.

### Acknowledgements

We thank Samson Shatashvili for suggesting the study of 7D abelian
Chern–Simons theory in the quantum BV-BFV formalism, now in [21, Section 6.3],
which was the original motivation out of which this paper grew. We also thank
Francesco Bonechi, Ivan Contreras, Philippe Mathieu, Nicolai Reshetikhin,
Pavel Safronov, Michele Schiavina, Stephan Stolz, Alan Weinstein, Ping Xu and
Donald Youmans for useful discussions.

##  2\. Hamilton–Jacobi for nondegenerate actions

In this section we collect several known facts on the Hamilton–Jacobi action,
which will be used in the rest of the paper (for a general reference, see,
e.g., [3, Sections 47-48]). We only focus on actions of the form

$S[p,q]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-H(p,q)\mathrm{d}t),$

where a sum over repeated indices is understood. Here $(p,q)$ denotes a point
(or a path) in the cotangent bundle $T^{*}M$ of a fixed manifold $M$, and $H$
is a smooth function on $T^{*}M$. By abuse of notation, we denote by
$\mathrm{d}q^{i}$ both the coordinate 1-form on $M$ and its pullback by the
map $q\colon[t_{a},t_{b}]\to M$ (which then actually means
$\dot{q}^{i}\mathrm{d}t$). We will later be interested in some
generalizations, like adding constraints and working with an infinite-
dimensional target.

Recall that the Euler–Lagrange (EL) equations for the above action are
Hamilton’s equations

$\dot{q}^{i}=\frac{\partial H}{\partial
p_{i}},\quad\dot{p}_{i}=-\frac{\partial H}{\partial q^{i}}.$

Next assume that for every $q_{a},q_{b}\in M$ these equations have a unique
solution, which we will denote by $(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$,
satisfying $q_{q_{a},q_{b}}(t_{a})=q_{a}$ and $q_{q_{a},q_{b}}(t_{b})=q_{b}$.
Then one defines the Hamilton–Jacobi (HJ) action (a.k.a. Hamilton’s principal
function)

$S_{\text{HJ}}(q_{a},q_{b})\mathrel{\mathop{\mathchar
58\relax}}=S[p^{q_{a},q_{b}},q_{q_{a},q_{b}}].$

Jacobi’s theorem then, in particular, asserts that

(1) $p^{q_{a},q_{b}}_{i}(t_{b})=\frac{\partial S_{\text{HJ}}}{\partial
q^{i}_{b}},\quad p^{q_{a},q_{b}}_{i}(t_{a})=-\frac{\partial
S_{\text{HJ}}}{\partial q^{i}_{a}}.$

In other words, $S_{\text{HJ}}$ is a generating function for the graph of the
hamiltonian flow of $H$ (from $t_{a}$ to $t_{b}$), see Appendix A (in
particular, Remark A.32). Let us briefly review the proof because we will need
some generalization thereof in the following. The variation of the action
reads

$\delta S=\text{\sl EL}+p_{i}(t_{b})\delta q^{i}(t_{b})-p_{i}(t_{a})\delta
q^{i}(t_{a}),$

where EL denotes the bulk term responsible for the EL equations. If we insert
a solution of the EL equations, the EL term vanishes. In particular, inserting
the solution $(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ yields

$\delta S_{\text{HJ}}=p^{q_{a},q_{b}}_{i}(t_{b})\delta
q^{i}_{b}-p^{q_{a},q_{b}}_{i}(t_{a})\delta q_{a}^{i},$

which implies (1).

######  Remark 2.1 (Time dependency).

Above we have considered the time endpoints $t_{a}$ and $t_{b}$ as fixed, but
one can let them vary as well. We put the explicit dependence in the notation:

$S[p,q](t_{a},t_{b})=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-H(p,q)\mathrm{d}t).$

The time-dependent HJ action is then

(2) $S_{\text{HJ}}(q_{a},q_{b};t_{a},t_{b})\mathrel{\mathop{\mathchar
58\relax}}=S[p^{q_{a},q_{b}},q_{q_{a},q_{b}}](t_{a},t_{b}),$

where the solution is still required to satisfy $q_{q_{a},q_{b}}(t_{a})=q_{a}$
and $q_{q_{a},q_{b}}(t_{b})=q_{b}$. As a result,
$(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ also depends on $t_{a}$ and $t_{b}$ (even
though we do not include this in the notation). In addition to the formulae in
(1), we now also have888Equations (1) and (3) are also important because they
imply that $S_{\text{HJ}}$ is a solution of the Hamilton–Jacobi equation.

(3) $\frac{\partial S_{\text{HJ}}}{\partial
t_{b}}=-H(p^{q_{a},q_{b}}(t_{b}),q_{b}),\quad\frac{\partial
S_{\text{HJ}}}{\partial t_{a}}=H(p^{q_{a},q_{b}}(t_{a}),q_{a}).$

To get these formulae, we have to observe that, e.g., a variation $\delta
t_{b}$ of $t_{b}$ produces not only a variation in the final endpoint of the
integral, which yields
$(p^{q_{a},q_{b}}_{i}(t_{b})\dot{q}_{q_{a},q_{b}}^{i}(t_{b})-H(p^{q_{a},q_{b}}(t_{b}),q_{b}))\delta
t_{b}$, but also an induced variation $-\dot{q}_{q_{a},q_{b}}(t_{b})\delta
t_{b}$ of $q$ at the final endpoint.

######  Remark 2.2 (Uniqueness).

For simplicity we have assumed that there is a unique solution
$(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ satisfying $q_{q_{a},q_{b}}(t_{a})=q_{a}$
and $q_{q_{a},q_{b}}(t_{b})=q_{b}$. This might not be true. Generically one
may exclude focal points $q_{a},q_{b}$ connected by a continuous family of
solutions,999For instance, if the target $M$ is a Riemannian manifold with
metric $g$ and we take $H(p,q)=\frac{\|p\|_{g}^{2}}{2m}$, then the solutions
to the EL equations are cotangent lifts of parametrized geodesics. If $M$ is
the sphere with round metric and we take antipodal points $q_{a}$ and $q_{b}$,
then we have a continuous family of solutions joining them. However, it is
enough to perturb one of them, to get a discrete family. but typically we may
expect a discrete, or finite, family to exist. In this case, the above
procedure is actually meant to work in a neighborhood of a given nonfocal pair
$(q_{a},q_{b})$ for a choice of one solution in the discrete set. When one
varies $q_{a}$ or $q_{b}$ (or the time interval), it is understood that one
follows the variation of the chosen solution. The resulting HJ action is then
a local generating function. To get the whole graph of the hamiltonian flow of
$H$ one needs the various, possibly infinitely many, HJ actions associated to
the different solutions.

######  Example 2.3.

A simple example where uniqueness of solutions with given endpoints is not
satisfied is the free particle on $S^{1}$, the unit circle.101010The
nonuniqueness in this example is due to the fact that $S^{1}$ is not simply
connected, but it is possible to find examples of nonuniqueness on simply
connected configuration spaces as well. The hamiltonian is simply
$H(p,q)=\frac{\,p^{2}}{2m}$. The EL equations are $\dot{p}=0$ and
$\dot{q}=\frac{p}{m}$. The graph of the hamiltonian flow of $H$ is then

$L\mathrel{\mathop{\mathchar 58\relax}}=\left\\{(p^{a},q_{a},p^{b},q_{b})\in
T^{*}S^{1}\times T^{*}S^{1}\ |\ p^{b}=p^{a},\
q_{b}=q_{a}+\frac{p^{a}}{m}(t_{b}-t_{a})\bmod 2\pi\right\\}.$

On the other hand, if we fix $q_{a}$ and $q_{b}$, we have the solutions

$\begin{split}p(t)&=\frac{m(q_{b}-q_{a}+2k\pi)}{t_{b}-t_{a}},\\\
q(t)&=q_{a}+(q_{b}-q_{a}+2k\pi)\frac{t-t_{a}}{t_{b}-t_{a}}\bmod
2\pi,\end{split}$

for each $k\in\mathbb{Z}$. It follows that, for a given $k$, the HJ action
reads

$S_{\text{HJ},k}(q_{a},q_{b})=\frac{1}{2}m\frac{(q_{b}-q_{a}+2k\pi)^{2}}{t_{b}-t_{a}},$

which is indeed a generating function for $L$ in a neighborhood of
$(q_{a}^{0},p_{0},q_{b}^{0},p_{0})$ with
$p_{0}=\frac{m(q_{b}^{0}-q_{a}^{0}+2k\pi)}{t_{b}-t_{a}}$. To generate the
whole $L$, one needs all the $S_{\text{HJ},k}$s.

######  Remark 2.4 (Semiclassical approximation).

In the path integral approach to quantum mechanics (say, on the configuration
space $M=\mathbb{R}^{n}$), the integral kernel $K(q_{a},q_{b})$ for the
evolution operator is, formally, given by the path integral [26, 28, 29]

(4) $K(q_{a},q_{b})=\int_{\begin{subarray}{c}q(t_{a})=q_{a}\\\
q(t_{b})=q_{b}\end{subarray}}Dp\,Dq\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S[p,q]}.$

Assuming for simplicity that there is a unique solution
$(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ satisfying $q_{q_{a},q_{b}}(t_{a})=q_{a}$
and $q_{q_{a},q_{b}}(t_{b})=q_{b}$, one usually makes the affine change of
variables (translation)

(5) $p\mapsto p^{q_{a},q_{b}}+\hat{p},\quad q\mapsto q_{q_{a},q_{b}}+\hat{q},$

getting

$K(q_{a},q_{b})=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S_{\text{HJ}}(q_{a},q_{b})}\int_{\hat{q}(t_{a})=\hat{q}(t_{b})=0}D\hat{p}\,D\hat{q}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\hat{S}_{(q_{a},q_{b})}[\hat{p},\hat{q}]}$

with

$\hat{S}_{(q_{a},q_{b})}[\hat{p},\hat{q}]=S[p^{q_{a},q_{b}}+\hat{p},q_{q_{a},q_{b}}+\hat{q}]-S_{\text{HJ}}(q_{a},q_{b}).$

Since $(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ is a solution,
$\hat{S}_{(q_{a},q_{b})}$ starts with a quadratic term in $(\hat{p},\hat{q})$,
so one can define the path integral perturbatively. One then sees that the HJ
action yields the semiclassical approximation to $K$. The crucial point here
is that the Hessian of $S$ at a solution is nondegenerate (one also says that
the action or the lagrangian is regular). This does not happen for degenerate
actions (e.g., in gauge theories). In this case one has to use some technique
to gauge fix the action, but one also has to take into account that the
symplectic space of initial conditions is defined via some symplectic
reduction. A cohomological way to resolve these issues is to use the BV
formalism [7] in the bulk and the BFV formalism [6, 30] on the boundary. In
[19] it was shown how to reconcile the two formalisms, into what was termed
the quantum BV-BFV formalism, and it was in particular observed that, in
general, extending boundary values to a solution of the EL equations is
incompatible with the formalism (we will review this in Sections 9.2.3 and
9.2.6). The above argument to show that the semiclassical contribution to the
integral kernel of the evolution operator is the HJ action is therefore no
longer valid. We will see in Theorem 11.4 that the result still holds, yet the
HJ action is recovered not by evaluating the classical action on a solution
but via resumming trees in the Feynman diagrams expansion.

######  Remark 2.5 (Changing the endpoint conditions).

Fixing $q$ at the endpoints yields the EL equations as critical points for
$S$. One may however be interested in choosing different endpoint conditions.
In general, they have to correspond to a lagrangian submanifold of $T^{*}M$,
not necessarily the zero section, but one has to adapt the one-form
$p_{i}\mathrm{d}q^{i}$. For simplicity, we now assume $M=\mathbb{R}^{n}$. Let
$(P_{i},Q^{i})$ be another set of Darboux coordinates on $T^{*}\mathbb{R}^{n}$
and assume (see also Example A.33 for a more general viewpoint) that we have a
generating function $f(q,Q)$ for the transfomation: namely,

$p_{i}\mathrm{d}q^{i}=P_{i}\mathrm{d}Q^{i}+\mathrm{d}f,$

i.e.,

$p_{i}=\frac{\partial f}{\partial q^{i}},\quad P_{i}=-\frac{\partial
f}{\partial Q^{i}}.$

Assume we want to fix $Q$ at the final endpoint and $q$ at the initial
endpoint. We then change the action by a boundary term:

$S^{f}[p,q]\mathrel{\mathop{\mathchar
58\relax}}=-f(q(t_{b}),Q(p(t_{b}),q(t_{b})))+\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-H(p,q)\mathrm{d}t).$

When we take a variation, we now get

$\delta S^{f}=\text{\sl EL}+P_{i}(t_{b})\delta Q^{i}(t_{b})-p_{i}(t_{a})\delta
q^{i}(t_{a}),$

with the same EL term as before. Therefore, the solutions of the EL equations
are the critical points of $S^{f}$ with fixed $q$ at the initial endpoint and
$Q$ at the final endpoint. We define the corresponding HJ action as

$S^{f}_{\text{HJ}}(q_{a},Q_{b})\mathrel{\mathop{\mathchar
58\relax}}=S^{f}[p^{q_{a},Q_{b}},q_{q_{a},Q_{b}}],$

where $(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})$ is the (assumedly unique) solution
ot the EL equations with $q_{q_{a},Q_{b}}(t_{a})=q_{a}$ and
$Q(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b}))=Q_{b}$. Exactly as in the
case discussed at the very beginning of this section, inserting a solution
into the variation of $S^{f}$ kills the $EL$ term, so we get

$P_{i}(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})(t_{b})=\frac{\partial
S^{f}_{\text{HJ}}}{\partial Q^{i}_{b}},\quad
p^{q_{a},Q_{b}}_{i}(t_{a})=-\frac{\partial S^{f}_{\text{HJ}}}{\partial
q^{i}_{a}}.$

In other words, $S^{f}_{\text{HJ}}$ is a generating function for the graph of
the hamiltonian flow of $H$ (from $t_{a}$ to $t_{b}$) with respect to the new
set of coordinates (see again Remark A.32). The dependency on time can be
studied repeating the steps of Remark 2.1 verbatim.

######  Example 2.6.

Consider the case when $H=0$, i.e.,
$S[p,q]=\int_{t_{a}}^{t_{b}}p_{i}\mathrm{d}q^{i}$. Since the EL equations have
constant solutions, we have $S[p^{q_{a},Q_{b}},q_{q_{a},Q_{b}}]=0$ and
$q(t_{b})=q_{a}$. Therefore

(6) $S^{f}_{\text{HJ}}(q_{a},Q_{b})=-f(q_{a},Q_{b}).$

######  Remark 2.7.

For simplicity, we only discuss the change of the final endpoint conditions,
but of course the same discussion may be repeated for the initial endpoint and
for both endpoints at the same time.

######  Example 2.8.

A very common case is when one wants to fix $q$ at the initial point and $p$
at the final point. This fits in the above construction by choosing $Q=p$ and
$P=-q$ and $f(q,Q)=Q_{i}q^{i}$. In this case, we simply have

$S^{f}[p,q]\mathrel{\mathop{\mathchar
58\relax}}=-p_{i}(t_{b})q^{i}(t_{b})+\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-H(p,q)\mathrm{d}t),$
$\delta S^{f}=\text{\sl EL}-q^{i}(t_{b})\delta p_{i}(t_{b})-p_{i}(t_{a})\delta
q^{i}(t_{a}),$

and

(7) $q_{q_{a},p^{b}}^{i}(t_{b})=-\frac{\partial S^{f}_{\text{HJ}}}{\partial
p_{i}^{b}},\quad p^{q_{a},p^{b}}_{i}(t_{a})=-\frac{\partial
S^{f}_{\text{HJ}}}{\partial q^{i}_{a}}.$

If we apply this to the Example 2.3 of the free particle on the circle, we get
a unique solution for a given $(q_{a},p^{b})$ and the HJ action

$S^{f}_{\text{HJ}}(q_{a},p^{b})=-p^{b}q_{a}-\frac{(p^{b})^{2}}{2m}(t_{b}-t_{a}).$

If we instead consider the case $H=0$ of Example 2.6, we get, as a particular
case of (6),

(8) $S^{f}_{\text{HJ}}(q_{a},p^{b})=-p^{b}q_{a}.$

######  Remark 2.9 (Legendre transform).

One can pass from $S^{f}_{\text{HJ}}(q_{a},p^{b})$ to
$S_{\text{HJ}}(q_{a},q_{b})$, and back, by the Legendre transform:

$S_{\text{HJ}}(q_{a},q_{b})=\lambda^{\text{crit}}_{i}q_{b}^{i}+S^{f}_{\text{HJ}}(q_{a},\lambda^{\text{crit}}),$

where $\lambda^{\text{crit}}$ is the critical point (assumed to be unique) of
the variable $p^{b}$ for the function $f(p^{b})\mathrel{\mathop{\mathchar
58\relax}}=p^{b}_{i}q_{b}^{i}+S^{f}_{\text{HJ}}(q_{a},p^{b})$:

$\frac{\partial f}{\partial p^{b}}=0\text{\ at\ }p^{b}=\lambda^{\text{crit}}.$

We will return to this in Section 6.2.

