Quantum mechanics of 4-derivative theories

Alberto Salvioa and Alessandro Strumiab

a Departamento de Física Teórica, Universidad Autónoma de Madrid
and Instituto de Física Teórica IFT-UAM/CSIC, Madrid, Spain
b Dipartimento di Fisica dell’Università di Pisa and INFN, Italia
CERN, Theory Division, Geneva, Switzerland

Abstract

> A renormalizable theory of gravity is obtained if the dimension-less
> 4-derivative kinetic term of the graviton, which classically suffers from
> negative unbounded energy, admits a sensible quantization. We find that a
> 4-derivative degree of freedom involves a canonical coordinate with unusual
> time-inversion parity, and that a correspondingly unusual representation
> must be employed for the relative quantum operator. The resulting theory has
> positive energy eigenvalues, normalizable wave functions, unitary evolution
> in a negative-norm configuration space. We present a formalism for quantum
> mechanics with a generic norm.

> ###### Contents

  1. 1 Introduction
  2. 2 The Ostrogradski classical canonical formalism
  3. 3 Quantum mechanics with negative norm
  4. 4 4 derivatives want Dirac-Pauli
  5. 5 Path-integral quantization
  6. 6 Interactions, Quantum Field Theory, probability
  7. 7 Conclusions (so far)

##  1 Introduction

Newton invented classical mechanics putting two time derivatives in his
equation $F=m\ddot{x}$, which corresponds to a kinetic energy with 2 time
derivatives, $m\dot{x}^{2}/2$. Later Ostrogradski proofed a no-go theorem:
non-degenerate classical systems with more than two time derivatives contain
arbitrarily negative energies and develop fatal run-away instabilities [1].
Classically, they do not make sense.

The discovery that nature is relativistic and quantum opened the quest for an
extension of Newtonian gravity. A century ago Einstein and Hilbert found the
classical theory of relativistic gravity. However, its quantum version is not
renormalizable in 3+1 space-time dimensions. Sticking to the observed number
of space-time dimensions, a renormalizable extension of general relativity is
found by adding terms quadratic in the curvature tensor to the Einstein-
Hilbert Lagrangian, such that the graviton acquires a 4-derivative kinetic
term. Stelle proposed and dismissed this extension [2] (see also [3, 4, 5, 6,
7, 8, 9]).

Recently the Higgs mass hierarchy problem brought interest to dimension-less
theories. In this context, gravitons must have dimension 0 (being a dimension-
less metric) and thereby must have a 4-derivative kinetic term. If these
theories could make sense at quantum level, despite the negative classical
energy, a great deal would be gained: relativistic quantum gravity, plus
hierarchies among dynamically generated mass scales [8], plus inflation [10,
11].

Quantization can eliminate arbitrarily negative classical energies. The
following example is well known: the classical relativistic spin 1/2 field is
described by a spinor $\Psi(x)$ with Dirac Lagrangian
$\mathscr{L}=\bar{\Psi}(i\not{\partial}-m)\Psi$ containing one time
derivative. Treating $\Psi$ as a classical field (as initially proposed by
Schrödinger), and inserting the plane-wave expansion

$\Psi(x)=\int\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E_{p}}}[a_{p,s}u_{p,s}e^{-ip\cdot
x}+b^{\dagger}_{p,s}v_{p,s}e^{ip\cdot x}]$ (1)

in the Hamiltonian one finds negative energies in half of the configurations
space:111Unlike in the case of the Hydrogen atom emphasized by Woodard [12],
where the instability eliminated by quantum mechanics occurs only in one point
of the configuration space.

$H=\int\frac{d^{3}p}{(2\pi)^{3}}\,E_{p}[a_{p,s}^{\dagger}a_{p,s}-b_{p,s}b_{p,s}^{\dagger}],\qquad
E_{p}=\sqrt{m^{2}+\vec{p}^{2}}$ (2)

This classical arbitrarily negative energy is avoided by quantization with
anti-commutators, if the vacuum state is appropriately chosen. Indeed, the
two-state solution to $\\{b,b^{\dagger}\\}=1$ shows that one can switch
annihilation with creation operators by choosing the vacuum to be the state
with lower energy.

The spin 0 and spin 1 relativistic fields (described by dimension-1 fields
with 2 derivatives) do not have this issue: the negative-frequency solutions
to the Klein-Gordon equation correspond to Hamiltonians with positive energy.

The general lesson is that quantization depends on the number of time
derivatives.

The goal of this study is describing if/how systems with 4 derivatives can be
quantized obtaining a consistent theory, in particular of quantum gravity. We
will find that a unique structure emerges, that again involves switching
annihilation and creation operators.

This will bring us into the territory of negative norm quanta, avoided like a
plague by serious theorists that call them ‘ghosts’, and explored only by
notorious crackpots such as Dirac [13], Pauli [14], Heisenberg [15], Pais,
Uhlenbeck [16], Lee, Wick [17], Cutkosky [18], Coleman [19], Feynman [20],
Gross [21], Hawking [22], ’t Hooft [23] and others, also more recently [24,
25, 26, 27, 28, 29, 30, 31, 32]. These works sometimes contain bizarre and
confusing statements and obsolete motivations, together with interesting ideas
and ad-hoc prescriptions.

This paper is structured as follows. In section 2 we review the canonical
Ostrogradski formalism. In section 3 we present negative norm quantum
mechanics, the negative norm harmonic oscillator (section 3.3), and the
associated negative-norm representation of a canonical coordinate (section
3.4), with unusual parity under time-inversion $T$. In section 4 we recall
that a 4-derivative degree of freedom $q(t)$ is described by two canonical
coordinates: $q_{1}=q$ and $q_{2}=\dot{q}$. While $q_{1}$ is $T$-even as
usual, $q_{2}$ is $T$-odd: we argue that thereby it naturally follows the
negative-norm representation. The resulting quantum theory is unitary: time
evolution preserves the negative norm. The path-integral formulation is
discussed in section 5. In section 6 we discuss the interacting theory,
outline the extension to quantum field theory, and discuss the issue of giving
a sensible interpretation to negative norms, via a postulate that generalizes
the Born rule. Conclusions are given in section 7.

##  2 The Ostrogradski classical canonical formalism

Let us now introduce the main issues in the simplest relevant case. Our final
goal will be 4-derivative gravity; however the graviton components can be
Fourier expanded into modes with given momentum and 4 time derivatives, and at
leading order in the perturbative expansion one has decoupled harmonic
oscillators. So, we start considering a single mode $q(t)$, described by the
Lagrangian

$\mathscr{L}=-\frac{\ddot{q}^{2}}{2}+(\omega_{1}^{2}+\omega_{2}^{2})\frac{\dot{q}^{2}}{2}-\omega_{1}^{2}\omega_{2}^{2}\frac{q^{2}}{2}-V(q)=-\frac{1}{2}q(\frac{d^{2}}{dt^{2}}+\omega_{1}^{2})(\frac{d^{2}}{dt^{2}}+\omega_{2}^{2})q-V(q)+\mbox{
total derivatives}.$

where $V(q)$ is some interaction. We assume real $\omega_{1}$, $\omega_{2}$,
because we are interested in ghosts (negative kinetic and potential energy),
not in tachyonic instabilities (potential unstable with respect to the kinetic
term). The $-$ sign means that the ghost is the state with larger $\omega$; we
choose $\omega_{1}>\omega_{2}$ and don’t explicitly discuss here the
degenerate case $\omega_{1}=\omega_{2}$.

Ostrogradski introduced an auxiliary coordinate $q_{2}$ that allows to
describe the 4-derivative oscillator in canonical Hamiltonian form (see also
ref. [16] for a review of this method):

$\begin{array}[]{ll}q_{1}=q,&\displaystyle
p_{1}=\frac{\delta\mathscr{L}}{\delta\dot{q}_{1}}=(\omega_{1}^{2}+\omega_{2}^{2})\dot{q}+\dddot{q},\\\\[14.22636pt]
q_{2}=\lambda\dot{q},&\displaystyle
p_{2}=\frac{\delta\mathscr{L}}{\delta\dot{q}_{2}}=-\frac{\ddot{q}}{\lambda},\end{array}$
(3)

where for a generic variable $x$ we have introduced the variational derivative

$\frac{\delta\mathscr{L}}{\delta x}=\frac{\partial\mathscr{L}}{\partial
x}-\frac{d}{dt}\frac{\partial\mathscr{L}}{\partial\dot{x}}+\frac{d^{2}}{dt^{2}}\frac{\partial\mathscr{L}}{\partial\ddot{x}}+\cdots.$
(4)

While Ostrogradski assumed $\lambda=1$, we introduced an arbitrary constant
$\lambda$. The system in eq. (3) can be solved for $q$ and its time
derivatives,

$q=q_{1},\qquad\dot{q}=\frac{q_{2}}{\lambda},\qquad\ddot{q}=-\lambda
p_{2},\qquad\dddot{q}=p_{1}-\left(\omega_{1}^{2}+\omega_{2}^{2}\right)\frac{q_{2}}{\lambda},$
(5)

and the Hamiltonian turns out to be

$H=\sum_{i=1}^{2}p_{i}\dot{q}_{i}-\mathscr{L}=\frac{p_{1}q_{2}}{\lambda}-\frac{\lambda^{2}}{2}p_{2}^{2}-\frac{\omega_{1}^{2}+\omega_{2}^{2}}{2\lambda^{2}}q_{2}^{2}+\frac{\omega_{1}^{2}\omega_{2}^{2}}{2}q_{1}^{2}+V(q_{1}).$
(6)

In view of its first term, the classical Hamiltonian $H$ has no minimal energy
configuration: this is the essence of the Ostrogradski no-go classical
theorem. Using the Poisson parentheses $\\{~{},~{}\\}$ one computes the
Hamiltonian equations of motion:

$\left\\{\begin{array}[]{ll}\displaystyle\dot{q}_{1}=\\{q_{1},H\\}=\frac{\partial
H}{\partial
p_{1}}=\frac{q_{2}}{\lambda},&\displaystyle\dot{p}_{1}=\\{p_{1},H\\}=-\frac{\partial
H}{\partial
q_{1}}=-\omega_{1}^{2}\omega_{2}^{2}q_{1}-V^{\prime}(q_{1}),\\\\[14.22636pt]
\displaystyle\dot{q}_{2}=\\{q_{2},H\\}=\frac{\partial H}{\partial
p_{2}}=-\lambda^{2}p_{2},&\displaystyle\dot{p}_{2}=\\{p_{2},H\\}=-\frac{\partial
H}{\partial
q_{2}}=-\frac{p_{1}}{\lambda}+\frac{\omega_{1}^{2}+\omega_{2}^{2}}{\lambda^{2}}q_{2}.\end{array}\right.$
(7)

For any $\lambda$ they imply the classical Lagrangian equation of motion.
Setting $V=0$, it is

$(\frac{d^{2}}{dt^{2}}+\omega_{1}^{2})(\frac{d^{2}}{dt^{2}}+\omega_{2}^{2})q=\frac{d^{4}q}{dt^{4}}+(\omega_{1}^{2}+\omega^{2}_{2})\frac{d^{2}q}{dt^{2}}+\omega_{1}^{2}\omega_{2}^{2}q=0.$
(8)

The corresponding classical solution, for given initial conditions
$q_{0},\dot{q}_{0},\ddot{q}_{0},\dddot{q}_{0}$ at $t=0$, is

$q(t)=-\frac{\omega_{2}^{2}q_{0}+\ddot{q}_{0}}{\omega_{1}^{2}-\omega_{2}^{2}}\cos(\omega_{1}t)+\frac{\omega_{1}^{2}q_{0}+\ddot{q}_{0}}{\omega_{1}^{2}-\omega_{2}^{2}}\cos(\omega_{2}t)-\frac{\omega_{2}^{2}\dot{q}_{0}+\dddot{q}_{0}}{\omega_{1}(\omega_{1}^{2}-\omega_{2}^{2})}\sin(\omega_{1}t)+\frac{\omega_{1}^{2}\dot{q}_{0}+\dddot{q}_{0}}{\omega_{2}(\omega_{1}^{2}-\omega_{2}^{2})}\sin(\omega_{2}t).$
(9)

This is a well behaved oscillator without run-away issues because the
positive-energy and negative-energy components are decoupled. Run-away
solutions appear when they interact through a generic interaction, such as a
$V\neq 0$.