######  Remark 2.10.

In terms of the path-integral computation of the evolution operator, see
Remark 2.4, changing the endpoint conditions, as discussed in Remark 2.5,
corresponds to assigning different Hilbert space representations,
corresponding to different choices of polarization, to the endpoints. For
example, fixing $p$ at the final endpoint, as in Example 2.8, corresponds to
computing the integral kernel $\hat{K}(q_{a},p^{b})$ for the position
representation at the initial endpoint and the momentum representation at the
final one. This is the Fourier transform of $K(q_{a},q_{b})$ with respect to
the second argument. One can see a classical track of this in the
$p_{i}(t_{b})q^{i}(t_{b})$ term in $S^{f}$ (and in the Legendre transform of
the previous remark).

######  Example 2.11.

Consider

$\hat{K}(q_{a},p^{b})=\int_{\begin{subarray}{c}q(t_{a})=q_{a}\\\
p(t_{b})=p^{b}\end{subarray}}Dp\,Dq\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S^{f}[p,q]}.$

with $S^{f}[p,q]=-p(t_{b})q(t_{b})+S[p,q]$ for the simple action
$S[p,q]=\int_{t_{a}}^{t_{b}}p\,\mathrm{d}q$. (Nothing changes if we have many
$p$ and $q$ variables: just think of a hidden scalar product
$p\mathrm{d}q=p_{i}\mathrm{d}q^{i}$.) As the EL equations imply that the
solutions are constant, the change of variables (5) is now

$p\mapsto p^{b}+\hat{p},\quad q\mapsto q_{a}+\hat{q},$

with $\hat{p}(t_{b})=0$ and $\hat{q}(t_{a})=0$. Since
$S^{f}[p,q]=-p^{b}(q_{a}+\hat{q}(t_{b}))+\int_{t_{a}}^{t_{b}}(p^{b}+\hat{p})\mathrm{d}\hat{q}$,
we have

$\hat{K}(q_{a},p^{b})=\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}p^{b}q_{a}}\int_{\begin{subarray}{c}\hat{q}(t_{a})=0\\\
\hat{p}(t_{b})=0\end{subarray}}D\hat{p}\,D\hat{q}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\left(-p^{b}\hat{q}(t_{b})+\int_{t_{a}}^{t_{b}}p^{b}\mathrm{d}\hat{q}+\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}\right)}.$

This is a Gaussian integral with quadratic term
$\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}$ and no $\hat{p}$-sources, so it
simply yields a constant (which we normalize to one), and we get

(9) $\hat{K}(q_{a},p^{b})=\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}p^{b}q_{a}},$

the integral kernel for the Fourier transform, whose exponent is given by the
HJ action (8).

######  Example 2.12 (Formal quantum change of polarization).

The computation of Example 2.11 may be generalized to the general case
discussed in Remark 2.5. Namely, we have

$\hat{K}^{f}(q_{a},Q_{b})=\int_{\begin{subarray}{c}q(t_{a})=q_{a}\\\
Q(p(t_{b}),q(t_{b}))=Q_{b}\end{subarray}}Dp\,Dq\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S^{f}[p,q]},$

with $S^{f}[p,q]=-f(q(t_{b}),Q(p(t_{b}),q(t_{b})))+S[p,q]$. Again we consider
$S[p,q]=\int_{t_{a}}^{t_{b}}p\,\mathrm{d}q$. The change of variables (5) now
becomes

$p\mapsto p^{0}+\hat{p},\quad q\mapsto q_{a}+\hat{q},$

with the constant solution $p^{0}$ given by

$p^{0}=\frac{\partial f}{\partial q}{(q_{a}+\hat{q}(t_{b}),Q_{b})},$

and endpoint conditions $\hat{p}(t_{b})=0$ and $\hat{q}(t_{a})=0$.111111Note
that the endpoint fluctuations of $p$ and $q$ are constrained by the condition
$Q(p(t_{b}),q(t_{b}))=Q_{b}$. We have arranged the change of variables so
that, while $p^{0}$ may fluctuate at the endpoint, $\hat{p}$ does not. In
general this change of variables is no longer affine, since $p^{0}$ depends on
$\hat{q}$, possibly in a nonlinear way. On the other hand, the Jacobian
determinant of the transformation is (formally) $1$ (the Jacobian is lower
triangular with “$1$”s on the diagonal). We now have

$\hat{K}^{f}(q_{a},Q_{b})=\int_{\begin{subarray}{c}\hat{q}(t_{a})=0\\\
\hat{p}(t_{b})=0\end{subarray}}D\hat{p}\,D\hat{q}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\left(-f(q_{a}+\hat{q}(t_{b}),Q_{b})+\int_{t_{a}}^{t_{b}}p^{0}\mathrm{d}\hat{q}+\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}\right)},$

which is again a Gaussian integral with quadratic term
$\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}$ and no $\hat{p}$-sources,
Therefore,

(10)
$\hat{K}^{f}(q_{a},Q_{b})=\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(q_{a},Q_{b})},$

the exponential of the HJ action (6).

######  Digression 2.13 (Unitarity).

The result (10) of the previous Example 2.12 is correct insofar as the
starting point is the quantization of the action $S^{f}$ as a partition
“function.” This has to be corrected if we understand the integral kernel as a
half-density in the boundary variables, which is more appropriate for
composition. The idea is that the functional measure is the product over
points of the measure $\mathrm{d}p\,\mathrm{d}q/(2\pi\hbar)$ (for simplicity
we discuss this for the target $T^{*}\mathbb{R}$). At the end points we then
retain only the square root of the fixed boundary variable, as the other
corresponding square root will come from another integral kernel we want to
compose with. For example, in passing from the $q$\- to the
$p$-representation, the integral kernel half-density will be

$\mathbb{K}(q_{a},p^{b})=\sqrt{\mathrm{d}q_{a}}\;\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}p^{b}q_{a}}\;\sqrt{\frac{\mathrm{d}p^{b}}{2\pi\hbar}}$

instead of (9). Note that $\mathbb{K}$ is unitary:

$\int_{\\{p^{b}\\}}\mathbb{K}(q,p^{b})\,\overline{\mathbb{K}}(p^{b},q^{\prime})=\sqrt{\mathrm{d}q}\;\left(\int_{\\{p^{b}\\}}\mathrm{e}^{\frac{\mathrm{i}}{\hbar}p^{b}(q^{\prime}-q)}\,\frac{\mathrm{d}p^{b}}{2\pi\hbar}\right)\;\sqrt{\mathrm{d}q^{\prime}}=\sqrt{\mathrm{d}q}\,\delta(q^{\prime}-q)\,\sqrt{\mathrm{d}q^{\prime}}.$

Unitarity is somehow expected, since reversing the orientation of the interval
on which the theory is defined produces
$\overline{\mathbb{K}}(p^{b},q^{\prime})$, and the composition of the quantum
evolution on an interval with the reversed one should be the identity. In
Example 2.12, we could follow the same reasoning and expect unitarity (indeed,
at the classical level, composition of the generating functions
$f(q^{\prime},Q_{b})$ and $-f(q,Q_{b})$, see Remark A.37, yields a generating
function for the identity); however, in general, this is not the case at the
quantum level as we will see in a moment. In this example as well we split the
fields in terms of the initial $q$ and final $p$ variables, but then we
expressed the final values in terms of $Q_{b}$. The change of variables we
perfomed there entails in particular

(11) $p(t_{b})=\frac{\partial f}{\partial q}{(q_{a}+\hat{q}(t_{b}),Q_{b})}$

and therefore

$\mathrm{d}p(t_{b})=\frac{\partial^{2}f}{\partial q\,\partial
Q}{(q_{a}+\hat{q}(t_{b}),Q_{b})}\,\mathrm{d}Q_{b}+\frac{\partial^{2}f}{\partial
q^{2}}{(q_{a}+\hat{q}(t_{b}),Q_{b})}\,(\mathrm{d}q_{a}+\mathrm{d}\hat{q}(t_{b})).$

The terms in $\mathrm{d}q_{a}$ and $\mathrm{d}\hat{q}(t_{b})$ can be
neglected, since they already appear in the functional measure. Therefore, we
only retain the first term, of which we eventually have to take the square
root, getting the integral kernel half-density

$\begin{split}\mathbb{K}^{f}(q_{a},Q_{b})=\sqrt{\mathrm{d}q_{a}}\;\Bigg{(}\int_{\begin{subarray}{c}\hat{q}(t_{a})=0\\\
\hat{p}(t_{b})=0\end{subarray}}D\hat{p}\,D\hat{q}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\left(-f(q_{a}+\hat{q}(t_{b}),Q_{b})+\int_{t_{a}}^{t_{b}}p^{0}\mathrm{d}\hat{q}+\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}\right)}\\\
\sqrt{\frac{\partial^{2}f}{\partial q\,\partial
Q}{(q_{a}+\hat{q}(t_{b}),Q_{b})}}\Bigg{)}\;\sqrt{\frac{\mathrm{d}Q_{b}}{2\pi\hbar}},\end{split}$

where we have assumed the nonzero $\frac{\partial^{2}f}{\partial q\,\partial
Q}$ term to be actually positive. Again, since this is a Gaussian integral
with quadratic term $\int_{t_{a}}^{t_{b}}\hat{p}\mathrm{d}\hat{q}$ and no
$\hat{p}$-sources, we get

(12)
$\mathbb{K}^{f}(q_{a},Q_{b})=\sqrt{\mathrm{d}q_{a}}\;\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(q_{a},Q_{b})}\sqrt{\frac{\partial^{2}f}{\partial
q\,\partial Q}{(q_{a},Q_{b})}}\;\sqrt{\frac{\mathrm{d}Q_{b}}{2\pi\hbar}}$

instead of (10). In a neighborhood of the diagonal $q=q^{\prime}$,
$\mathbb{K}^{f}$ is almost unitary: i.e.,

$\int_{\\{Q_{b}\\}}\mathbb{K}^{f}(q,Q_{b})\,\overline{\mathbb{K}}^{f}(Q_{b},q^{\prime})=\sqrt{\mathrm{d}q}\,(1+O(\hbar))\delta(q^{\prime}-q)\,\sqrt{\mathrm{d}q^{\prime}},$

where the $O(\hbar)$ term is a differential operator. This may be seen by the
($q$ and $q^{\prime}$ dependent) change of variable

$\lambda(Q_{b})=\frac{f(q^{\prime},Q_{b})-f(q,Q_{b})}{q^{\prime}-q}$

in the $\mathrm{d}Q_{b}$ integral.121212In some cases, $\mathbb{K}^{f}$ is
exactly unitary. This happens, e.g., when $f$ has product form:
$f(q,Q)=\alpha(q)\beta(Q)$. (In particular, this happens when we have a linear
symplectomorphism, in which case, moreover, the correction term
$\frac{\partial^{2}f}{\partial q\,\partial Q}$ is constant.) In this case, we
have
$\int_{\\{Q_{b}\\}}\mathbb{K}^{f}(q,Q_{b})\,\overline{\mathbb{K}}^{f}(Q_{b},q^{\prime})=\sqrt{\mathrm{d}q}\;\left(\int_{\\{Q_{b}\\}}\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\beta(Q_{b})(\alpha(q^{\prime})-\alpha(q))}\sqrt{\alpha^{\prime}(q)\alpha^{\prime}(q^{\prime})}\beta^{\prime}(Q)\frac{\mathrm{d}Q_{b}}{2\pi\hbar}\right)\;\sqrt{\mathrm{d}q^{\prime}},$
where we have assumed the nonzero $\alpha^{\prime}$ and $\beta^{\prime}$ terms
to be actually positive. By the change of variable $\mu=\beta(Q_{b})$, we get
$\begin{split}\int_{\\{Q_{b}\\}}\mathbb{K}^{f}(q,Q_{b})\,\overline{\mathbb{K}}^{f}(Q_{b},q^{\prime})&=\sqrt{\mathrm{d}q}\;\left(\int_{\\{\mu\\}}\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\mu(\alpha(q^{\prime})-\alpha(q))}\sqrt{\alpha^{\prime}(q)\alpha^{\prime}(q^{\prime})}\frac{\mathrm{d}\mu}{2\pi\hbar}\right)\;\sqrt{\mathrm{d}q^{\prime}}\\\
&=\sqrt{\mathrm{d}q}\;\delta(\alpha(q^{\prime})-\alpha(q))\sqrt{\alpha^{\prime}(q)\alpha^{\prime}(q^{\prime})}\;\sqrt{\mathrm{d}q^{\prime}}=\sqrt{\mathrm{d}q}\,\delta(q^{\prime}-q)\,\sqrt{\mathrm{d}q^{\prime}}.\end{split}$
Note that (12) resembles the starting point of the WKB approximation (see,
e.g., [44, Appendix], [25, Section 15], and references therein). Indeed, one
possible way to get corrections to it to make it unitary is to regard, if
possible, $f$ as the Hamilton–Jacobi action for some hamiltonian system (on
the other hand, the corrections are not uniquely determined, since the choice
of this hamiltonian system is not unique).

##  3\. Systems with one constraint

Our first generalization is an action of the form131313If one requires
$e\not=0$ everywhere, this action may also be regarded as the first-order
version of one-dimensional gravity, described by the coframe $e$, coupled to
matter, described by $q$. Its BV-BFV quantization is studied in [22].

$S[p,q,e]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-eH(p,q)),$

where the new field $e$, the Lagrange multiplier, is a one-form on the
interval $[t_{a},t_{b}]$.

We will first consider the case when $e$ is different from zero everywhere.
This restriction will allow us to deal with endpoint conditions on the $q$s.
Next we will consider the general case, which, however, will force us to pick
different endpoint conditions.

###  3.1. Nondegenerate Lagrange multiplier

We assume $e$ to be different from zero everywhere. A variation of the action
yields

$\delta S=\text{\sl EL}+p_{i}(t_{b})\delta q^{i}(t_{b})-p_{i}(t_{a})\delta
q^{i}(t_{a})$

with the EL equations

$\mathrm{d}q^{i}=e\frac{\partial H}{\partial
p_{i}},\quad\mathrm{d}p_{i}=-e\frac{\partial H}{\partial q^{i}},\quad
H(p,q)=0.$

The last equation, the variation with respect to $e$, is imposed at all times.
Note that the endpoint symplectic structures are the same as in the
nondegenerate case and depend only on $p$ and $q$. This in particular means
that no endpoint condition on $e$ has to be imposed.

To solve the EL equations, we first define

$\tau(t)\mathrel{\mathop{\mathchar 58\relax}}=\int_{t_{a}}^{t}e.$

The first two sets of EL equations now read

$\frac{\mathrm{d}q^{i}}{\mathrm{d}\tau}=\frac{\partial H}{\partial
p_{i}},\quad\frac{\mathrm{d}p_{i}}{\mathrm{d}\tau}=-\frac{\partial H}{\partial
q^{i}},$

so they are the usual Hamilton equations for the hamiltonian $H$ in the time
variable $\tau\in[0,T]$, with $T\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t_{b}}e$. The last EL equation, $H=0$, is
automatically satisfied at any point if we impose it at the initial point (or
at the final point). We then have that the endpoint values for $(p,q)$
corresponding to solutions form the set

$L=\\{(p^{a},q_{a},p^{b},q_{b})\in T^{*}M\times T^{*}M\ |\ H(p^{b},q_{b})=0\\\
\text{ and }\exists T\in\mathbb{R}_{\not=0}\mathrel{\mathop{\mathchar
58\relax}}(p^{b},q_{b})=\Phi^{H}_{T}(p^{a},q_{a})\\},$

where $\Phi^{H}_{\tau}$ denotes the hamiltonian flow of $H$ at time $\tau$.
Note that in this case $L$ is not the graph of a map $T^{*}M\to T^{*}M$, but
just a subset of $T^{*}M\times T^{*}M$. We will call it the evolution relation
of the system. (See Remark A.38 for a more general perspective.)

For a fixed $e$, let $(p^{q_{a},q_{b}},q_{q_{a},q_{b}})$ denote the (assumedly
unique) solution to the first two sets of EL equations (i.e., we do not impose
the constraint $H=0$) satisfying $q_{q_{a},q_{b}}(t_{a})=q_{a}$ and
$q_{q_{a},q_{b}}(t_{b})=q_{b}$. Since we may rewrite the action as
$\int_{0}^{T}(p_{i}\mathrm{d}q^{i}-H(p,q)\mathrm{d}\tau)$, we get that the HJ
action is

$S_{\text{HJ}}(q_{a},q_{b})[e]\mathrel{\mathop{\mathchar
58\relax}}=S[p^{q_{a},q_{b}},q_{q_{a},q_{b}},e]=S^{H}_{\text{HJ}}(q_{a},q_{b};0,T),$

where the last term denotes the HJ action for the system with hamiltonian $H$
over the time interval $[0,T]$.

The first remark is that the HJ action depends on $e$ only via its integral
$T$; we will then simply write

(13) $\hat{S}_{\text{HJ}}(q_{a},q_{b};T)\mathrel{\mathop{\mathchar
58\relax}}=S_{\text{HJ}}(q_{a},q_{b})[e],\quad\text{with
}T=\int_{t_{a}}^{t_{b}}e,$

so

$\hat{S}_{\text{HJ}}(q_{a},q_{b};T)=S^{H}_{\text{HJ}}(q_{a},q_{b};0,T).$

The second remark, which follows immediately from (1) and (3), is that the
evolution relation $L$ is determined by the equations

$p^{b}_{i}=\frac{\partial\hat{S}_{\text{HJ}}}{\partial q^{i}_{b}},\quad
p^{a}_{i}=-\frac{\partial\hat{S}_{\text{HJ}}}{\partial
q^{i}_{a}},\quad\frac{\partial\hat{S}_{\text{HJ}}}{\partial T}=0.$

In other words, $\hat{S}_{\text{HJ}}$ is a generalized generating
function141414In the terminology of [9, Section 4.3], the triple
$(M\times\mathbb{R}_{\not=0},\pi_{1},\hat{S}_{\text{HJ}})$, where $\pi_{1}$ is
the projection to the first factor, is a Morse family generating $L$. for the
lagrangian submanifold $L$, see Appendix A (in particular, Remark A.11 for the
linear case and Remark A.27 for the general case).

###  3.2. The general case

The assumption $e\not=0$ everywhere is only needed to ensure that the map
$t\mapsto\tau$ is a diffeomorphism and hence to relate $\hat{S}_{\text{HJ}}$
to the HJ function for the system with hamiltonian $H$. The existence of a HJ
action, however, holds also in the general case where we make no assumptions
on $e$, provided we choose the endpoint conditions more carefully.

First observe that the first two sets of EL equations, which we will call the
evolution equations,

$\mathrm{d}q^{i}=e\frac{\partial H}{\partial
p_{i}},\quad\mathrm{d}p_{i}=-e\frac{\partial H}{\partial q^{i}},$

can be uniquely solved, for fixed $e$, in terms of initial conditions
$p^{a},q_{a}$. In fact, in the regions (closed intervals) where $e=0$, the
solution is constant and on the other regions we can make a change of
variables $t\mapsto\tau$ as above. Moreover, the function $H$ is constant when
evaluated on a solution to the evolution equations, which means that it is
enough to impose this constraint at the initial (or final) endpoint to have it
satisfied at every time.

Let $L$ still denote the evolution relation of the system, i.e., the set of
endpoint values for solutions to the EL equations. Note that, for a given
$e=\underline{e}\,\mathrm{d}t$, we have to compute the “flow”
$\Phi^{\tilde{H}}_{T}$ of the time-dependent hamiltonian
$\tilde{H}=\underline{e}H$ up to time $T=\int_{t_{a}}^{t_{b}}e$. We then have

$L=\\{(p^{a},q_{a},p^{b},q_{b})\in T^{*}M\times T^{*}M\ |\ H(p^{b},q_{b})=0\\\
\text{ and }\exists e\mathrel{\mathop{\mathchar
58\relax}}(p^{b},q_{b})=\Phi^{\tilde{H}}_{T}(p^{a},q_{a})\\}.$

######  Remark 3.1.

We can have a better geometric description if we assume that the differential
of $H$ is different from zero at every point of the zero locus $C$ of $H$.
This ensures that $C$ is a submanifold of $T^{*}M$ and that the hamiltonian
vector field $X^{H}$ of $H$ restricted to $C$ has no zeros, so it defines a
distribution of lines. Note that $C$ is an example of what is called a
coisotropic submanifold and $\operatorname{span}X^{H}|_{C}$ is called its
characteristic distribution (see Definition A.19 and Remark A.25 for more on
this). In this geometric framework, the evolution relation $L\subset
T^{*}M\times T^{*}M$ consists of pairs of points on $C$ that can be connected
by a path along the characteristic distribution. (It is therefore lagrangian
by Lemma A.29.)

Now observe that fixing $q_{a}$ and $q_{b}$ is problematic, as for $e=0$ we
have no solution unless $q_{a}=q_{b}$. For this reason we have to choose
different endpoint conditions, as in Remark 2.5; for instance, we can fix $p$
at one endpoint as in Example 2.8.

We denote by $(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}},e)$ a solution
$(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})$ to the evolution equations for the given
$e$ (but we do not impose the constraint $H=0$) satisfying
$q_{q_{a},Q_{b}}(t_{a})=q_{a}$ and
$Q(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b}))=Q_{b}$. We then set

$S^{f}_{\text{HJ}}(q_{a},Q_{b})[e]\mathrel{\mathop{\mathchar
58\relax}}=S^{f}[p^{q_{a},Q_{b}},q_{q_{a},Q_{b}},e]$

with

$S^{f}[p,q,e]=-f(q(t_{b}),Q(p(t_{b}),q(t_{b})))+\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-eH(p,q)).$

We then have

(14) $\begin{split}\delta S^{f}_{\text{HJ}}&=P_{i}(t_{b})\delta
Q^{i}_{b}-p_{i}(t_{a})\delta q^{i}_{a}-\int_{t_{a}}^{t_{b}}\delta
e\,H(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})\\\ &=P_{i}(t_{b})\delta
Q^{i}_{b}-p_{i}(t_{a})\delta q^{i}_{a}-\left(\int_{t_{a}}^{t_{b}}\delta
e\right)H(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b}))\\\
&=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta q^{i}_{a}-\delta
T\,H(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),\end{split}$

where, as above, we set $T=\int_{t_{a}}^{t_{b}}e$. Note that
$S^{f}_{\text{HJ}}$ is invariant under variations of $e$ with vanishing
integral, so we may define $\hat{S}^{f}_{\text{HJ}}$ like in (13):

$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};T)\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b})[e],\quad\text{with
}T=\int_{t_{a}}^{t_{b}}e.$

Then the above equation reads

$\delta\hat{S}^{f}_{\text{HJ}}=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\delta T\,H(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),$

and we see that the equations

$P_{i}(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})(t_{b})=\frac{\partial\hat{S}^{f}_{\text{HJ}}}{\partial
Q^{i}_{b}},\quad p^{a}_{i}=-\frac{\partial\hat{S}^{f}_{\text{HJ}}}{\partial
q^{i}_{a}},\quad\frac{\partial\hat{S}^{f}_{\text{HJ}}}{\partial T}=0$

define $L$.

> _The HJ action, obtained inserting a solution of the evolution equations but
> ignoring the constraint, is a generalized generating function for the
> evolution relation determined by all EL equations._

######  Example 3.2 (Linear case).

Consider $M=\mathbb{R}^{n}$ and suppose $H$ is linear on
$T^{*}\mathbb{R}^{n}$:

$H(p,q)=p_{i}v^{i}+w_{i}q^{i},$

where the $(v^{i},w_{i})$s are given constants. In this case the evolution
equations simply read

$\mathrm{d}q^{i}=ev^{i},\quad\mathrm{d}p_{i}=-ew_{i}.$

They can be easily solved in terms of the initial conditions, so we get the
evolution relation

$\displaystyle L=$ $\displaystyle\\{(p^{a},q_{a},p^{b},q_{b})\in
T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}\ |$ $\displaystyle|\ \exists
T\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}p^{b}_{i}v^{i}+w_{i}(q^{i}_{a}+Tv^{i})=0,\ p^{a}=p^{b}+Tw,\
q_{b}=q_{a}+Tv\\},$

where $T$ corresponds to $\int_{t_{a}}^{t_{b}}e$, and we have evaluated the
constraint $H=0$ at the final endpoint. As in Example 2.8 we now fix $q_{a}$
and $p^{b}$. The evolution equations can then be easily integrated for the
given endpoint conditions as

$\begin{split}q_{q_{a},p^{b}}^{i}(t)&=q_{a}^{i}+\int_{t_{a}}^{t}ev^{i}=q_{a}^{i}+\tau(t)v^{i},\\\
p^{q_{a},p^{b}}_{i}(t)&=p^{b}_{i}+\int_{t}^{t_{b}}ew_{i}=p^{b}_{i}+(T-\tau(t))w_{i},\end{split}$

with $\tau(t)=\int_{t_{a}}^{t}e$. For this choice of endpoint conditions we
consider

$S^{f}[p,q,e]\mathrel{\mathop{\mathchar
58\relax}}=-p_{i}(t_{b})q^{i}(t_{b})+\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-eH(p,q)).$

Now observe that

$\displaystyle
p^{q_{a},p^{b}}_{i}\mathrm{d}q_{q_{a},p^{b}}^{i}(t)-eH(p^{q_{a},p^{b}},q_{q_{a},p^{b}})$
$\displaystyle=p^{q_{a},p^{b}}_{i}ev^{i}-ep^{q_{a},p^{b}}_{i}v^{i}-ew_{i}q_{q_{a},p^{b}}^{i}$
$\displaystyle=-\mathrm{d}\tau w_{i}(q_{a}^{i}+\tau v^{i}),$

which implies

$\int_{t_{a}}^{t_{b}}(p^{q_{a},p^{b}}_{i}\mathrm{d}q_{q_{a},p^{b}}^{i}(t)-eH(p^{q_{a},p^{b}},q_{q_{a},p^{b}}))=-Tw_{i}q_{a}^{i}-\frac{T^{2}}{2}w_{i}v^{i}.$

On the other hand,

$p^{q_{a},p^{b}}_{i}(t_{b})q_{q_{a},p^{b}}^{i}(t_{b})=p^{b}_{i}(q_{a}^{i}+Tv^{i}).$

In conclusion, we get

(15)
$\hat{S}^{f}_{\text{HJ}}(q_{a},p^{b};T)=-p^{b}_{i}q_{a}^{i}-T(p^{b}_{i}v^{i}+w_{i}q_{a}^{i})-\frac{T^{2}}{2}w_{i}v^{i},$

which is readily seen to be a generalized generating function for $L$.

######  Remark 3.3.

Sometimes it is possible to solve for $T$ the constraints and then obtain a
standard (i.e., without parameters) generating function for the evolution
relation $L$. For instance, in the linear case of Example 3.2, the constraint
is

$p^{b}_{i}v^{i}+w_{i}q_{a}^{i}+Tw_{i}v^{i}=0,$

which can be solved if $w_{i}v^{i}\not=0$. In this case we get

$T=-\frac{p^{b}_{i}v^{i}+w_{i}q_{a}^{i}}{w_{i}v^{i}},$

which can be inserted back into $\hat{S}^{f}$ to get the standard generating
function

$\tilde{S}^{f}(q_{a},p^{b})=-p^{b}_{i}q_{a}^{i}+\frac{(p^{b}_{i}v^{i}+w_{i}q_{a}^{i})^{2}}{2w_{i}v^{i}}.$

In the one-dimensional case, $n=1$, we get

$\tilde{S}^{f}(q_{a},p^{b})=-p^{b}q_{a}+\frac{(p^{b}v+wq_{a})^{2}}{2wv}=\frac{1}{2}\frac{v}{w}(p^{b})^{2}+\frac{1}{2}\frac{w}{v}(q_{a})^{2},$

which correctly generates the evolution relation

$L=\\{(p^{a},q_{a},p^{b},q_{b})\in T^{*}\mathbb{R}\times T^{*}\mathbb{R}\ |\
p^{a}v+wq_{a}=0\text{ and }p^{b}v+wq_{b}=0\\}.$

Note that in this very particular example the evolution plays no role and $L$
has the product form $L=L^{\prime}\times L^{\prime}$, with
$L^{\prime}=\\{(p,q)\in T^{*}\mathbb{R}\ |\ pv+wq=0\\}$.

######  Remark 3.4 (Integrable systems I).

The case $n=1$ can also be regarded as a one-dimensional integrable system
with vanishing hamiltonian. In this case, another expression for the HJ action
can be derived as follows. Around a point $(p_{0},q_{0})$ where
$\mathrm{d}H\neq 0$, by the Carathéodory–Lie–Jacobi theorem151515The
Carathéodory–Lie–Jacobi theorem says that if $H_{1},\ldots,H_{k}$ are
functions on a symplectic manifold $(M,\omega)$ such that
$\\{H_{i},H_{j}\\}=0$ and $\mathrm{d}H_{1},\ldots,\mathrm{d}H_{k}$ are
linearly independent at some $p\in M$, then there exists a Darboux chart
$(q_{i},p^{i})$ containing $p$ with $q_{1}=H_{1},\ldots,q_{k}=H_{k}$. See, for
instance, [40, Theorem 13.4.1]. we may extend $H$ to a Darboux chart $(P,Q)$
with $P=H$ ($P$ is what is called the action variable and $Q$ the
corresponding angle variable). Let $f(q,Q)$ be the generating function
satisfying $p\,\mathrm{d}q=P\,\mathrm{d}Q+\mathrm{d}f$. We may then write

$S^{f}[p,q,e]\mathrel{\mathop{\mathchar
58\relax}}=-f(q(t_{b}),Q(t_{b}))+\int_{t_{a}}^{t_{b}}(p\,\mathrm{d}q-eH)=-f(q(t_{a}),Q(t_{a}))+\int_{t_{a}}^{t_{b}}(P\,\mathrm{d}Q-eP),$

where $P=P(p,q),Q=Q(p,q)$. The evolution equations now simply read
$\mathrm{d}P=0,\mathrm{d}Q=e$. Evaluating on a solution of the evolution
equations, the action vanishes and we get

(16) $S^{f}_{\text{HJ}}(q_{a},Q_{b};T)=-f(q_{a},Q_{b}-T)$

for $(q_{a},Q_{b})$ in a neighborhood of $(q_{0},Q(p_{0},q_{0}))$.
Alternatively, if $g(q,P)$ is the generating function satisfying
$p\,\mathrm{d}q={-}Q\,\mathrm{d}P+\mathrm{d}g$, we can express the HJ action
with final endpoint condition on $P$ by

(17) $S^{g}_{\text{HJ}}(q_{a},P^{b};T)=-TP^{{b}}-g(q_{a},P^{{b}}).$

Notice also that (17) is the Legendre transform of (16). We will revisit the
case of integrable systems in Remarks 4.5 and 5.1.

##  4\. Systems with several constraints

From now on we consider an action of the form161616Such an action functional
is the classical part of an AKSZ theory [2] on the interval $[t_{a},t_{b}]$;
see Section 8 for a review. See also [5] for the study of such a theory in the
Dirac formalism.

(18)
$S[p,q,e]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-e^{\alpha}H_{\alpha}(p,q)),$

where the $e^{\alpha}$s (the Lagrange multipliers) are one-forms on the
interval $[t_{a},t_{b}]$ and the $H_{\alpha}$s are given functions (the
constraints). (The sum over $\alpha$, like that over $i$, is now understood.)
The EL equations are

$\mathrm{d}q^{i}=e^{\alpha}\frac{\partial H_{\alpha}}{\partial
p_{i}},\quad\mathrm{d}p_{i}=-e^{\alpha}\frac{\partial H_{\alpha}}{\partial
q^{i}},\quad H_{\alpha}(p,q)=0.$

Again we will call the first two sets of equations the evolution equations and
the last set the constraints. Evaluating the constraints on a solution of the
evolution equations yields

$\mathrm{d}H_{\alpha}=\frac{\partial H_{\alpha}}{\partial
p_{i}}\mathrm{d}p_{i}+\frac{\partial H_{\alpha}}{\partial
q^{i}}\mathrm{d}q^{i}=-\\{H_{\alpha},H_{\beta}\\}e^{\beta},$

where $\\{\ ,\ \\}$ denotes the Poisson bracket.171717We define the Poisson
bracket as $\\{f,g\\}=\frac{\partial f}{\partial p_{i}}\frac{\partial
g}{\partial q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial
p_{i}}$. If we assume, as we will do from now on, that the constraints are in
involution,181818Using Dirac’s terminology, one also says that they are first-
class constraints. i.e., that there are structure functions
$f_{\alpha\beta}^{\gamma}$ such that

(19) $\\{H_{\alpha},H_{\beta}\\}=f_{\alpha\beta}^{\gamma}H_{\gamma},$

then the previous equations become

(20) $\mathrm{d}H_{\alpha}=-f_{\alpha\beta}^{\gamma}e^{\beta}H_{\gamma}.$

This implies that, if the constraints are imposed at the initial (or final)
endpoint, they will be satisfied at every time.