###  2.1 Quantizing the Ostrogradski Hamiltonian

The classical equation differs from the usual 2-derivative equation
$d(q+ip)/dt=i\omega(q+ip)$, so that, trying to quantize the theory, we do not
define the usual annihilation operator $a_{i}\propto q_{i}+ip_{i}$. Rather, it
is convenient to define the operators $a_{i}$ as the coefficients of given
frequency:

$q(t)=a_{1}e^{-i\omega_{1}t}+a_{2}e^{-i\omega_{2}t}+\hbox{h.c.}$ (10)

The $a_{1},a_{2}$ can be expressed in terms of canonical Hamiltonian
coordinates:

$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda\omega_{1}\omega_{2}^{2}q_{1}-i\omega_{1}^{2}q_{2}+ip_{1}\lambda-\omega_{1}p_{2}\lambda^{2}}{2\lambda\omega_{1}(\omega_{2}^{2}-\omega_{1}^{2})},$
(11) $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{\lambda\omega_{1}^{2}\omega_{2}q_{1}-i\omega_{2}^{2}q_{2}+ip_{1}\lambda-\omega_{2}p_{2}\lambda^{2}}{2\lambda\omega_{2}(\omega_{1}^{2}-\omega_{2}^{2})}.$
(12)

Using the canonical quantization $[q_{i},p_{j}]=i\delta_{ij}$ one finds the
commutation relations for the $a_{i}$:

$[\tilde{a}_{1},\tilde{a}^{\dagger}_{1}]=-1,\qquad[\tilde{a}_{2},\tilde{a}^{\dagger}_{2}]=1,\qquad\hbox{all
other commutators vanish.}$ (13)

having normalised
$\tilde{a}_{1}=\sqrt{2\omega_{1}(\omega_{1}^{2}-\omega_{2}^{2})}a_{1}$ and
$\tilde{a}_{2}=\sqrt{2\omega_{2}(\omega_{1}^{2}-\omega_{2}^{2})}a_{2}$. The
Hamiltonian is

$H=-\omega_{1}\frac{\tilde{a}_{1}\tilde{a}_{1}^{\dagger}+\tilde{a}_{1}^{\dagger}\tilde{a}_{1}}{2}+\omega_{2}\frac{\tilde{a}_{2}\tilde{a}_{2}^{\dagger}+\tilde{a}_{2}^{\dagger}\tilde{a}_{2}}{2}.$
(14)

The state 1 with higher frequency $\omega_{1}>\omega_{2}$ is a ghost.

As better discussed later in section 3.3, this system can be quantized in two
different ways:

  * 1.

Positive norm, negative energy. One redefines
$\tilde{a}^{\prime}_{1}=\tilde{a}_{1}^{\dagger}$, such that it has the usual
commutation $[\tilde{a}^{\prime}_{1},\tilde{a}^{\prime\dagger}_{1}]=1$. The
vacuum state $|\tilde{0}\rangle$ is defined as usual by
$\tilde{a}^{\prime}_{1}|\tilde{0}\rangle=0$ and
$\tilde{a}_{2}|\tilde{0}\rangle=0$. By solving this condition as a
differential equation for $\psi_{\tilde{0}}(q_{1},q_{2})=\langle
q_{1},q_{2}|\tilde{0}\rangle$ with $p_{i}=-i\partial/\partial q_{i}$ one
obtains the ground state wave function:

$\psi_{\tilde{0}}(q_{1},q_{2})=\exp\bigg{(}-\frac{q_{1}^{2}\omega_{1}\omega_{2}+q_{2}^{2}/\lambda^{2}}{2}(\omega_{1}-\omega_{2})+iq_{1}\frac{q_{2}}{\lambda}\omega_{1}\omega_{2}\bigg{)}.$
(15)

  * 2.

Negative norm, positive energy. The vacuum is now defined as
$a_{1}|0\rangle=0$ and $a_{2}|0\rangle=0$. Using $p_{i}=-i\partial/\partial
q_{i}$ one obtains the ground-state wave function

$\psi_{0}(q_{1},q_{2})\propto\exp\bigg{(}\frac{-q_{1}^{2}\omega_{1}\omega_{2}+q_{2}^{2}/\lambda^{2}}{2}(\omega_{1}+\omega_{2})-iq_{1}\frac{q_{2}}{\lambda}\omega_{1}\omega_{2}\bigg{)}.$
(16)

If $\lambda=1$ the situation is bad, as emphasized by [12]: the positive-norm
quantization gives a normalizable wave-function $\psi_{\tilde{0}}$ but
negative energies; the negative-norm quantization gives a ground state wave
function not normalizable in $q_{2}=\dot{q}$. Excited states have the same
problem.

However, as we will show in section 4, consistency demands the negative-norm
Dirac-Pauli representation of a canonical coordinate which roughly amounts to
choosing an imaginary $\lambda$, e.g. $\lambda=-i$. One then obtains positive
energy, negative norm, and a wave function $\psi_{0}(q_{1},q_{2})$
normalizable in $q_{1}$ and $q_{2}=-i\dot{q}$. As we will now discuss, despite
the strange $i$ factor, $\dot{q}=iq_{2}$ as well as the Ostrogradski
Hamiltonian $H=iq_{2}p_{1}+\cdots=\dot{q}p_{1}+\cdots$ are self-adjoint, so
that time evolution is unitary.