######  Remark 4.1.

As in Remark 3.1, it may be convenient to assume that the differentials of the
$H_{\alpha}$s are linearly independent at every point of the common zero locus
$C$ of the $H_{\alpha}$s. This ensures that $C$ is a submanifold of $T^{*}M$
and that the hamiltonian vector fields $X^{H_{\alpha}}$ restricted to $C$ are
linearly independent, so they define a regular distribution. It follows that
$C$ is also an example of what is called a coisotropic submanifold and
$\operatorname{span}\\{X^{H}_{\alpha}|_{C}\\}$ is called its characteristic
distribution (see Definition A.19 and Remark A.25 for more on this).191919The
characteristic distribution is also the same as the kernel of the restriction
of the symplectic form to $C$. As a consequence of (19), the characteristic
distribution is involutive (and hence, by Frobenius’ theorem, integrable). In
this geometric framework, the evolution relation $L\subset T^{*}M\times
T^{*}M$ consists of pairs of points on $C$ that can be connected by a path
along the characteristic distribution and is therefore lagrangian in
$\overline{T^{*}M}\times T^{*}M$, where the bar denotes that one takes the
opposite symplectic form.

Like in the case with a single constraint, fixing $q_{a}$ and $q_{b}$ is
problematic, as for $e^{\alpha}=0$ for all $\alpha$ we have no solution unless
$q_{a}=q_{b}$. Therefore, we choose different endpoint conditions, as in
Remark 2.5; for instance, we can fix $p$ at one endpoint as in Example 2.8.

We will denote by $(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}},e)$ a solution
$(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}})$ to the evolution equations with the given
$e$ (but we do not impose the constraints $H_{\alpha}=0$) satisfying
$q_{q_{a},Q_{b}}(t_{a})=q_{a}$ and
$Q(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b}))=Q_{b}$. We then set

$S^{f}_{\text{HJ}}(q_{a},Q_{b})[e]\mathrel{\mathop{\mathchar
58\relax}}=S^{f}[p^{q_{a},Q_{b}},q_{q_{a},Q_{b}},e]$

with

$S^{f}[p,q,e]=-f(q(t_{b}),Q(p(t_{b}),q(t_{b})))+\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-e^{\alpha}H_{\alpha}(p,q)).$

By taking a variation, we get

$\delta S^{f}_{\text{HJ}}=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\int_{t_{a}}^{t_{b}}\delta
e^{\alpha}\,H_{\alpha}(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}}).$

This shows that $S^{f}_{\text{HJ}}$ is indeed a generalized generating
function for the evolution relation $L$. However, there is a lot of redundancy
because the variations with respect to the $e^{\alpha}(t)$s yield the
constraints for every $t\in[t_{a},t_{b}]$, whereas we have observed that it is
enough to impose them at the initial (or final) endpoint. This is related to
the fact that, thank to (20) and upon integration by parts,
$S^{f}_{\text{HJ}}$ is invariant under a variation of $e$ of the form

(21) $\delta
e^{\alpha}=\mathrm{d}\gamma^{\alpha}+f_{\alpha\beta}^{\gamma}e^{\alpha}\gamma^{\beta},$

with the $\gamma$s arbitrary functions on $[t_{a},t_{b}]$ that vanish at both
endpoints. We call a transformation as in (21) a gauge transformation. We
therefore see that $S^{f}_{\text{HJ}}$ only depends on the $e^{\alpha}(t)$s
via their gauge classes. Analogously to the case of equation (13), we define

(22) $\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};[e])\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b},e),\quad\text{with }e\in[e].$

In summary, we have the following

######  Theorem 4.2.

The generalized HJ action $\hat{S}^{f}_{\text{HJ}}$, obtained by inserting
into the classical action $S^{f}$ the solution of the evolution equations, is
a generalized generating functions for the evolution relation, which is
defined in terms of all the EL equations (evolution and constrains).

In the next subsections we will discuss more explicitly the gauge classes and
how the HJ action depends on them. We will proceed in increasing order of
difficulty. In Section 4.1, we will consider the strictly involutive case
where all the structure functions $f_{\alpha\beta}^{\gamma}$ vanish. This
case, which is a straightforward generalization of the single-constraint case,
comprises the case where the constraints $H_{\alpha}$ are linear, see Example
4.3, and will be important for abelian Chern–Simons theory. Next, in Section
4.2, we will consider the case when the structure functions are actually
constant, so that there is an underlying Lie algebra structure (the
$H_{\alpha}$s are in this case the components of an equivariant momentum map).
This case will be important for nonabelian Chern–Simons theory. Finally, for
completeness, we will treat the general case in Section 4.3: this case
requires knowledge of Lie algebroids and Lie groupoids, see [42] for a general
introduction, but will not be used in the rest of the paper, so it may be
safely skipped.

###  4.1. The strictly involutive case

Suppose that all the structure functions $f_{\alpha\beta}^{\gamma}$ vanish;
i.e., $\\{H_{\alpha},H_{\beta}\\}=0$ for all $\alpha$ and $\beta$. This
implies that, evaluating on a solution to the evolution equations, we have
$\mathrm{d}H_{\alpha}=0$ for all $\alpha$. If we define

$\tau^{\alpha}(t)\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t}e^{\alpha},$

we then have $e^{\alpha}=\mathrm{d}\tau^{\alpha}$ and
$\tau^{\alpha}(t_{a})=0$. Observe that the computation in (14) can be repeated
verbatim yielding

$\delta S^{f}_{\text{HJ}}=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\delta
T^{\alpha}\,H_{\alpha}(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),$

with $T^{\alpha}=\tau^{\alpha}(t_{b})=\int_{t_{a}}^{t_{b}}e^{\alpha}$. Again,
this first implies that $S^{f}_{\text{HJ}}$ is invariant under variations of
the $e^{\alpha}$s with vanishing integral, so we may define
$\hat{S}^{f}_{\text{HJ}}$ as in (22) as

$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};T)\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b})[e],\quad\text{with
}T^{\alpha}=\int_{t_{a}}^{t_{b}}e^{\alpha}.$

Second, the above computation shows that $\hat{S}^{f}_{\text{HJ}}$ is a
generalized generating function for $L$.

######  Example 4.3 (Linear case).

Assume

$H_{\alpha}=p_{i}v^{i}_{\alpha}+w_{i\alpha}q^{i}=pv_{\alpha}+w_{\alpha}q,$

where the $(v^{i}_{\alpha},w_{i\alpha})$s are given constants and we use the
product of row and column vectors to make the notation a bit lighter. The
Poisson bracket of any two constraints is a constant function. The
involutivity conditions (19) then imply that all these constant functions
vanish, so we actually have

$\\{H_{\alpha},H_{\beta}\\}=0.$

This is equivalent to the conditions $w_{\alpha}v_{\beta}=w_{\beta}v_{\alpha}$
for all $\alpha$ and $\beta$. For future convenience we define the
(symmetric!) matrix

(23) $A_{\alpha\beta}\mathrel{\mathop{\mathchar
58\relax}}=w_{\alpha}v_{\beta}=w_{\beta}v_{\alpha}.$

We now want to compute $\hat{S}^{f}_{\text{HJ}}$ explicitly as a function of
$q_{a}$ and $p^{b}$. First observe that the evolution equations read

$\mathrm{d}q=e^{\alpha}v_{\alpha},\quad\mathrm{d}p=-e^{\alpha}w_{\alpha},$

and can be easily integrated for the given endpoint conditions:

$\begin{split}q_{q_{a},p^{b}}(t)&=q_{a}+\tau^{\alpha}(t)v_{\alpha},\\\
p^{q_{a},p^{b}}(t)&=p^{b}+(T^{\alpha}-\tau^{\alpha}(t))w_{\alpha}.\end{split}$

Moreover, we have

$\begin{split}p^{q_{a},p^{b}}\mathrm{d}q_{q_{a},p^{b}}(t)-e^{\alpha}H_{\alpha}(p^{q_{a},p^{b}},q_{q_{a},p^{b}})&=e^{\alpha}p^{q_{a},p^{b}}v_{\alpha}-e^{\alpha}p^{q_{a},p^{b}}v_{\alpha}-e^{\alpha}w_{\alpha}q_{q_{a},p^{b}}\\\
&=-\mathrm{d}\tau^{\alpha}w_{\alpha}(q_{a}+\tau^{\beta}v_{\beta})=-\mathrm{d}\tau^{\alpha}w_{\alpha}q_{a}-\mathrm{d}\tau^{\alpha}\tau^{\beta}A_{\alpha\beta}\\\
&=-\mathrm{d}\tau^{\alpha}w_{\alpha}q_{a}-\frac{1}{2}\mathrm{d}(\tau^{\alpha}\tau^{\beta})A_{\alpha\beta}\end{split}$

with $A_{\alpha\beta}$ the symmetric matrix defined in (23). This implies

$\int_{t_{a}}^{t_{b}}(p^{q_{a},p^{b}}\mathrm{d}q_{q_{a},p^{b}}(t)-e^{\alpha}H_{\alpha}(p^{q_{a},p^{b}},q_{q_{a},p^{b}}))=-T^{\alpha}w_{\alpha}q_{a}-\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta}.$

On the other hand,

$p^{q_{a},p^{b}}(t_{b})q_{q_{a},p^{b}}(t_{b})=p^{b}(q_{a}+T^{\alpha}v_{\alpha}).$

In conclusion, we get

(24)
$\hat{S}^{f}_{\text{HJ}}(q_{a},p^{b},T)=-p^{b}q_{a}-T^{\alpha}(p^{b}v_{\alpha}+w_{\alpha}q_{a})-\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta},$

which generalizes (15) to the the case of several linear constraints.

######  Remark 4.4 (General change of endpoint conditions).

The above example is simple enough for the answer to be computable for a
general change of endpoint conditions as in Remark 2.5. Let $f(q,Q)$ be a
generating function for the change and fix $q_{a}$ and $Q_{b}$. By inspection
in the previous computation, we see that to compute $\hat{S}^{f}_{\text{HJ}}$
we actually only have to solve $q(t)$ for the given initial condition $q_{a}$
(this is independent of $Q_{b}$), which has already been done above. We then
get in general

(25)
$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};T)=-f(q_{a}+T^{\alpha}v_{\alpha},Q_{b})-T^{\alpha}w_{\alpha}q_{a}-\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta}.$

Note that this may also be obtained as the composition of the generating
functions (24) and (6), following Remark A.36. Namely, let us write

$S_{1}(q_{a},p^{b};T)=-p^{b}q_{a}-T^{\alpha}(p^{b}v_{\alpha}+w_{\alpha}q_{a})-\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta},$

and

$S_{2}(q_{a},Q_{b})=-f(q_{a},Q_{b}).$

Let

$S_{3}(q_{a},p^{b},\bar{q}_{a},Q_{b};T)\mathrel{\mathop{\mathchar
58\relax}}=S_{1}(q_{a},p^{b},T)+S_{2}(\bar{q}_{a},Q_{b})+p^{b}\bar{q}_{a}.$

The composition of $S_{1}$ and $S_{2}$ is the evaluation of $S_{3}$ at its
critical point in $(p^{b},\bar{q}_{a})$:

$\bar{q}_{a}+\frac{\partial S_{1}}{\partial p^{b}}=0,\quad
p^{b}+\frac{\partial S_{2}}{\partial\bar{q}_{a}}=0.$

One easily sees that the result is (25).

######  Remark 4.5 (Integrable systems II).

As a generalization of Remark 3.4, we may consider the case where we have
exactly $n$ constraints $H_{1},\ldots,H_{n}$ in strict involution - an
$n$-dimensional integrable system with vanishing hamiltonians. Around a point
$(p_{0},q_{0})$ where $\mathrm{d}H_{1},\ldots,\mathrm{d}H_{n}$ are linearly
independent, again by the Carathéodory–Jacobi–Lie theorem, see footnote 15, we
may find a Darboux chart $(P,Q)$ with $P_{i}=H_{i}$ (the action-angle
coordinates). By the same computation as in Remark 3.4, in this chart the HJ
action is expressed as

(26) $S^{f}_{\text{HJ}}(q_{a},Q_{b};T)=-f(q_{a},Q_{b}-T)$

and

(27) $S^{g}_{\text{HJ}}(q_{a},P^{b};T)=-T^{i}P_{i}^{b}-{g}(q_{a},P^{b}).$

###  4.2. The Lie algebra case

We now suppose that all the $f_{\alpha\beta}^{\gamma}$s are constant (but not
necessarily zero). They can be viewed as structure constants of some Lie
algebra $\mathfrak{g}$. In turn the $e^{\alpha}$s may be viewed as the
components of a $\mathfrak{g}$ valued one-form $e$ on $[t_{a},t_{b}]$ or,
equivalently, as a connection one-form. We may also view the $H_{\alpha}$s as
the components of a map $H\colon T^{*}M\to\mathfrak{g}^{*}$, where $T^{*}M$ is
the target symplectic manifold. The involutivity conditions (19) say that $H$
is an equivariant momentum map, whereas (20) may be read as the condition
$\mathrm{d}_{e}H=0$ that the composition of $H$ with a solution to the
evolution equations is covariantly constant. We rewrite the action as

$S[p,q,e]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-\langle H(p,q),e\rangle),$

where $\langle\ ,\ \rangle$ denotes the pairing of $\mathfrak{g}^{*}$ with
$\mathfrak{g}$. Next we define

(28) $h(t)=P\mathrm{e}^{\int_{t_{a}}^{t}e}\in G,$

where $P$ denotes the path ordered product and $G$ is the simply connected Lie
group with Lie algebra $\mathfrak{g}$. Note that $h(t_{a})=1$ and
$e=h^{-1}\mathrm{d}h$.202020We use for simplicity the matrix notation. If $G$
is not a matrix Lie group, the correct formula is
$e(t)=(\mathrm{d}_{1}l_{h})^{-1}\mathrm{d}_{t}h\colon T_{t}I\to\mathfrak{g},$
where $1$ denotes the unit of $G$ and $l_{h}\colon G\to G$ denotes the left
translation $g\mapsto hg$. We write

$g\mathrel{\mathop{\mathchar
58\relax}}=h(t_{b})=P\mathrm{e}^{\int_{t_{a}}^{t_{b}}e}.$

Finally, we define $\tilde{H}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Ad}_{h}^{*}H$. Observe that, when evaluated on a
solution to the evolution equations, $\tilde{H}$ is constant. We now repeat
the computation as in (14):

$\begin{split}\delta S^{f}_{\text{HJ}}&=P_{i}(t_{b})\delta
Q^{i}_{b}-p_{i}(t_{a})\delta q^{i}_{a}-\int_{t_{a}}^{t_{b}}\langle
H(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}}),\delta e\rangle\\\ &=P_{i}(t_{b})\delta
Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\int_{t_{a}}^{t_{b}}\langle\tilde{H}(p^{q_{a},Q_{b}},q_{q_{a},Q_{b}}),\operatorname{Ad}_{h}\delta
e\rangle\\\ &=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\left\langle\tilde{H}(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),\int_{t_{a}}^{t_{b}}\mathrm{d}(\delta
hh^{-1})\right\rangle\\\ &=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\left\langle\tilde{H}(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),\delta
gg^{-1}\right\rangle\\\ &=P_{i}(t_{b})\delta Q^{i}_{b}-p_{i}(t_{a})\delta
q^{i}_{a}-\left\langle
H(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b})),g^{-1}\delta
g\right\rangle.\end{split}$

This explicitly shows that $S^{f}_{\text{HJ}}$ is invariant under variations
of $e$ that do not change $g$. As in (22), we define

(29) $\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b})[e],\quad\text{with
}g=P\mathrm{e}^{\int_{t_{a}}^{t_{b}}e},$

and it immediately follows that $\hat{S}^{f}_{\text{HJ}}$ is a generalized
generating function for the evolution relation $L$.