##  3 Quantum mechanics with negative norm

We here discuss quantum mechanics with negative norm from a general point of
view. Negative norm states require putting some minus sign here and there. It
is convenient to be more general and consider a Hilbert-like space with
generic, possibly negative, constant norm (called Krein space by
mathematicians) and develop a basis-independent formalism. This will let us to
clarify confusions, in particular about self-adjoint operators that are
represented (in some basis) by non-hermitian matrices, allowing us to
understand the unusual imaginary $\lambda$ introduced in the previous section.

We follow the notations used in general relativity, rewriting the quantum
state metric as $\langle_{n}|_{m}\rangle=\eta_{nm}$ and defining the inverse
metric $(\eta)^{nm}\equiv(\eta^{-1})_{nm}$, the contro-variant ket
$|^{n}\rangle=\eta^{nm}|_{m}\rangle$ such that
$\langle^{n}|^{m}\rangle=\eta^{nm}$ and
$\langle^{n}|_{m}\rangle=\delta^{n}_{m}=\langle_{n}|^{m}\rangle$. Summations
over repeated indexes are implicit. As usual, bras denote complex conjugate of
kets.

A generic state $|\psi\rangle$ can be expanded in either the ‘covariant’ or
the ‘controvariant’ basis:

$\psi_{n}\equiv\langle_{n}|\psi\rangle,\qquad\psi^{n}\equiv\langle^{n}|\psi\rangle.$
(17)

Then

$|\psi\rangle=\psi^{n}|_{n}\rangle=\psi_{n}|^{n}\rangle.$ (18)

A generic linear operator $A$ can be written as a matrix in 4 different ways:

$A_{nm}\equiv\langle_{n}|A|_{m}\rangle,\qquad
A^{nm}\equiv\langle^{n}|A|^{m}\rangle,\qquad
A_{n}{}^{m}\equiv\langle_{n}|A|^{m}\rangle,\qquad
A^{n}{}_{m}\equiv\langle^{n}|A|_{m}\rangle.$ (19)

Then

$A={A}^{nm}|_{n}\rangle\langle_{m}|={A}_{nm}|^{n}\rangle\langle^{m}|={A}_{n}{}^{m}|^{n}\rangle\langle_{m}|={A}^{n}{}_{m}|_{n}\rangle\langle^{m}|.$
(20)

The components of the matrices are related by
$A_{n}{}^{m}=\eta_{nn^{\prime}}A^{n^{\prime}}{}_{m^{\prime}}\eta^{m^{\prime}m}$
which is an iso-spectral transformation: the eigenvalues do not change because
the matrix $A$ gets left-multiplied by $\eta$ and right-multiplied by its
inverse.

The unity operator is represented by $1_{nm}=\eta_{nm}$ and
${1}^{nm}={\eta}^{nm}$ and expanded as

$1=\eta^{nm}|_{n}\rangle\langle_{m}|=\eta_{nm}|^{n}\rangle\langle^{m}|=|_{n}\rangle\langle^{n}|=|^{n}\rangle\langle_{n}|.$
(21)

Operator multiplication becomes, in components,
$(AB)_{nm}=A_{nn^{\prime}}\eta^{n^{\prime}m^{\prime}}B_{m^{\prime}m}$.
Expectation values are given by
$\langle\psi|A|\psi\rangle/\langle\psi|\psi\rangle$.

The adjoint $A^{\dagger}$ of an operator $A$ is defined, as usual, as the
operator such that $|\psi^{\prime}\rangle=A|\psi\rangle$ implies
$\langle\psi^{\prime}|=\langle\psi|A^{\dagger}$. Thereby for generic matrix
elements one has
$\langle\psi_{2}|A^{\dagger}|\psi_{1}\rangle\equiv\langle\psi_{1}|A|\psi_{2}\rangle^{*}$,
and the relation for the components

$(A^{\dagger})_{nm}=A_{mn}^{*}\qquad\hbox{i.e.}\qquad(A^{\dagger})^{nm}=A^{mn*}\qquad\hbox{i.e.}\qquad(A^{\dagger})_{n}{}^{m}=(A^{m}{}_{n})^{*}.$
(22)

The covariant components of a self-adjoint operator $A$ satisfy the usual
condition: a self-adjoint operator is described by a hermitian matrix,
$A_{nm}$. The same result holds for the contro-variant matrix $A^{nm}$. The
mixed components satisfy a different condition, where complex conjugation is
supplemented by an iso-spectral transformation: $A_{n}{}^{m}=(\eta
A^{*T}\eta^{-1})_{n}{}^{m}$.222Here $T$ denotes the matrix transpose; $*$
denotes complex conjugation; $\dagger$ denotes the adjoint operation, that
generalizes the usual hermitian conjugation and reduces to it in the positive
norm case. We never use $\dagger$ to denote hermitian conjugation of a matrix.

A self-adjoint operator, $A^{\dagger}=A$ has real expectation values
$\langle\psi|A|\psi\rangle/\langle\psi|\psi\rangle$, although the matrix
$A^{m}{}_{n}$ that represents it can be anti-hermitian.

The mixed components directly enter into the eigenvector equation
$A|\psi\rangle=A_{\psi}|\psi\rangle$:

$A_{n}{}^{m}\psi_{m}=A_{nm^{\prime}}\eta^{m^{\prime}m}\psi_{m}=A_{\psi}\psi_{n}\qquad\hbox{or}\qquad
A^{n}{}_{m}\psi^{m}=\eta^{nn^{\prime}}A_{n^{\prime}m}\psi^{m}=A_{\psi}\psi^{n}$
(23)

where $A_{\psi}$ is the eigenvalue. Let us now consider a self-adjoint
operator $H$ (later it will be the Hamiltonian), with eigenstates
$|E_{n}\rangle$ and eigenvalues $E_{n}$. The identity

$\langle E_{n}|H|E_{m}\rangle=\langle E_{n}|E_{m}\rangle
E_{m}=E^{*}_{n}\langle E_{n}|E_{m}\rangle$ (24)

tells that $H$ can have three different kinds of eigenstates:

  * $+$)

orthogonal eigenstates $\langle E_{n}|E_{m}\rangle=0$ with real $E_{n}$ and
norm $\langle E_{n}|E_{n}\rangle=+1$;

  * $-$)

orthogonal eigenstates $\langle E_{n}|E_{m}\rangle=0$ with real $E_{n}$ and
norm $\langle E_{n}|E_{n}\rangle=-1$;

  * 0)

pairs of complex conjugated eigenvalues, $E_{n}=E_{m}^{*}$ with $\langle
E_{n}|E_{m}\rangle\neq 0$ and zero norm, $\langle E_{n}|E_{n}\rangle=0$.

In the classical analogue, the latter possibility corresponds to a ghost which
is also a tachyon, which is a different kind of instability, to be avoided
even in absence of ghosts.

It is often convenient to choose a basis of eigenstates of $H$:
$|_{n}\rangle=|E_{n}\rangle$. The associated contro-variant states
$|^{n}\rangle$ then satisfy $H|^{n}\rangle=E_{n}^{*}|^{n}\rangle$. In this
basis the space splits into two sectors: positive norm and negative norm, plus
the possible pairs of zero-norm states. The two sectors experience a joint
dynamics only if the initial state has a quantum entanglement among them.

###  3.1 Unitary time evolution

The evolution equation $i\partial_{t}|\psi\rangle=H|\psi\rangle$ becomes

$i\frac{\partial}{\partial
t}\psi_{n}=H_{nm}\eta^{mm^{\prime}}\psi_{m^{\prime}}\qquad\hbox{or}\qquad
i\frac{\partial}{\partial
t}\psi^{n}=\eta^{nn^{\prime}}H_{n^{\prime}m}\psi^{m}.$ (25)

The norm of any state $|\psi(t)\rangle$ is conserved by time evolution if $H$
is self-adjoint:

$i\frac{\partial}{\partial
t}\langle\psi^{\prime}(t)|\psi(t)\rangle=\langle\psi^{\prime}|H-H^{\dagger}|\psi\rangle=0.$
(26)

A self-adjoint Hamiltonian $H$ leads to unitary time evolution. The explicit
solution can be written as $|\psi(t)\rangle=U(t)|\psi(0)\rangle$ with
$U(t)={\rm T}e^{-i\int H(t)\,dt}$, where T is the usual time-ordering. In
components,

$\psi_{n}(t)=U_{n}{}^{m}\psi_{m}(0)=U_{nm^{\prime}}\eta^{m^{\prime}m}\psi_{m^{\prime}}(0)\qquad\hbox{or}\qquad\psi^{n}(t)=U^{n}{}_{m}\psi^{m}(0)=\eta^{nn^{\prime}}U_{n^{\prime}m}\psi^{m}(0).$
(27)

Having written generic-metric quantum mechanics in an abstract formalism that
resembles as much as possible the usual positive-norm formalism, let us now
emphasize the key differences. For simplicity, let us consider a time-
independent $H$. One can then expand
$U=e^{-iHt}=\sum_{n=0}^{\infty}(-iHt)^{n}/n!$.

  * •

Writing $U$ in mixed components, $U_{n}{}^{m}$ is the naive exponentiation of
the matrix $H_{n}{}^{m}$. However, the mixed components of a self-adjoint $H$
do not form a hermitian matrix. Rather, the self-adjoint condition in eq. (22)
dictates that they are hermitian up to an iso-spectral transformation.

  * •

The covariant components of a self-adjoint $H$ satisfy the usual Hermiticity
$H^{*}_{nm}=H_{mn}$. However, the covariant components $U_{nm}$ are not given
by the naive matrix exponentiation of $H_{nm}$. Rather, extra metric factors
appear to covariantize the expansion:

$U_{nm}=\eta_{nm}+\eta_{nn^{\prime}}(-iHt)^{n^{\prime}m^{\prime}}\eta_{m^{\prime}m}+\frac{1}{2}\eta_{nn^{\prime}}(-iHt)^{n^{\prime}r^{\prime}}\eta_{r^{\prime}s^{\prime}}(-iHt)^{s^{\prime}m^{\prime}}\eta_{m^{\prime}m}+\cdots$
(28)

Correspondingly, the unitarity condition $U^{\dagger}U=1$ written in covariant
components is
$U_{n^{\prime}n}^{*}\eta^{n^{\prime}m^{\prime}}U_{m^{\prime}m}=\eta_{nm}$,
while in mixed components one gets the usual
$U^{k*}_{~{}\,n}\,U_{k}{}^{m}=\delta_{m}^{n}$.

Practical computations often employ perturbation theory, which can now be
easily generalized to generic norm. Decomposing $H=H_{0}+V(t)$, the
$I$nteraction picture is related to the Schroedinger picture as
$A_{I}=e^{iH_{0}t}Ae^{-iH_{0}t}$ where $A$ is any operator (including $V$).
Time evolution is given by

$U_{I}(t_{i},t_{f})={\rm
T}\,e^{-i\int_{t_{i}}^{t_{f}}dt\,V_{I}(t)}=1-i\int_{t_{i}}^{t_{f}}dt^{\prime}\,V_{I}(t^{\prime})-\int_{t_{i}}^{t_{f}}dt^{\prime}\int_{t_{i}}^{t^{\prime}}dt^{\prime\prime}\,V_{I}(t^{\prime})V_{I}(t^{\prime\prime})+\cdots.$
(29)

The above explicit form of $U_{I}$ shows that the energy conserved by quantum
evolution (up to the usual quantum uncertainty $\Delta t\,\Delta E\geq\hbar$)
are the eigenvalues of $H$. Let us consider for example a time-independent
interaction $V$ and an initial state and a final state which are energy
eigenstates with eigenvalues $E_{i}$ and $E_{f}$. Defining
$V^{f}{}_{i}=\langle^{f}|V|_{i}\rangle$, at first order one has

$|\langle^{f}|U|_{i}\rangle|^{2}\simeq\bigg{|}\int_{0}^{t}dt^{\prime}~{}e^{i(E_{f}-E_{i})t^{\prime}}V^{f}{}_{i}\bigg{|}^{2}=\frac{4|V^{f}{}_{i}|^{2}}{|E_{f}-E_{i}|^{2}}\sin^{2}\frac{(E_{f}-E_{i})t}{2}\stackrel{{\scriptstyle
t\to\infty}}{{\simeq}}2\pi t|V^{f}{}_{i}|^{2}\delta(E_{f}-E_{i}).$ (30)

This means that energy conservation reads $E_{f}=E_{i}$, up to the usual
quantum uncertainty $1/(t_{f}-t_{i})$. Higher order corrections give the usual
sum over intermediate quasi-on-shell states.