######  Example 4.6 (Biaffine case).

Consider a target $T^{*}V=V^{*}\oplus V\ni(p,q)$ and constraints that are
affine both on $V$ and on $V^{*}$; namely,

$H_{\alpha}(p,q)=-p_{i}(\rho_{\alpha})^{i}_{j}q^{j}+p_{i}v^{i}_{\alpha}+w_{i\alpha}q^{i}=-p\rho_{\alpha}q+pv_{\alpha}+w_{\alpha}q,$

where, for each $\alpha$, $\rho_{\alpha}$ is a given endomorphism of $V$,
$v_{\alpha}$ is a given vector in $V$ and $w_{\alpha}$ is a given vector in
$V^{*}$ (we denote the pairing between $V^{*}$ and $V$ simply by
juxtaposition). The involutivity conditions (19) are equivalent to the
relations

(30)
$\begin{split}[\rho_{\alpha},\rho_{\beta}]&=f_{\alpha\beta}^{\gamma}\rho_{\gamma},\\\
\rho_{\alpha}v_{\beta}-\rho_{\beta}v_{\alpha}&=f_{\alpha\beta}^{\gamma}v_{\gamma},\\\
w_{\alpha}\rho_{\beta}-w_{\beta}\rho_{\alpha}&=f_{\alpha\beta}^{\gamma}w_{\gamma},\\\
w_{\beta}v_{\alpha}-w_{\alpha}v_{\beta}&=0.\end{split}$

The first relation says that $\rho$ is a representation of the Lie algebra
$\mathfrak{g}$ on $V$, the second and the third that $v\colon\mathfrak{g}\to
V$ and $w\colon\mathfrak{g}\to V^{*}$ are Lie algebra $1$-cocycles, and the
last is what we already saw in Example 4.3. The evolution equations read

$\mathrm{d}_{e}q=e^{\alpha}v_{\alpha}=v(e),\quad\mathrm{d}_{e}p=-e^{\alpha}w_{\alpha}=-w(e),$

where we introduced the covariant derivatives

$\mathrm{d}_{e}q\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{d}q+e^{\alpha}\rho_{\alpha}q,\quad\mathrm{d}_{e}p\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{d}p-e^{\alpha}p\rho_{\alpha}.$

Let $R\colon G\to\operatorname{Aut}(V)$ denote the representation that
integrates $\rho$ to the simply connected Lie group $G$. If we introduce
$\tilde{q}(t)\mathrel{\mathop{\mathchar 58\relax}}=R_{h(t)}q(t)$ with $h$
defined in (28), then the first evolution relation becomes

$\mathrm{d}\tilde{q}=R_{h}v(e),$

which has solution, for the the inital condition
$\tilde{q}(t_{a})=q(t_{a})=q_{a}$, given by
$\tilde{q}(t)=q_{a}+\int_{t_{a}}^{t}R_{h}v(e)$. Therefore,

$q_{q_{a},p^{b}}(t)=R_{h(t)}^{-1}\left(q_{a}+\int_{t_{a}}^{t}R_{h}v(e)\right).$

Similarly, we get

$p^{q_{a},p^{b}}(t)=\bar{R}_{h(t)}^{-1}\left(p^{b}+\int_{t}^{t_{b}}\bar{R}_{h}w(e)\right),$

where $\bar{R}_{h}\mathrel{\mathop{\mathchar 58\relax}}=(R_{h}^{-1})^{*}$ is
the dual representation on $V^{*}$. For later convenience we introduce the
$1$-forms on $G$ (valued in $V$ and $V^{*}$, respectively)

$\alpha(h)\mathrel{\mathop{\mathchar
58\relax}}=R_{h}v(e),\quad\beta(h)\mathrel{\mathop{\mathchar
58\relax}}=\bar{R}_{h}w(e).$

As a consequence of the second and third relations of (30) (and of the fact
that the Maurer–Cartan $1$-form $e$ is flat), we get that $\alpha$ and $\beta$
are closed. This implies that $\int_{\gamma}\alpha$ and $\int_{\gamma}\beta$
are invariant under homotopies of the path $\gamma\colon[t_{a},t_{b}]\to G$
with fixed endpoints. In particular, if $\gamma$ is a path joining the
identity in $G$ to some element $g$, then we define

$\Phi(g)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\gamma}\alpha,\quad\Psi(g)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\gamma}\beta.$

Note that, since $G$ is simply connected, we get well-defined maps $\Phi\colon
G\to V$ and $\Psi\colon G\to V^{*}$.212121One can easily show that
$\Phi(g_{1}g_{2})=\Phi(g_{1})+R_{g_{1}}\Phi(g_{2})$ and
$\Psi(g_{1}g_{2})=\Psi(g_{1})+\bar{R}_{g_{1}}\Psi(g_{2})$; i.e., $\Phi$ and
$\Psi$ are $1$-cocycles on $G$ with values in $V$ and $V^{*}$, respectively.
Our formulae give an explicit integration of the Lie algebra $1$-cocycles to
the corresponding Lie group $1$-cocycles, which is guaranteed by general
arguments, see [41, Lemma 2.13]. We now want to compute the HJ action. First
observe that

$\begin{split}p^{q_{a},p^{b}}\mathrm{d}q_{q_{a},p^{b}}-e^{\alpha}H_{\alpha}(p^{q_{a},p^{b}},q_{q_{a},p^{b}})&=p^{q_{a},p^{b}}\mathrm{d}_{e}q_{q_{a},p^{b}}-p^{q_{a},p^{b}}v(e)-w(e)q_{q_{a},p^{b}}\\\
&=-w(e)q_{q_{a},p^{b}}=-w(e)R_{h}^{-1}\left(q_{a}+\int_{t_{a}}^{\bullet}R_{h}v(e)\right),\end{split}$

so

$\int_{t_{a}}^{t_{b}}(p^{q_{a},p^{b}}\mathrm{d}q_{q_{a},p^{b}}-e^{\alpha}H_{\alpha}(p^{q_{a},p^{b}},q_{q_{a},p^{b}}))=-\Psi(g)q_{a}-\text{WZW}(g)$

with $g=h(t_{b})$. We introduced the following notation (which will become the
WZW term in the case of Chern–Simons theory):

(31) $\text{WZW}(g)\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{1}<t_{2}}\beta_{2}\,\alpha_{1}$

with $\alpha_{i}\mathrel{\mathop{\mathchar 58\relax}}=\pi_{i}^{*}\alpha$,
$\beta_{i}\mathrel{\mathop{\mathchar 58\relax}}=\pi_{i}^{*}\beta$ and
$\pi_{i}\colon[t_{a},t_{b}]^{2}\to G$, $\pi_{i}(t_{1},t_{2})=\gamma(t_{i})$,
where $\gamma$ is a(ny) path joining the identity to $g$ (e.g., the path
$h(t)$). The fact that $\text{WZW}(g)$ is invariant under homotopies of
$\gamma$ with fixed endpoints follows from the fact that $\alpha$ and $\beta$
are closed and from the fourth relation in (30). Finally, we have

$p^{q_{a},p^{b}}(t_{b})q_{q_{a},p^{b}}(t_{b})=p^{b}R_{g}^{-1}\left(q_{a}+\Phi(g)\right),$

so

(32)
$\hat{S}^{f}_{\text{HJ}}(q_{a},p^{b};g)=-p^{b}R_{g}^{-1}q_{a}-p^{b}R_{g}^{-1}\Phi(g)-\Psi(g)q_{a}-\text{WZW}(g),$

which generalizes (24) to the case of biaffine constraints.

######  Remark 4.7 (Exponential map).

We now want to rewrite the WZW term (31) of the last example to facilitate the
comparison with the case of Chern–Simons theory under the assumption that the
exponential map $\mathfrak{g}\to G$ is surjective. To simplify the notation,
we also assume $t_{a}=0$ and $t_{b}=1$. In this case, given
$g=\mathrm{e}^{\xi}$, we may choose $h(t)=\mathrm{e}^{t\xi}$. It follows that
$e=\xi\mathrm{d}t$. We introduce

(33) $\begin{split}W_{2}(g)&\mathrel{\mathop{\mathchar
58\relax}}=\Psi(g)\Phi(g)=\int_{t_{1}<t_{2}}\beta_{2}\,\alpha_{1}+\int_{t_{2}<t_{1}}\beta_{2}\,\alpha_{1},\\\
W_{3}(g)&\mathrel{\mathop{\mathchar
58\relax}}=\int_{0}^{1}\Psi(h)\rho_{e}\Phi(h)=\int_{t_{1}<t_{2},t_{3}<t_{2}}\beta_{3}\rho_{e_{2}}\alpha_{1}.\end{split}$

We have

$\begin{split}W_{3}(g)&=\int_{t_{1}<t_{2},t_{3}<t_{2}}w(e_{3})R_{h_{3}}^{-1}\rho_{e_{2}}\alpha_{1}=\int_{t_{1}<t_{2},t_{3}<t_{2}}w(\xi)R_{\mathrm{e}^{t_{3}\xi}}^{-1}\rho(\xi)\mathrm{d}t_{3}\mathrm{d}t_{2}\alpha_{1}\\\
&=\int_{t_{1}<t_{2},t_{3}<t_{2}}w(\xi)\rho(\xi)R_{\mathrm{e}^{t_{3}\xi}}^{-1}\mathrm{d}t_{3}\mathrm{d}t_{2}\alpha_{1}=-\int_{t_{1}<t_{2},t_{3}<t_{2}}w(\xi)\frac{\mathrm{d}}{\mathrm{d}t_{3}}R_{\mathrm{e}^{t_{3}\xi}}^{-1}\mathrm{d}t_{3}\mathrm{d}t_{2}\alpha_{1}\\\
&=-\int_{t_{1}<t_{2}}w(\xi)R_{\mathrm{e}^{t_{2}\xi}}^{-1}\mathrm{d}t_{2}\alpha_{1}+\int_{t_{1}<t_{2}}w(\xi)\mathrm{d}t_{2}\alpha_{1}\\\
&=-\int_{t_{1}<t_{2}}\beta_{2}\,\alpha_{1}+\int_{t_{1}<t_{2}}w(\xi)\mathrm{d}t_{2}\alpha_{1}.\end{split}$

Therefore,

$W_{2}(g)-W_{3}(g)=2\text{WZW}(g)+\left(\int_{t_{2}<t_{1}}\beta_{2}\,\alpha_{1}-\int_{t_{1}<t_{2}}w(\xi)\mathrm{d}t_{2}\alpha_{1}\right).$

We now want to show that the term in brackets vanishes. In fact,

$\int_{t_{2}<t_{1}}\beta_{2}\,\alpha_{1}=\int_{t_{2}<t_{1}}w(\xi)R_{\mathrm{e}^{(t_{1}-t_{2})\xi}}v(\xi)\mathrm{d}t_{2}\mathrm{d}t_{1}\underset{r=t_{1}}{\overset{s=t_{1}-t_{2}}{=}}\int_{s<r}w(\xi)R_{\mathrm{e}^{s\xi}}v(\xi)\mathrm{d}r\mathrm{d}s.$

On the other hand,

$\int_{t_{1}<t_{2}}w(\xi)\mathrm{d}t_{2}\alpha_{1}=\int_{t_{1}<t_{2}}w(\xi)R_{\mathrm{e}^{t_{1}\xi}}v(\xi)\mathrm{d}t_{2}\mathrm{d}t_{1}\underset{r=t_{2}}{\overset{s=t_{1}}{=}}\int_{s<r}w(\xi)R_{\mathrm{e}^{s\xi}}v(\xi)\mathrm{d}r\mathrm{d}s.$

In conclusion,

(34) $\text{WZW}(g)=\frac{1}{2}W_{2}(g)-\frac{1}{2}W_{3}(g).$

######  Example 4.8 (Adjoint representation).

We now specialize the above example and remark to the case when
$(V,\rho)=(\mathfrak{g},\mathrm{ad})$ is the adjoint representation. Note that
the second relation of (30) now says that $v$ is a derivation of
$\mathfrak{g}$. As such it may be extended to the universal enveloping algebra
as a derivation222222This is well-defined because, if $v$ is a Lie algebra
derivation and is extended to the tensor algebra as a derivation, then it
preserves the ideal generated by elements of the form
${\xi\otimes\eta-\eta\otimes\xi-[\xi,\eta]}$. and then to (the image of the
exponential map in) the group. We have

$v(\mathrm{e}^{\xi})=\int_{0}^{1}\mathrm{e}^{\xi
t}v(\xi)\mathrm{e}^{\xi(1-t)}\mathrm{d}t.$

In this case, for $g=\mathrm{e}^{\xi}$, we get

$\Phi(g)=v(g)g^{-1},$

where again for notational simplicity we have assumed to have a matrix Lie
group. We further assume that $\mathfrak{g}$ is a quadratic Lie algebra, i.e.,
it is equipped with an invariant nondegenerate pairing $\langle\ ,\ \rangle$,
which we can use to identify $\mathfrak{g}^{*}$ with $\mathfrak{g}$. We will
then denote by $\bar{q}$ the image of $p$ under this isomorphism (and we will
then take the target to be $\mathfrak{g}\oplus\mathfrak{g}$ with symplectic
structure induced from the pairing) and by $\bar{v}$ the composition of $w$
with the isomorphism. We can then rewrite (32) as

(35)
$\hat{S}^{f}_{\text{HJ}}(q_{a},p^{b};g)=-\langle\bar{q}^{b},{g}^{-1}q_{a}g\rangle-\langle\bar{q}^{b},{g}^{-1}v(g)\rangle-\langle\bar{v}(g)g^{-1},q_{a}\rangle-\text{WZW}(g)$

and, using (33) and (34), the WZW term as

(36)
$\text{WZW}(g)=\frac{1}{2}\langle\bar{v}(g)g^{-1},v(g)g^{-1}\rangle+\frac{1}{2}\int_{0}^{1}\langle\bar{v}(h)h^{-1},[v(h)h^{-1},\mathrm{d}hh^{-1}]\rangle.$

######  Remark 4.9 (A trivial extension).

The last example may be generalized to the case when $V=\mathfrak{g}\otimes
Z$, with $Z$ a trivial representation of $\mathfrak{g}$. We keep denoting the
representation by $\mathrm{ad}$: $\mathrm{ad}_{x}(y\otimes z)=[x,y]\otimes z$.
Now $v\colon\mathfrak{g}\to\mathfrak{g}\otimes Z$ is a derivation in the sense
that $v([x,y])=\mathrm{ad}_{x}v(y)-\mathrm{ad}_{y}v(x)$ for all
$x,y\in\mathfrak{g}$. We identify $V^{*}$ with $\mathfrak{g}\otimes Z^{*}$
using the nondegenerate pairing on $\mathfrak{g}$ and denote by $\bar{q}$ and
$\bar{v}$ the composition of $p$ and $w$ with this isomorphism. Note that
$\bar{v}$ is also a derivation in the above sense. The HJ action is again
given by (35) and (36) with the obvious understanding of notations.

######  Remark 4.10 (General solution via the exponential map).

In the general case, but under the assumption that the exponential map
$\mathfrak{g}\to G$ is surjective, the HJ action can also be characterized as
follows. As in (29), we set $g=P\mathrm{e}^{\int_{t_{a}}^{t_{b}}e}$. We then
take $T\in\mathfrak{g}$ with $\mathrm{e}^{T}=g$ and define

$h(t)=\mathrm{e}^{-\frac{t-t_{a}}{t_{b}-t_{a}}T}P\mathrm{e}^{\int_{t_{a}}^{t}e}.$

We have $e=h^{-1}\mathrm{d}h+h^{-1}\frac{T\,\mathrm{d}t}{t_{b}-t_{a}}h$, which
shows that we can reduce $e$ to $e_{0}\mathrel{\mathop{\mathchar
58\relax}}=\frac{T\,\mathrm{d}t}{t_{b}-t_{a}}$ by a gauge transformation.
Since (29) does not depend on which representative of $[e]$ we take, we may
pick $e_{0}$. With the change of variable $s=\frac{t}{t_{b}-t_{a}}$, the
action then becomes

$S[p,q,e_{0}]=\int_{0}^{1}(p_{i}\mathrm{d}q^{i}-\tilde{H}(p,q)\mathrm{d}s),\qquad\tilde{H}\mathrel{\mathop{\mathchar
58\relax}}={\langle H,T\rangle},$

As a result,

(37)
$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)=S_{\text{HJ}}^{f,\tilde{H}}(q_{a},Q_{b};0,1),$

where $S_{\text{HJ}}^{f,\tilde{H}}$ denotes the HJ action for a system with
hamiltonian $\tilde{H}$. This formula is very general, but, unlike the
previous ones for the biaffine case, not explicit.

###  4.3. The general case

We briefly discuss the general case, where we make no simplifying assumptions
on the structure functions $f_{\alpha\beta}^{\gamma}$, only for completeness,
as this will not be needed for the rest of the paper. As a consequence, this
section, which requires knowledge of Lie algebroids and Lie groupoids, may be
safely skipped by the not interested reader.