###  3.2 Example: the indefinite-norm two-state system

Let us consider a two-state system: $|_{+}\rangle$ with positive unit norm,
and $|_{-}\rangle$ with negative unit norm. Without loss of generality, by
redefining the relative phase of the two states and adding a constant overall
energy, one can trivially write the most generic self-adjoint Hamiltonian as

$H=\frac{1}{2}\bordermatrix{&|_{+}\rangle&|_{-}\rangle\cr\langle_{+}|&E_{R}&-iE_{I}\cr\langle_{-}|&iE_{I}&E_{R}}=\frac{1}{2}\bordermatrix{&|^{+}\rangle&|^{-}\rangle\cr\langle_{+}|&E_{R}&iE_{I}\cr\langle_{-}|&iE_{I}&-E_{R}}$
(31)

having used $|^{\pm}\rangle=\pm|_{\pm}\rangle$. We see that the $H_{nm}$
components are hermitian, unlike the $H_{n}{}^{m}$ components. The eigenvalues
of $H$ are $E_{\pm}=\pm E$ with $E=\sqrt{E_{R}^{2}-E_{I}^{2}}/2$. The
corresponding eigenstates are

$|_{E_{+}}\rangle=\sqrt{\frac{\gamma+1}{2}}|_{+}\rangle-i\sqrt{\frac{\gamma-1}{2}}|_{-}\rangle,\qquad|_{E_{-}}\rangle=i\sqrt{\frac{\gamma-1}{2}}|_{+}\rangle+\sqrt{\frac{\gamma+1}{2}}|_{-}\rangle.$
(32)

where $\gamma=1/\sqrt{1-E_{I}^{2}/E_{R}^{2}}$ is a ‘boost factor’ that
substitutes the usual mixing angle.

  * •

If $E_{I}<E_{R}$ the eigenvalues of $H$ are real, the orthogonal eigenvectors
satisfy $\langle_{E_{\pm}}|_{E_{\pm}}\rangle=\pm 1$, and tend to get closer to
the ‘light-cone’ of zero-norm states as $E_{I}$ increases. The components of
$U=e^{-iHt}$ oscillate in time:

$U=\bordermatrix{&|^{+}\rangle&|^{-}\rangle\cr\langle_{+}|&\cos(Et)-i\gamma\sin(Et)&\sqrt{\gamma^{2}-1}\sin(Et)\cr\langle_{-}|&\sqrt{\gamma^{2}-1}\sin(Et)&\cos(Et)+i\gamma\sin(Et)}.$
(33)

The unusual feature is that $|\langle_{\pm}|U|_{\pm}\rangle|^{2}$ oscillates
between 1 and $\gamma^{2}\geq 1$.

  * •

In the critical case, $E_{R}=E_{I}$, such that $\gamma=\infty$, the two
eigenstates become degenerate with energy $E=0$. The two eigenvectors also
become degenerate, and parallel to the zero-norm state
$\propto|_{+}\rangle+i|_{-}\rangle$. The evolution operator is

$U=\bordermatrix{&|^{+}\rangle&|^{-}\rangle\cr\langle_{+}|&1-iE_{R}t/2&E_{R}t/2\cr\langle_{-}|&E_{R}t/2&1+iE_{R}t/2}.$
(34)

This exemplifies a more general result: zero-norm eigenstates with complex
eigenvalues appear when, increasing the interaction, a level crossing between
a positive-norm and a negative-norm eigenstate takes place; the Hamiltonian
becomes degenerate at the critical transition.

  * •

If the interaction $E_{I}$ is strong enough, $E_{I}>E_{R}$, one has a pair of
complex conjugated eigenvalues, with zero-norm eigenvectors that satisfy
$\langle_{E_{+}}|_{E_{-}}\rangle=1$ and describe tachyonic ghosts: their time-
evolution factor $e^{-iE_{\pm}t}$ also contains a real exponential, in analogy
to tachyonic states present in positive-norm theories. In the extreme limit
$E_{I}\gg E_{R}$ the eigenvalues of $H$ are $\pm iE_{I}/2$, and the time
evolution operator is:

$U=\bordermatrix{&|^{+}\rangle&|^{-}\rangle\cr\langle_{+}|&\cosh(E_{I}t/2)&\sinh(E_{I}t/2)\cr\langle_{-}|&\sinh(E_{I}t/2)&\cosh(E_{I}t/2)}.$
(35)

This runaway happens whenever $H$ has a pair of complex eigenvalues
$E_{+}=E_{-}^{*}$, as clear writing time evolution in terms of energy
eigenstates,
$|\psi(t)\rangle=\psi^{E_{+}}e^{-iE_{+}t}|_{E_{+}}\rangle+\psi^{E_{-}}e^{-iE_{-}t}|_{E_{-}}\rangle$.
Both the norm of $|\psi(t)\rangle$ and the real expectation value of $H$ are
preserved by time evolution:

$\frac{\langle\psi(t)|H|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle}=\frac{E_{+}\psi^{E_{+}}\psi^{E_{-}*}+\hbox{c.c.}}{\psi^{E_{+}}\psi^{E_{-}*}+\hbox{c.c.}}.$
(36)

###  3.3 The negative-norm harmonic oscillator

We here study the concrete system that lies at the basis of perturbative
Quantum Field Theory: the harmonic oscillator. As discussed by Lee and Wick
[17] it admits two inequivalent quantizations: positive norm, and indefinite
norm.

Let us first recall the standard oscillator, described (up to irrelevant
constants) by the Hamiltonian $H=\frac{1}{2}(p^{2}+q^{2})$ with $[q,p]=i$.
Defining

$a=\frac{q+ip}{\sqrt{2}},\qquad a^{\dagger}=\frac{q-ip}{\sqrt{2}}$ (37)

one has $[a,a^{\dagger}]=1$ and $H=(aa^{\dagger}+a^{\dagger}a)/2$.

Let us next consider a more general system described by the following
Hamiltonian and commutation relations:

$H=s_{H}\frac{a^{\dagger}a+aa^{\dagger}}{2},\qquad[a,a^{\dagger}]=s.$ (38)

For $s=s_{H}=1$ this reduces to the usual oscillator. We now show that
$s=s_{H}=-1$ defines another consistent positive-energy theory. The symbol
$a^{\dagger}$ here indicates the adjoint of $a$, which generalizes the
Hermitian conjugate to negative norm.

We again define the vacuum as $a|0\rangle=0$ and the excited states as
$|_{n}\rangle=a^{\dagger}|_{n-1}\rangle/\sqrt{n}=(a^{\dagger})^{n}|0\rangle/\sqrt{n!}$.
Its inverse is $a|_{n}\rangle=s\sqrt{n}|_{n-1}\rangle$. The state metric is
$\eta_{nm}\equiv\langle_{m}|_{n}\rangle=s^{n}\delta_{nm}$. The norm is
determined by the dynamics, and odd states have negative norm for $s=-1$. The
inverse metric is $\eta^{nm}=s^{-n}\delta_{nm}$ and the contro-variant states
are $|^{n}\rangle=s^{-n}|_{n}\rangle$. In components one has:

$a=\bordermatrix{&|0\rangle&|_{1}\rangle&|_{2}\rangle&|_{3}\rangle&\cdots\cr\langle
0|&0&s&0&0&\cdots\cr\langle_{1}|&0&0&\sqrt{2}s^{2}&0&\cdots\cr\langle_{2}|&0&0&0&\sqrt{3}s^{3}&\cdots\cr\langle_{3}|&0&0&0&0&\cdots\cr\vdots&\vdots&\vdots&\vdots&\vdots&\ddots}=\bordermatrix{&|0\rangle&|^{1}\rangle&|^{2}\rangle&|^{3}\rangle&\cdots\cr\langle
0|&0&1&0&0&\cdots\cr\langle_{1}|&0&0&\sqrt{2}&0&\cdots\cr\langle_{2}|&0&0&0&\sqrt{3}&\cdots\cr\langle_{3}|&0&0&0&0&\cdots\cr\vdots&\vdots&\vdots&\vdots&\vdots&\ddots}$
(39)
$a^{\dagger}=\bordermatrix{&|0\rangle&|_{1}\rangle&|_{2}\rangle&|_{3}\rangle&\cdots\cr\langle
0|&0&0&0&0&\cdots\cr\langle_{1}|&s&0&0&0&\cdots\cr\langle_{2}|&0&\sqrt{2}s^{2}&0&0&\cdots\cr\langle_{3}|&0&0&\sqrt{3}s^{3}&0&\cdots\cr\vdots&\vdots&\vdots&\vdots&\vdots&\ddots}=s\bordermatrix{&|0\rangle&|^{1}\rangle&|^{2}\rangle&|^{3}\rangle&\cdots\cr\langle
0|&0&0&0&0&\cdots\cr\langle_{1}|&1&0&0&0&\cdots\cr\langle_{2}|&0&\sqrt{2}&0&0&\cdots\cr\langle_{3}|&0&0&\sqrt{3}&0&\cdots\cr\vdots&\vdots&\vdots&\vdots&\vdots&\ddots}$
(40)

In components the commutation relations read

$[a,a^{\dagger}]_{nm}=(a\cdot\eta\cdot a^{\dagger}-a^{\dagger}\cdot\eta\cdot
a)_{nm}=s^{n+1}\delta_{nm}=s\eta_{nm}\qquad\hbox{i.e.}\qquad[a,a^{\dagger}]=s\sum_{n}|_{n}\rangle\langle^{n}|=s1$
(41)

and the Hamiltonian is:

$H_{nm}=(n+\frac{1}{2})s_{H}s^{n+1}\delta_{nm}=E_{n}\eta_{nm}\qquad\hbox{i.e.}\qquad
H=\sum_{n=0}^{\infty}E_{n}|_{n}\rangle\langle^{n}|$ (42)

where $E_{n}=(n+\frac{1}{2})ss_{H}$ are the Hamiltonian eigenvalues,
$H|_{n}\rangle=E_{n}|_{n}\rangle$. We see that positive-energy eigenvalues are
obtained for $s=s_{H}=1$ (the usual case with positive $H$ and positive norm),
but also for $s=s_{H}=-1$ (negative $H$ and negative norm).

Concerning the negative-norm case, $s=-1$, notice that the harmonic oscillator
does not predict tachyonic ghosts with zero norm. Furthermore the matrix
elements $a_{n}{}^{m}$ are not the hermitian conjugates of
$(a^{\dagger})_{n}{}^{m}$, such that the operators
$q=(a+a^{\dagger})/\sqrt{2}$ and $p=i(a^{\dagger}-a)/\sqrt{2}$ are represented
by matrices $q_{n}{}^{m}$ and $p_{n}{}^{m}$ which are not Hermitian. This is
why various authors who look at these matrices improperly speak of ‘anti-
Hermitian’ operators. Nevertheless, $q$ and $p$ are self-adjoint operators. We
will now find their coordinate representation.