We will assume that the differentials of the $H_{\alpha}$s on their common
zero locus $C$ are linearly independent. This in particular implies that $C$
is a submanifold of $N\mathrel{\mathop{\mathchar 58\relax}}=T^{*}M$ and, as
observed in Remark 4.1, that the hamiltonian vector fields
$X_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}=X^{H_{\alpha}}$ restricted to
$C$ are linearly independent, so they define a regular distribution: the
characteristic distribution of the coisotropic submanifold $C$. (See
Definition A.19 and Remark A.25.)

The first remark is that the structure functions $f_{\alpha\beta}^{\gamma}$
and the hamiltonian vector fields $X_{\alpha}$ of the constraints $H_{\alpha}$
define a Lie algebroid $A$ over $C$. As a vector bundle, $A$ is the trivial
product $C\times\mathbb{R}^{k}$, where $k$ is the number of constraints. The
anchor map $\rho\colon A\to TC$ is given by the vector fields $X_{\alpha}$:

$\rho(x,u_{\alpha})=X_{\alpha}(x),$

where $(u_{1},\dots,u_{k})$ is the standard basis of $\mathbb{R}^{k}$. We can
expand a section $\sigma\in\Gamma(A)$ as $\sigma^{\alpha}u_{\alpha}$, where
now the $u_{\alpha}$s are regarded as constant sections of $A$. We then have
$\rho(\sigma)=\sigma^{\alpha}X_{\alpha}$. The Lie bracket is defined as

$[\sigma,\tau]^{\gamma}=f_{\alpha\beta}^{\gamma}\sigma^{\alpha}\tau^{\beta}+\sigma^{\alpha}X_{\alpha}(\tau^{\gamma})-\tau^{\alpha}X_{\alpha}(\sigma^{\gamma}).$

######  Remark 4.11.

The Lie algebroid $A$ is isomorphic to the characteristic distribution of $C$
viewed as a Lie subalgebroid of $TC$. Note that in general the conormal bundle
$N^{*}C$ of a coisotropic submanifold of a symplectic manifold $N$ has a
canonical Lie algebroid structure, isomorphic to the characteristic
distribution, which makes it into a Lie subalgebroid of $T^{*}N$. In our
particular case, since $C$ is defined by constraints, its normal bundle, and
therefore its conormal bundle, can be trivialized, and this is what have done
above.

The Lie algebroid $A\to C$ can be extended to the vector bundle $\hat{A}\to
N=T^{*}M$ with $\hat{A}=N\times\mathbb{R}^{k}$. We can then view the
$H_{\alpha}$s as a section $H$ of $\hat{A}^{*}$. It is also possible to extend
the anchor map and the bracket as defined above, but in general $\hat{A}$ will
not be a Lie algebroid (it is, though, when the structure functions are
constant).

Next we denote the map $(p,q)\colon I\to T^{*}M$, with $I$ the interval
$[t_{a},t_{b}]$, as $x\colon I\to N$. The field $e$ may be regarded as a
section of $T^{*}I\otimes x^{*}\hat{A}$. The constraints $H_{\alpha}$s
composed with $x$ define a section of $x^{*}\hat{A}^{*}$, which we keep
denoting as $H$. As a consequence, we may rewrite the action as

$S[p,q,e]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-\langle H(p,q),e\rangle),$

where $\langle\ ,\ \rangle$ denotes the pairing of $\hat{A}^{*}$ with
$\hat{A}$. A solution $(x,e)$ of the evolution equations is the same as an
anchor-preserving morphism $TI\to\hat{A}$.

The generalized HJ action $\hat{S}^{f}_{\text{HJ}}$ is defined as in (13) and,
by Theorem 4.2, is a generalized generating function for the evolution
relation.

It is possible to give a more transparent expression for
$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};[e])$, analogous to the one we have in
the Lie algebra case, when the $x$ component of the assumedly unique solution
of the evolution equations specified by $q_{a}$, $Q_{b}$, and $e$ has image in
$C$ (note that, if $x(t)$ is in $C$ for some $t$, then $x(t)$ is in $C$ for
every $t$). In fact, in this case we may view $(x,e)$ as an anchor-preserving
morphism $x\colon TI\to A$, which is the same as a Lie algebroid morphism
$TI\to A$. Denote by $\mathcal{G}$ the source simply connected Lie groupoid of
$A$ (i.e., up to isomorphism, the monodromy groupoid of the characteristic
distribution). The Lie algebroid morphism $(x,e)$, with initial condition
$x(t_{a})=x_{a}\in C$, may be then uniquely integrated to a Lie groupoid
morphism $E\colon I\times I\to\mathcal{G}$. If we denote by $\alpha$ and
$\beta$ the source and target maps of $\mathcal{G}$, then we have, in
particular,

$x(s)=\alpha(E(s,t))\ \forall t\in I,\qquad x(t)=\beta(E(s,t))\ \forall s\in
I.$

Define $h(t)\mathrel{\mathop{\mathchar 58\relax}}=E(t_{a},t)$. We then have

$h(t_{a})=1_{x_{a}},\quad\alpha(h(t))=x_{a}\ \forall t\in I,$

where $1_{x}$ denotes the unit of $\mathcal{G}$ at $x\in N$. In particular,
$h$ is a map $I\to\alpha^{-1}(x_{a})$.

We may now recover $e$ from $h$, generalizing footnote 20, as follows. First
consider the linear map $\mathrm{d}_{t}h\colon T_{t}I\to
T_{h(t)}\alpha^{-1}(x_{a})$. Next consider the left translation
$l_{h}\colon\alpha^{-1}(\beta(h))\to\alpha^{-1}({\alpha(h)})$, $l_{h}g=hg$.
Since $h1_{\beta(h)}=h$, we get the linear map

$\mathrm{d}_{1_{\beta(h)}}l_{h}\colon A_{\beta(h)}\to
T_{h}\alpha^{-1}(\alpha(h)).$

Using $\beta(h(t))=x(t)$ and $\alpha(h(t))=x_{a}$, we get the linear map

$e(t)=(\mathrm{d}_{1_{x(t)}}l_{h})^{-1}\mathrm{d}_{t}h\colon T_{t}I\to
A_{x(t)},$

which may be shown to be the value at $t$ of our section $e$ of $T^{*}I\otimes
x^{*}\hat{A}$.

The rest of the computation is now like in the Lie algebra case, and we have

$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)=S^{f}_{\text{HJ}}(q_{a},Q_{b})[e]\quad\text{with
}g=h(t_{b})\in\alpha^{-1}(x(t_{a}))\cap\beta^{-1}(x(t_{b})).$

Finally, we can take variations with respect to $q_{a}$, $Q_{b}$, and $g$,
getting

$\delta\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)=P_{i}(t_{b})\delta
Q^{i}_{b}-p_{i}(t_{a})\delta q^{i}_{a}-\left\langle
H(x(t_{b})),(\mathrm{d}_{{1_{x(t_{b})}}}l_{g})^{-1}\delta g\right\rangle,$

with $x(t_{b})=(p^{q_{a},Q_{b}}(t_{b}),q_{q_{a},Q_{b}}(t_{b}))$. This is
consistent with $\hat{S}^{f}_{\text{HJ}}$ being a generalized generating
function for the evolution relation $L$ (it is in general a weaker statement,
as we are now only allowed to take variations in the class of variables for
which the $x$ component of a solution lies in $C$).

######  Remark 4.12.

The case of constant structure functions, hence corresponding to a Lie algebra
$\mathfrak{g}$, fits into the general case as the action Lie algebroid
$T^{*}M\times\mathfrak{g}$ with the action Lie groupoid $T^{*}M\times G$ as
its integration. In this case, the Lie algebroid and the Lie groupoid are
actually defined over the whole of $T^{*}M$ and not only over $C$, so the
formula for $\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)$ has no restrictions.

##  5\. Systems with nontrivial evolution and constraints

For completeness, we briefly discuss here the case of systems which, in
addition to constraints $H_{\alpha}$, $\alpha=1\dots,k$, also have a
nontrivial evolution with hamiltonian $H$. Namely, we consider an action of
the form

$S[p,q,e]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-\mathrm{d}t\,H-e^{\alpha}H_{\alpha}(p,q)).$

In addition to assuming involutivity of the constraints, as in (19), we also
assume them to be constants of motion for $H$: namely, $\\{H,H_{\alpha}\\}=0$
for all $\alpha$.

This system can actually be treated as above simply adding $H$ to the set of
the, now $k+1$, constraints, say, as $H_{0}$. The only difference is that the
one-form $e^{0}$ instead of being free will be set to be $\mathrm{d}t$. This
is a possible choice, as the Lie group $\tilde{G}$ (in the case of structure
constants $f_{\alpha\beta}^{\gamma}$; otherwise, more generally, the Lie
groupoid $\tilde{\mathcal{G}}$) corresponding to the rank-$(k+1)$ Lie algebra
(Lie algebroid) factorizes as $\mathbb{R}\times G$
($\mathbb{R}\times\mathcal{G}$), so we can fix $t_{b}-t_{a}$ in the first
factor.

The HJ action, which serves as a generalized generating function, is then, in
the Lie algebra case,

$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b})[\mathrm{d}t,e],\quad\text{with
}g=P\mathrm{e}^{\int_{t_{a}}^{t_{b}}e}.$

In the Lie algebroid case, we get, more generally,

$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};g)\mathrel{\mathop{\mathchar
58\relax}}=S^{f}_{\text{HJ}}(q_{a},Q_{b})[\mathrm{d}t,e],$

with $g\mathrel{\mathop{\mathchar
58\relax}}=\pi_{\mathcal{G}}(\tilde{h}(t_{b}))\in\alpha^{-1}(x(t_{a}))\cap\beta^{-1}(x(t_{b}))$
and $\pi_{\mathcal{G}}$ the projection $\tilde{\mathcal{G}}\to\mathcal{G}$.

###  5.1. Classical mechanics

A more conceptual way is to add two new canonically conjugated variables
$(E,\mathbb{t})$ and consider instead the action

$S[p,q,E,\mathbb{t},e^{0}]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}-E\mathrm{d}\mathbb{t}-e^{0}(H-E)-e^{\alpha}H_{\alpha}(p,q)).$

Note that the constraints $H_{\alpha}$, $\alpha=0\dots,k$, with $H_{0}=H-E$,
are still first class. This is then a system as the ones we have studied in
the previous sections. One of the evolution equations is now $\mathrm{d}E=0$,
which says that eventually $E$ will be a constant (the energy). Another
important evolution equation is $\mathrm{d}\mathbb{t}=e^{0}$, which can be
used to substitute $\mathrm{d}\mathbb{t}$ to $e^{0}$, so that we may regard
$\mathbb{t}$ as time. Note that, if we set endpoint conditions
$\mathbb{t}(t_{a})=t_{a}$ and $\mathbb{t}(t_{b})=t_{b}$, we actually get the
time evolution from $t_{a}$ to $t_{b}$.

In the following, we focus on the case $H_{\alpha}=0$ for all $\alpha>0$,
i.e.,

$S[p,q,E,\mathbb{t},e^{0}]=\int_{t_{a}}^{t_{b}}(p_{i}\mathrm{d}q^{i}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}E\mathrm{d}\mathbb{t}-e^{0}(H-E)).$

The evolution equations are

$\mathrm{d}q^{i}=e^{0}\frac{\partial H}{\partial
p_{i}},\quad\mathrm{d}p_{i}=-e^{0}\frac{\partial H}{\partial
q^{i}},\quad\mathrm{d}E=0,\quad\mathrm{d}\mathbb{t}=e^{0},$

and the constraint is $H(p,q)=E$. The evolution relation is then

(38)
$L=\\{(p^{a},E^{a},q_{a},\mathbb{t}_{a},p^{b},E^{b},q_{b},\mathbb{t}_{b})\ |\
(p,q)_{b}=\phi^{H}_{\mathbb{t}_{b}-\mathbb{t}_{a}}((p,q)_{a}),\
E^{b}=E^{a}=H(p^{a},q_{a})\\}$

where $\phi^{H}_{T}$ denotes the hamiltonian flow of $H$ for time $T$.

We now consider endpoint conditions $(q_{a},\mathbb{t}_{a},q_{b},E^{b})$. We
thus evaluate the modified action
$S^{f}[p,q,E,\mathbb{t},e^{0}]\mathrel{\mathop{\mathchar
58\relax}}=E(t_{b})\mathbb{t}(t_{b})+S[p,q,E,\mathbb{t},e^{0}]$ on the
solution, with components $(\tilde{p},\tilde{q},\tilde{\mathbb{t}})$, of the
evolution equations for fixed $(q_{a},\mathbb{t}_{a},q_{b},E^{b})$ and fixed
$e^{0}$. The first remark is that $-E\mathrm{d}\mathbb{t}+e^{0}E$ vanishes on
a solution. Therefore,

$S^{f}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},E^{b},e^{0})=E^{b}\,\tilde{\mathbb{t}}(t_{b})+\int_{t_{a}}^{t_{b}}(\tilde{p}_{i}\mathrm{d}\tilde{q}^{i}-\mathrm{d}\tilde{\mathbb{t}}\,H(\tilde{p},\tilde{q})).$

We can actually pick $e^{0}$ constant or at least of constant sign (by a gauge
transformation). With this choice we are sure that the map
$t\mapsto\tilde{\mathbb{t}}(t)=\mathbb{t}_{a}+\int_{t_{a}}^{t}e^{0}$ is a
diffeomorphism. We can then make this change of variable in the integral,
getting

$\hat{S}^{f}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b};T)=E^{b}\,(\mathbb{t}_{a}+T)+\int_{\mathbb{t}_{a}}^{\mathbb{t}_{a}+T}(\check{p}_{i}\mathrm{d}\check{q}^{i}-\mathrm{d}\tilde{\mathbb{t}}\,H(\check{p},\check{q})),$

with $\check{p}(\tilde{\mathbb{t}})\mathrel{\mathop{\mathchar
58\relax}}=\tilde{p}(t(\mathbb{t}))$,
$\check{q}(\tilde{\mathbb{t}})\mathrel{\mathop{\mathchar
58\relax}}=\tilde{q}(t(\mathbb{t}))$ and $T\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t_{b}}e^{0}$. But now we recognize in the second term
on the right hand side the time-dependent HJ action (2) for the hamiltonian
$H$ from time $\mathbb{t}_{a}$ to time $\mathbb{t}_{a}+T$. Therefore, we have

(39)
$\begin{split}\hat{S}^{f}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},E^{b};T)&=S_{\text{HJ}}^{H}(q_{a},q_{b};\mathbb{t}_{a},\mathbb{t}_{a}+T)+E^{b}(\mathbb{t}_{a}+T)\\\
&=S_{\text{HJ}}^{H}(q_{a},q_{b};0,T)+E^{b}(\mathbb{t}_{a}+T).\end{split}$

One can easily verify that $\hat{S}^{f}_{\text{HJ}}$ is a generating function
for the evolution relation (38).

The HJ action $\hat{S}_{\text{HJ}}$ for endpoint conditions
$(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})$ can be obtained from (39) by
composing, following Remark A.36, $\hat{S}^{f}_{\text{HJ}}$ with the
generating function $-E^{b}\mathbb{t}_{b}$ of the identity map, and reducing
with respect to the intermediate variable $E^{b}$; namely,
$\hat{S}_{\text{HJ}}$ is the evaluation of
$\hat{S}^{f}_{\text{HJ}}-E^{b}\mathbb{t}_{b}$ at its critical point in
$E^{b}$. We get

(40)
$\hat{S}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})=S_{\text{HJ}}^{H}(q_{a},q_{b};\mathbb{t}_{a},\mathbb{t}_{b}).$

One can easily verify that $\hat{S}_{\text{HJ}}$ is also a generating function
for the evolution relation (38). Observe that the dependency on $T$ has
disappeared.

Note that we could not have computed $\hat{S}_{\text{HJ}}$ directly because
fixing the endpoint conditions $(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})$
yields no solution in general. Namely, choosing $e^{0}$ arbitrarily would lead
to no solutions if
$\mathbb{t}_{b}-\mathbb{t}_{a}\not=T\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t_{b}}e^{0}$. A way to deal with such a situation
will be discussed in Section 6, see Remark 6.2. As a result, $T$ is fixed by
the endpoint conditions and therefore is no longer an independent variable for
$\hat{S}_{\text{HJ}}$.

The HJ action for the time-independent, constrained system (a parametrization
invariant theory) we considered in this section turns out to be the same as
the usual HJ action for a time dependent system describing a hamiltonian
evolution. In this case, time is restored via the endpoint conditions.

For simplicity we have obtained this result choosing $e^{0}$ constant, so that
we could make the change of variable globally. We might have also worked with
a generic $e^{0}$. In this case we would have made changes of variables in all
regions where it is different from zero and ignored the other regions. The
“time” variable $\tilde{\mathbb{t}}$ would have suffered slowing down,
speeding up, freezing and even rewinding, but with no change in the final
result described in (40) by $\hat{S}_{\text{HJ}}$.232323As the prince
observed, “Most people think time is like a river that flows swift and sure in
one direction, but I have seen the face of time and I can tell you they are
wrong. Time is an ocean in a storm,” yet “what is written in the timeline
cannot be changed.”

######  Remark 5.1 (Integrable systems III).

As a generalization of Remarks 3.4 and 4.5, let us consider the case of an
integrable system where we have exactly $n$ strictly involutive constraints
$H_{1},\ldots,H_{n}$ that are integrals of motion for a hamiltonian $H$, i.e.,
we have $\\{H,H_{i}\\}=\\{H_{i},H_{j}\\}=0$ for all $i,j$. Around a point
$(p_{0},q_{0})$ where the differentials
$\mathrm{d}H_{1},\ldots,\mathrm{d}H_{n}$ are linearly independent, again by
the Carathéodory–Jacobi–Lie theorem, see fotonote 15, we may find a Darboux
chart $(P_{i},Q^{i})$ with $P_{i}=H_{i}$. Changing coordinates from
$(p_{i},q^{i})$ to $(P_{i},Q^{i})$ in the integral we get

$S^{g}[p,q,E,\mathbb{t},e^{0},e]=-g(q(t_{a}),P(t_{a}))+\int_{t_{a}}^{t_{b}}(Q^{i}\mathrm{d}P_{i}-E\mathrm{d}\mathbb{t}-e^{0}(H-E)-e^{i}{P_{i}}).$

Notice that, since the constraints are integrals of motion, in $(P_{i},Q^{i})$
coordinates the hamiltonian $H$ depends only on $P_{i}$:
$0=\\{H,H_{i}\\}=\\{H,P_{i}\\}=\partial H/\partial Q_{i}$. On solutions, we
have $\mathrm{d}P_{i}=0$, $\mathrm{d}Q^{i}=e^{i}+e^{0}\frac{\partial
H}{\partial P_{i}}$ and $\mathrm{d}H=0$. The HJ action then reads

(41)
$\hat{S}^{g}_{\text{HJ}}(q_{a},P^{b},\mathbb{t}_{a},\mathbb{t}_{b};T)=-g(q_{a},P^{b})-(\mathbb{t}_{b}-\mathbb{t}_{a})H(P^{b})-T^{i}P_{i}$

in a neighborhood of $(p_{0},q_{0})$.242424Here $T$ stands for the collection
$T^{i}\mathrel{\mathop{\mathchar 58\relax}}=\int_{t_{a}}^{t_{b}}e^{i}$, $i>0$.
The parameter $T^{0}\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t_{b}}e^{0}$ does not appear, as in (40), because we
fix $\mathbb{t}_{a}$ and $\mathbb{t}_{b}$. We may also compute the HJ action
with $Q$ endpoint conditions. We have

$S^{f}[p,q,E,\mathbb{t},e^{0},e]=-f(q(t_{a}),Q(t_{a}))+\int_{t_{a}}^{t_{b}}(P_{i}\mathrm{d}Q^{i}-E\mathrm{d}\mathbb{t}-e^{0}(H-E)-e^{i}P_{i}).$

Solving the evolution equation for $Q^{i}$ we obtain the HJ action252525As in
footnote 24, $T$ stands for the collection $T^{i}$, $i>0$.

(42)
$\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b},\mathbb{t}_{a},\mathbb{t}_{b};T)=-f\left(q_{a},Q_{b}-T-(\mathbb{t}_{b}-\mathbb{t}_{a})\frac{\partial
H}{\partial P}\right)+(\mathbb{t}_{b}-\mathbb{t}_{a})H(P(q_{a},Q_{b})).$

###  5.2. The free relativistic particle

The above discussion generalizes easily to the case of a free, relativistic
particle. In this case, there is no dynamics but only the constraint
$E^{2}=m^{2}+p^{2}$. (For simplicity, we omit indices. If we are not in one
dimension, $p^{2}$ means $\sum_{i}(p_{i})^{2}$. Similarly, in the action we
will simply write $p\mathrm{d}q$ instead of $p_{i}\mathrm{d}q^{i}$).
Therefore, we consider the action

$S[p,q,E,\mathbb{t},e^{0}]=\int_{t_{a}}^{t_{b}}\left(p\mathrm{d}q-E\mathrm{d}\mathbb{t}-\frac{e^{0}}{2}(p^{2}+m^{2}-E^{2})\right),$

where we have introduced the factor $\frac{1}{2}$ just for convenience. The
evolution equations are simply

$\mathrm{d}q=e^{0}p,\quad\mathrm{d}p=0,\quad\mathrm{d}E=0,\quad\mathrm{d}\mathbb{t}=e^{0}E,$

whereas the constraint is

$p^{2}+m^{2}-E^{2}=0.$

We first compute the HJ action, without fixing $e^{0}$, for endpoint
conditions given by $(q_{a},\mathbb{t}_{a},p^{b},E^{b})$. This means the we
have to insert a solution of the evolution equations into

$S^{f}[p,q,E,\mathbb{t},e^{0}]=E(t_{b})\mathbb{t}(t_{b})-p(t_{b})q(t_{b})+S[p,q,E,\mathbb{t},e^{0}].$

The first evolution equation implies $p\mathrm{d}q=e^{0}p^{2}$. The second
implies that $p(t)=p^{b}$ for all $t$, so we actually have
$p\mathrm{d}q=e^{0}(p^{b})^{2}$. The third equation implies $E(t)=E^{b}$ for
all $t$, which together with the last equation implies
$E\mathrm{d}\mathbb{t}=e^{0}(E^{b})^{2}$. Therefore,

$S|_{\text{solution}}=\int_{t_{a}}^{t_{b}}\frac{e^{0}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}}\,((p^{b})^{2}-m^{2}-(E^{b})^{2})=\frac{T}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}}\,((p^{b})^{2}-m^{2}-(E^{b})^{2}),$

where we have set $T\mathrel{\mathop{\mathchar
58\relax}}=\int_{t_{a}}^{t_{b}}e^{0}$. Solving the evolution equations
explicitly yields

$q(t_{b})=q_{a}+p^{b}T\quad\text{and}\quad\mathbb{t}(t_{b})=\mathbb{t}_{a}+E^{b}T,$

which implies

$p(t_{b})q(t_{b})=p^{b}q_{a}+(p^{b})^{2}T\quad\text{and}\quad
E(t_{b})\mathbb{t}(t_{b})=E^{b}\mathbb{t}_{a}+(E^{b})^{2}T$

on solutions. We then have

(43)
$\hat{S}^{f}_{\text{HJ}}(q_{a},\mathbb{t}_{a},p^{b},E^{b};T)=E^{b}\mathbb{t}_{a}-p^{b}q_{a}+\frac{(E^{b})^{2}-(p^{b})^{2}-m^{2}}{2}T,$

and one can easily verify that this is a generating funciton for the evolution
relation.

We may now easily pass to the HJ action for endpoint conditions
$(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})$ simply composing the generating
function $\hat{S}^{f}_{\text{HJ}}$ with the generating function
$p^{b}q_{b}-E^{b}\mathbb{t}_{b}$ of the identity map, and reducing with
respect to the intermediate variables $(p^{b},E^{b})$; namely, we evaluate
$\hat{S}^{f}_{\text{HJ}}+p^{b}q_{b}-E^{b}\mathbb{t}_{b}$ at its critical point
in $(p^{b},E^{b})$. Setting the derivatives with respect to $p^{b}$ and
$E^{b}$ to zero yields

$p^{b}=\frac{\Delta q}{T}\quad\text{and}\quad
E^{b}=\frac{\Delta\mathbb{t}}{T},$

where we have set

$\Delta q\mathrel{\mathop{\mathchar
58\relax}}=q_{b}-q_{a}\quad\text{and}\quad\Delta\mathbb{t}\mathrel{\mathop{\mathchar
58\relax}}=t_{b}-t_{a}.$

Inserting, we get

(44)
$\hat{S}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b};T)=\frac{(\Delta
q)^{2}-(\Delta\mathbb{t})^{2}}{2T}-\frac{1}{2}m^{2}T.$

We can finally try to reduce with respect to the variable $T$, i.e., to
evaluate $\hat{S}_{\text{HJ}}$ at its critical point in $T$. Setting the
derivative with respect to $T$ to zero yields

$T^{2}=\frac{(\Delta\mathbb{t})^{2}-(\Delta q)^{2}}{m^{2}}.$

This shows that we can perform this reduction if and only if the endpoint
conditions select a timelike trajectory: $(\Delta\mathbb{t})^{2}>(\Delta
q)^{2}$.262626This is actually a necessary condition for a solution to exist.
In fact, solving the evolution equations for $\mathbb{t}$ and $q$ yields
$\Delta\mathbb{t}=ET$ and $\Delta q=pT$, so $(\Delta\mathbb{t})^{2}-(\Delta
q)^{2}=(E^{2}-p^{2})T^{2}=m^{2}T^{2}\geq 0$. The case
$(\Delta\mathbb{t})^{2}=(\Delta q)^{2}$ would however imply $T=0$, so
$\Delta\mathbb{t}=0$ and $\Delta q=0$, which is excluded because the initial
and final conditions must be distinct in the HJ setting. There are two roots
in $T$ which yield, not unexpectedly,

(45) $S_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})=\mp
m\sqrt{(\Delta\mathbb{t})^{2}-(\Delta q)^{2}},$

i.e., the Minkowskian length up to the factor $\mp m$. Both choices of sign
give a generating function for the evolution relation.272727Note that taking
derivatives with respect to $\mathbb{t}_{a}$ and $\mathbb{t}_{b}$ yields
$E^{a}=E^{b}=\pm m\frac{\Delta\mathbb{t}}{\sqrt{(\Delta\mathbb{t})^{2}-(\Delta
q)^{2}}}$. In particular, for $\Delta q=0$, we get $E^{a}=E^{b}=\pm
m\operatorname{sgn}{\Delta\mathbb{t}}$. For $\Delta\mathbb{t}>0$, it is the
plus sign that yields the relation $E=m$ for the relativistic particle, the
minus sign corresponding to its antiparticle.

######  Remark 5.2.

In the massless case $m=0$, equation (44) becomes

$\hat{S}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b},T)=\frac{(\Delta
q)^{2}-(\Delta\mathbb{t})^{2}}{2T},$

so the last step—reduction in the variable $T$—is not possible (the critical
point in $T$ does not exist for general values of
$q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b}$). Still, the vanishing of the
derivative in $T$ correctly yields the “lightlike displacement” constraint:
$(\Delta q)^{2}-(\Delta\mathbb{t})^{2}=0$.

##  6\. Generalized generating functions for “bad” endpoint conditions

The presence of constraints is not the only source for the appearance of extra
parameters in the generating function for the evolution relation. Another
instance occurs when we choose “bad” endpoint conditions that do not ensure
existence of a solution.

###  6.1. No evolution and no constraints

We discuss a simple example here (more examples, also coupled to the presence
of constraints, will appear in the rest of the paper). Consider the action

$S[p,q]=\int_{t_{a}}^{t_{b}}p\dot{q}\,\mathrm{d}t.$

The EL equations are simply $\dot{p}=\dot{q}=0$, so the evolution relation is
just the diagonal in $T^{*}\mathbb{R}\times T^{*}\mathbb{R}$:

$L=\\{(p,q,p,q)\in T^{*}\mathbb{R}\times T^{*}\mathbb{R},(p,q)\in
T^{*}\mathbb{R}\\}.$

If we now choose endpoint conditions $q_{a}$ and $q_{b}$, we will get no
solutions unless $q_{a}=q_{b}$, in which case we get a whole family of
solutions, parametrized by $p$ solving $\dot{p}=0$. Nevertheless, there is a
generalized generating function for $L$:282828This generalized generating
function can also be obtained as the composition, in the sense of Remark A.37,
of the generating functions $\psi_{1}(q_{a},\lambda)=-\lambda q_{a}$ and
$\psi_{2}(\lambda,q_{b})=\lambda q_{b}$ that do and undo a $-\pi$-rotation in
phase space.

$S_{\text{gen}}(q_{a},q_{b};\lambda)=\lambda\,(q_{b}-q_{a}).$

In fact, $\frac{\partial S_{\text{gen}}}{\partial\lambda}=0$ yields the
condition $q_{a}=q_{b}$, whereas $p(t_{a})=-\frac{\partial
S_{\text{gen}}}{\partial q_{a}}=\lambda$ and $p(t_{b})=\frac{\partial
S_{\text{gen}}}{\partial q_{b}}=\lambda$ imply $p(t_{a})=p(t_{b})$.

This generating function may also be regarded as a sort of Hamilton–Jacobi
action, where we impose only one EL equation: namely, $\dot{p}=0$. (The other
equation, $\dot{q}=0$, cannot be imposed anyway because generically it has no
solutions.) The parameter $\lambda$ arises here as the constant value of $p$,
so it parametrizes the family of solutions.

We may also obtain the generating function
$S_{\text{gen}}(q_{a},q_{b},\lambda)$ by a partial Legendre transform, i.e.,
as

$S_{\text{gen}}(q_{a},q_{b};\lambda)=\lambda
q_{b}+S^{f}_{\text{HJ}}(q_{a},\lambda),$

where

$S^{f}_{\text{HJ}}(q_{a},p^{b})=-p^{b}q_{a}$

is the HJ action for the good choice $(q_{a},p^{b})$ of endpoint conditions.
Unlike the true Legendre transform (see Remark 2.9), we do not evaluate at
$\lambda^{\text{crit}}$, which in this case does not exist. We will put this
observation more in context in Section 6.2.

The generating function $S_{\text{gen}}$ may appear even more naturally from
the viewpoint of path integral quantization. In fact, since there is no
evolution at all, the evolution operator is the identity operator, so, for the
chosen polarizations, its integral kernel is a delta function:

$K(q_{a},q_{b})=\delta(q_{a}-q_{b}).$

By Fourier transform, we we can also write

$K(q_{a},q_{b})=\int\frac{\mathrm{d}\lambda}{2\pi\hbar}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S_{\text{gen}}(q_{a},q_{b},\lambda)}.$

One formal way to get this result directly from the path integral is as
follows. We start with (4) and write $p=\lambda+\hat{p}$, where $\lambda$ is a
constant (a solution to $\dot{p}=0$). Note that $\lambda$ here is still a
variable to be integrated out, so we have to make sure that
$\hat{p}\colon[t_{a},t_{b}]\to\mathbb{R}$ is in a complement to the space of
constant maps. For example, we may impose
$\int_{t_{a}}^{t_{b}}\hat{p}\,\mathrm{d}t=0$.292929Let $\mathfrak{c}$ be the
space of constant maps and $\mathfrak{d}$ the space of maps
$\hat{p}\colon[t_{a},t_{b}]\to\mathbb{R}$ with vanishing integral. We have
$C^{\infty}([t_{a},t_{b}])=\mathfrak{c}\oplus\mathfrak{d}$. In fact, if
$\lambda$ is constant and
$(t_{a}-t_{b})\lambda=\int_{t_{a}}^{t_{b}}\lambda\;\mathrm{d}t=0$, then
$\lambda$ vanishes, and we have shown that $\mathfrak{c}\cap\mathfrak{d}=0$.
On the other hand, every map $p$ can be written as $\lambda+\hat{p}$ with
$\lambda=\frac{1}{t_{b}-t_{a}}\int_{t_{a}}^{t_{b}}p(t)\mathrm{d}t$. It follows
immediately that $\int_{t_{a}}^{t_{b}}\hat{p}\,\mathrm{d}t=0$. Then we have

$S[\lambda+\hat{p},q]=\lambda\,(q_{b}-q_{a})+\int_{t_{a}}^{t_{b}}\hat{p}\dot{q}\,\mathrm{d}t.$

Inserting into (4), we get

$K(q_{a},q_{b})=\int\mathrm{d}\lambda\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\,\lambda\,(q_{b}-q_{a})}\int_{\begin{subarray}{c}q(t_{a})=q_{a}\\\
q(t_{b})=q_{b}\\\
\int_{t_{a}}^{t_{b}}\hat{p}\,\mathrm{d}t=0\end{subarray}}D\hat{p}\,Dq\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\int_{t_{a}}^{t_{b}}\hat{p}\dot{q}\,\mathrm{d}t}.$

The second integral formally does not depend on $q_{a}$ and $q_{b}$: in fact,
we can make the affine change of variables $q\mapsto\hat{q}$ with

$\hat{q}(t)=q(t)-\frac{t-t_{b}}{t_{a}-t_{b}}q_{a}-\frac{t-t_{a}}{t_{b}-t_{a}}q_{b}.$

Observe that
$\int_{t_{a}}^{t_{b}}\hat{p}\dot{q}\,\mathrm{d}t=\int_{t_{a}}^{t_{b}}\hat{p}\dot{\hat{q}}\,\mathrm{d}t$,
since $\dot{q}-\dot{\hat{q}}$ is constant, and that
$\hat{q}(t_{a})=\hat{q}(t_{b})=0$. Therefore,

$\begin{split}K(q_{a},q_{b})&=\int\mathrm{d}\lambda\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\,\lambda\,(q_{b}-q_{a})}\int_{\begin{subarray}{c}\hat{q}(t_{a})=\hat{q}(t_{b})=0\\\
\int_{t_{a}}^{t_{b}}\hat{p}\,\mathrm{d}t=0\end{subarray}}D\hat{p}\,D\hat{q}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\int_{t_{a}}^{t_{b}}\hat{p}\dot{\hat{q}}\,\mathrm{d}t}\\\
&\propto\int\frac{\mathrm{d}\lambda}{2\pi\hbar}\;\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S_{\text{gen}}(q_{a},q_{b},\lambda)}.\end{split}$

In this example, one might also think of $\lambda$ as parametrizing the vacua
of the theory. Instead of integrating over vacua, one may decide to select
just one, labeled by $\lambda$, and in this case interpret
$S_{\text{gen}}(q_{a},q_{b},\lambda)$ as the corresponding semiclassical
contribution, more in line with Hamilton–Jacobi.

######  Remark 6.1.

For an analysis of this example in the BV-BFV setting, see [19, Sect. 4.4]

######  Remark 6.2 (Classical mechanics).

We may apply the above considerations to the system with one constraint
described in Section 5.1. We fix $(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b})$
and let $T=\int_{t_{a}}^{t_{b}}e^{0}$. From the evolution equation
$\mathrm{d}\mathbb{t}=e^{0}$, we get that there is no solution unless
$T=\mathbb{t}_{b}-\mathbb{t}_{a}$. Therefore, we have to introduce a new
parameter $\lambda$ to fix this condition, and we get

$\check{S}_{\text{HJ}}(q_{a},\mathbb{t}_{a},q_{b},\mathbb{t}_{b};T,\lambda)=S_{\text{HJ}}^{H}(q_{a},q_{b};\mathbb{t}_{a},\mathbb{t}_{a}+T)+\lambda\,(T-\mathbb{t}_{b}+\mathbb{t}_{a}).$

Of course this generating function may be reduced. Setting the derivative with
respect to $\lambda$ to zero, we get back $T=\mathbb{t}_{b}-\mathbb{t}_{a}$,
which may be inserted into $\check{S}_{\text{HJ}}$ yielding
$\hat{S}_{\text{HJ}}$ as in (40).

###  6.2. The partial Legendre transform

Suppose our system has a unique solution for endpoint conditions
$(q_{a},p^{b})$, so that we have the HJ action
$S^{f}_{\text{HJ}}(q_{a},p^{b})$ as in Example 2.8. We define the partial
Legendre transform of $S^{f}_{\text{HJ}}$ as

$S_{\text{gen}}(q_{a},q_{b},\lambda)\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{i}q_{b}^{i}+S^{f}_{\text{HJ}}(q_{a},\lambda).$

It follows from (7) that $S_{\text{gen}}$ is a generalized generating function
for the same evolution relation as $S^{f}_{\text{HJ}}$ is. In fact,
$L_{S_{\text{gen}}}$ is determined by $S_{\text{gen}}$ via the equations

$p^{b}_{i}=\frac{\partial S_{\text{gen}}}{\partial q^{i}_{b}},\quad
p^{a}_{i}=-\frac{\partial S_{\text{gen}}}{\partial
q^{i}_{a}},\quad\frac{\partial S_{\text{gen}}}{\partial\lambda}=0.$

Since $\frac{\partial S_{\text{gen}}}{\partial q^{i}_{b}}=\lambda_{i}$, we get
$p^{b}=\lambda$. Moreover, by using (7), the last equation reads
$q_{b}-q_{q_{a},\lambda}(t_{b})=0$ and the middle equation yields
$p^{a}=p^{q_{a},\lambda}(t_{a})$. Therefore, $L_{S_{\text{gen}}}$ consists of
endpoint values $(p^{a},q_{a},p^{b},q_{b})$ of a solution and is thus the
evolution relation.303030It may of course happen that the last equation,
$q_{b}-q_{q_{a},\lambda}(t_{b})=0$, can be solved for a unique $\lambda$,
which we then denote as $\lambda^{\text{crit}}$. We may then insert this value
into $S_{\text{gen}}(q_{a},q_{b},\lambda)$ and get a generating function
without extra parameters, the true Legendre transform
$S_{\text{HJ}}(q_{a},q_{b})$ as in Remark 2.9.

The above argument extends immediately to the case with constraints. Namely,
suppose we have the HJ action $S^{f}_{\text{HJ}}(q_{a},p^{b})[e]$. Then we
define its partial Legendre transform as

$S_{\text{gen}}(q_{a},q_{b},\lambda)[e]\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{i}q_{b}^{i}+S^{f}_{\text{HJ}}(q_{a},\lambda)[e],$

and one immediately verifies, as above, that this is also a generalized
generating function for the evolution relation.

######  Example 6.3 (Partial Legendre transform for linear constraints).

Consider the case of a system with linear constraints in strict involution as
described in Example 4.3. We will suppress the target-space index $i$. If we
apply the partial Legendre transform to (24), we get

$\hat{S}_{\text{gen}}(q_{a},q_{b};\lambda,T)\mathrel{\mathop{\mathchar
58\relax}}=\lambda
q_{b}+\hat{S}^{f}_{\text{HJ}}(q_{a},\lambda;T)=\lambda(q_{b}-q_{a})-T^{\alpha}(\lambda
v_{\alpha}+w_{\alpha}q_{a})-\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta},$

which is also a generating function for the evolution relation. We may change
this expression a bit by observing that the derivative with respect to
$\lambda$ (i.e., $q_{b}-q_{a}-T^{\alpha}v_{\alpha}$) will have to be set to
zero to define the evolution relation. We can insert this relation into
$\hat{S}_{\text{gen}}$ (without having to solve with respect to $\lambda$).
Actually, multiplying the relation with $T^{\beta}w_{\beta}$ and using (23),
we get

$T^{\alpha}T^{\beta}A_{\alpha\beta}=T^{\beta}w_{\beta}q_{b}-T^{\beta}w_{\beta}q_{a}.$

We can use this to get rid of the quadratic term in the $T$s getting the
generalized generating function

(46)
$\begin{split}\tilde{S}_{\text{gen}}(q_{a},q_{b},\lambda,T)&=\lambda(q_{b}-q_{a})-T^{\alpha}\lambda
v_{\alpha}-\frac{1}{2}T^{\alpha}w_{\alpha}q_{a}-\frac{1}{2}T^{\alpha}w_{\alpha}q_{b}\\\
&=\left(\lambda-\frac{1}{2}T^{\alpha}w_{\alpha}\right)q_{b}-\left(\lambda+\frac{1}{2}T^{\alpha}w_{\alpha}\right)q_{a}-T^{\alpha}\lambda
v_{\alpha}.\end{split}$

##  7\. Infinite-dimensional targets

For simplicity, up to now we have only discussed examples where the target
symplectic manifold is finite-dimensional. The whole discussion can be
generalized, almost verbatim, to the case when the target is infinite-
dimensional and there are possibly infinitely many constraints. We will touch
upon more analytical issues in Appendix A, and in particular in A.3. Note that
the main problem is that, in principle, the evolution relation might fail to
be lagrangian.313131This is due to intrinsically analytical problems. See [16,
Sect. 5.9] for an example. In in this section we will only be considering
infinite-dimensional examples where this problem does not actually arise
(mainly thanks to the Hodge decomposition of differential forms).

###  7.1. Three-dimensional abelian Chern–Simons theory

As a warm up, we discuss here the case of abelian Chern–Simons theory in three
dimensions. The action for a cylinder $I\times\Sigma$, where $I=[t_{a},t_{b}]$
is an interval and $\Sigma$ is a closed, oriented surface, is

$S[A]=\frac{1}{2}\int_{I\times\Sigma}{A}\mathrm{d}{A}$

with $A\in\Omega^{1}(I\times\Sigma)$. A variation of the action yields

$\delta
S=\int_{I\times\Sigma}\delta{A}\mathrm{d}{A}-\frac{1}{2}\int_{\\{t_{b}\\}\times\Sigma}{A}\delta{A}+\frac{1}{2}\int_{\\{t_{a}\\}\times\Sigma}{A}\delta{A},$

from which we read off the EL equations $\mathrm{d}{A}=0$ and the boundary
symplectic structure:

$\mathcal{F}^{\partial}=\Omega^{1}(\Sigma),\quad\omega=-\frac{1}{2}\int_{\Sigma}\delta
A\delta A.$

We may split $A$ into its “horizontal” and “vertical” part

$A=A_{\Sigma}+A_{I}$

and regard $A_{\Sigma}$ as a map
$I\to\Omega^{1}(\Sigma)=\mathcal{F}^{\partial}$, with target symplectic form
$-\frac{1}{2}\int_{\Sigma}\delta A_{\Sigma}\delta A_{\Sigma}$, and $A_{I}$ as
a one-form on $I$ with values in $\Omega^{0}(\Sigma)$. Splitting also
$\mathrm{d}$ as $\mathrm{d}_{\Sigma}+\mathrm{d}_{I}$, we then have

$S^{\prime}[A_{\Sigma},A_{I}]\mathrel{\mathop{\mathchar
58\relax}}=S[A_{\Sigma}+A_{I}]=\int_{I\times\Sigma}\left(\frac{1}{2}A_{\Sigma}\mathrm{d}_{I}A_{\Sigma}+A_{I}\mathrm{d}_{\Sigma}A_{\Sigma}\right),$

from which we see that $A_{\Sigma}$ represents the dynamical variables,
whereas $A_{I}$ represents the Lagrange multipliers. Moreover, we now split
the EL equations into the evolution equations

$\mathrm{d}_{I}A_{\Sigma}=-\mathrm{d}_{\Sigma}A_{I}$

and the constraints

$\mathrm{d}_{\Sigma}A_{\Sigma}=0.$

This is a system with linear constraints (in strict involution, since
$\mathrm{d}^{2}=0$) that, almost, fits into Example 4.3. What is missing is a
splitting of the symplectic space of fields into $p$ and $q$ variables (or,
more precisely, a splitting in the sense of Remark A.6). We do it in the
complexified version
$\mathcal{F}^{\partial}_{\mathbb{C}}=\Omega^{1}(\Sigma)\otimes\mathbb{C}$,
which splits as $\Omega^{1,0}(\Sigma)\oplus\Omega^{0,1}(\Sigma)$ in terms of a
complex structure on $\Sigma$.323232 This actually corresponds to choosing a
complex polarization for the real Chern–Simons theory.  We write accordingly
$A_{\Sigma}=A^{1,0}+A^{0,1}$ and
$\mathrm{d}_{\Sigma}=\partial+\bar{\partial}$, so

$S^{\prime}[A_{\Sigma},A_{I}]=\int_{I\times\Sigma}\left(\frac{1}{2}A^{1,0}\mathrm{d}_{I}A^{0,1}+\frac{1}{2}A^{0,1}\mathrm{d}_{I}A^{1,0}+A_{I}(\bar{\partial}A^{1,0}+\partial
A^{0,1})\right).$

We can integrate by parts the second term, so that the action reduces, up to a
boundary term, to

$S^{\prime\prime}(A^{1,0},A^{0,1},A_{I})=\int_{I\times\Sigma}\left(A^{1,0}\mathrm{d}_{I}A^{0,1}+A_{I}(\bar{\partial}A^{1,0}+\partial
A^{0,1})\right),$

where we recognize $A^{1,0}$ as the $p$ variables and $A^{0,1}$ as the $q$
variables. The variation now, correctly, produces the boundary one-form
$\int_{\Sigma}A^{1,0}\delta A^{0,1}$.

We can finally apply the results of Example 4.3 to this case where, up to
signs, the matrix $v^{i}_{\alpha}$ is now replaced by the operator
$\bar{\partial}$ and the matrix $w_{i\beta}$ is replaced by the operator
$\partial$. We write $\sigma\mathrel{\mathop{\mathchar
58\relax}}=\int_{I}A_{I}\in\Omega^{0}(\Sigma)$ instead of $T$,
$\mathsf{A}^{0,1}_{a}$ instead of $q_{a}$ and $\mathsf{A}^{1,0}_{b}$ instead
of $p^{b}$, so equation (24) becomes

(47)
$\hat{S}^{f}_{\text{HJ}}=\int_{\Sigma}\left(\mathsf{A}^{1,0}_{b}\mathsf{A}^{0,1}_{a}+\sigma(\bar{\partial}\mathsf{A}^{1,0}_{b}+\partial\mathsf{A}^{0,1}_{a})+\frac{1}{2}\sigma\partial\bar{\partial}\sigma\right).$

Following Example 6.3 we may also get the generating function for the same
choice of both endpoint conditions. Namely, extending (46) to this case, we
get

(48)
$\tilde{S}_{\text{gen}}=\int_{\Sigma}\left(\mathsf{A}^{1,0}_{b}\left(\lambda+\frac{1}{2}\bar{\partial}\sigma\right)-\mathsf{A}^{1,0}_{a}\left(\lambda-\frac{1}{2}\bar{\partial}\sigma\right)+\lambda\partial\sigma\right)$

with $\lambda\in\Omega^{0,1}(\Sigma)$.

###  7.2. Nonabelian Chern–Simons theory

In this case the action is

$S[A]=\int_{I\times\Sigma}\left(\frac{1}{2}\langle{A},\mathrm{d}{A}\rangle+\frac{1}{6}\langle{A},[{A},{A}]\rangle\right),$

where now $A$ takes values in a quadratic Lie algebra $(\mathfrak{g},\langle\
,\ \rangle)$.

Proceeding as in the abelian case, we get

$S^{\prime\prime}(A^{1,0},A^{0,1},A_{I})=\int_{I\times\Sigma}\left(\langle
A^{1,0},\mathrm{d}_{I}A^{0,1}\rangle+\langle
A_{I},\bar{\partial}A^{1,0}+\partial A^{0,1}+[A^{0,1},A^{1,0}]\rangle\right),$

where we recognize the data of Example 4.8 and Remark 4.9 with $-A^{1,0}$ as
the $\bar{q}$, $A^{0,1}$ as the $q$, $A_{I}$ as the $e$ variables, $\partial$
as the derivation $\bar{v}$, and $\bar{\partial}$ as the derivation $v$.333333
To be more precise, the Lie algebra of Remark 4.9 is now
$\Omega^{0}(\Sigma,\mathfrak{g})$ with pointwise Lie bracket induced from that
of $\mathfrak{g}$, the trivial representation space $Z$ is
$\Omega^{0,1}(\Sigma)$, and their tensor product, over $\Omega^{0}(\Sigma)$,
is $V=\Omega^{0,1}(\Sigma,\mathfrak{g})$. The dual space $V^{*}$ is now
replaced by $\Omega^{1,0}(\Sigma,\mathfrak{g})$ with pairing to $V$ induced by
the paring $\langle\ ,\ \rangle$ on $\mathfrak{g}$ and integration on
$\Sigma$. We write $\mathsf{A}^{0,1}_{a}$ instead of $q_{a}$ and
$-\mathsf{A}^{1,0}_{b}$ instead of $\bar{q}^{b}$, so (35) becomes

$\hat{S}^{f}_{\text{HJ}}=\int_{\Sigma}\Big{(}\langle\mathsf{A}^{1,0}_{b},g^{-1}\,\mathsf{A}^{0,1}_{a}g\rangle+\langle\mathsf{A}^{1,0}_{b},g^{-1}\bar{\partial}g\rangle+\langle\mathsf{A}^{0,1}_{a},\partial
g\cdot g^{-1}\rangle\Big{)}+\mathrm{WZW}(g)$

with $g=P\mathrm{e}^{\int_{I}A_{I}}\in\Omega^{0}(\Sigma,G)$. For