###  3.4 The negative-norm coordinate representation

Starting from the harmonic oscillator, we now describe a more general
representation of a pair of canonical coordinate variables $q,p$. Parity flips
$q\to-q$ and $p\to-p$. In the harmonic oscillator case, this means $a\to-a$
and $a^{\dagger}\to-a^{\dagger}$: so eigenstates $|_{n}\rangle$ with even
(odd) $n$ are even (odd) under parity. In the negative norm quantization,
states with odd $n$ also have negative norm. Going to the coordinate wave-
function representation (we use the notation $x$ for the coordinate, that
later will become field space), this means that the norm is

$\langle\psi^{\prime}|\psi\rangle=\int dx\,[\psi^{\prime*}_{\rm
even}(x)\psi_{\rm even}(x)-\psi^{\prime*}_{\rm odd}(x)\psi_{\rm odd}(x)]=\int
dx\,\psi^{\prime*}(x)\psi(-x).$ (43)

The corresponding unit operator is $1=\int dx|-x\rangle\langle x|$. Switching
to the formalism appropriate for generic norm, one has the norm
$\langle_{x^{\prime}}|_{x}\rangle=\delta(x+x^{\prime})$. Thereby the
controvariant state is $|^{x}\rangle=|_{-x}\rangle$ and it satisfies the usual
$\langle^{x^{\prime}}|_{x}\rangle=\delta(x-x^{\prime})$. As already discussed
around eq. (17), a state can be expanded as $|\psi\rangle=\int
dx\,\psi(_{x})|^{x}\rangle=\int dx\,\psi(^{x})|_{x}\rangle$ with
$\psi(_{x})=\langle_{x}|\psi\rangle$ and
$\psi(^{x})=\langle^{x}|\psi\rangle=\psi(_{-x})$.

What is emerging from the harmonic oscillator computation is a more general
structure: a coordinate space representation of a pair $q,p$ of conjugated
canonical variables that differs from the usual positive-norm representation

${q|x\rangle=x|x\rangle,\qquad p|x\rangle=+i\frac{d}{dx}|x\rangle}$ (44)

which implies $\langle x|p|\psi\rangle=(-id/dx)\psi(x)$ so that it satisfies
$\langle x|[q,p]|\psi\rangle=i\langle x|\psi\rangle$.

The negative-norm coordinate representation, originally discussed by Dirac
[13] and Pauli [14], is

$\displaystyle q|_{x}\rangle=ix|_{x}\rangle,\qquad
p|_{x}\rangle=+\frac{d}{dx}|_{x}\rangle.$ (45)

Although $q$ looks anti-hermitian, taking into account the extra $i$ as well
as the negative norm, these unusual features combine to form a self-adjoint
$q$:

$\langle_{x^{\prime}}|q^{\dagger}|_{x}\rangle=\langle_{x}|q|_{x^{\prime}}\rangle^{*}=[ix^{\prime}\delta(x+x^{\prime})]^{*}=ix\delta(x+x^{\prime})=\langle_{x^{\prime}}|q|_{x}\rangle.$
(46)

This means that $\langle\psi|q|\psi\rangle=\int dq~{}\psi^{*}(-q)iq\,\psi(q)$
is real. A similar result holds for $p$. When acting on wave-functions one has
$\langle_{x}|q|\psi\rangle=-ix\psi(_{x})$ and
$\langle_{x}|p|\psi\rangle=(+d/dx)\psi(_{x})$, giving the desired $[q,p]=i$
commutator. Defining momentum eigenstates as $p|_{p}\rangle=ip|_{p}\rangle$
one finds $\langle_{q}|_{p}\rangle=e^{ipq}/\sqrt{2\pi}$,
$\langle_{p^{\prime}}|_{p}\rangle=\delta(p+p^{\prime})$. The operator $q$ acts
as $\langle_{q}|q|_{p}\rangle=(-d/dp)\langle_{q}|_{p}\rangle$. One can again
define $|^{p}\rangle=|_{-p}\rangle$ such that $1=\int
dp\,|^{p}\rangle\langle_{p}|$.

The $i$ factor that differentiates the usual representation from the Dirac-
Pauli representation has an impact on the time-inversion parity. As usual, a
positive energy spectrum demands that the time inversion symmetry is anti-
unitary. Then, in the Dirac-Pauli quantization $q$ is naturally $T$-odd and
$p$ is naturally $T$-even (while the opposite holds in the usual quantization,
unless $T$ is defined adding ad-hoc extra signs). This will play a key role in
section 4.

We are now ready to come back to the harmonic oscillator. Inserting into the
condition $\langle x|a|0\rangle=0$ the standard positive-norm representation
such that $a=(q+ip)/\sqrt{2}=(x+s\,d/dx)/\sqrt{2}$ gives a differential
equation which implies the ground-state wave function $\psi_{0}(x)\propto
e^{-sx^{2}/2}$. This is normalizable for $s=1$ (positive norm) and non-
normalizable for $s=-1$, where $s$ was defined in eq. (38). This problem was
emphaized e.g. by Woodard [12] that thereby dismissed the negative norm
quantization as purely formal. However, the problem arises because the
positive-norm representation of $q,p$ was used together with the negative-norm
oscillator: the problem is just a manifestation of the inconsistency of the
assumptions. Consistency demands that the negative norm harmonic oscillator
must be accompanied by the negative-norm Dirac-Pauli coordinate space
representation of the self-adjoint $q,p$ operators, eq. (45). Then, the
condition $\langle x|a|0\rangle=0$ leads to a normalizable wave function for
the ground state $\psi_{0}\propto e^{-x^{2}/2}$, as well as for the excited
states. The Dirac-Pauli choice thereby provides a self-consistent description
of the negative-norm oscillator. Furthermore, as discussed in the next
section, in the 4-derivative case the Pauli-Dirac representation is demanded
by simple considerations.

$\begin{array}[]{c|cc|cc|c}\hbox{norm}&\langle
x|q|\psi\rangle&T\hbox{-parity}&\langle
x|p|\psi\rangle&T\hbox{-parity}&\hbox{harmonic oscillator with $E>0$}\\\
\hline\cr\hbox{positive}&x\psi(x)&\hbox{even}&-i\,d\psi/dx&\hbox{odd}&\hbox{$\psi_{0}(q)\propto
e^{-q^{2}/2}$ and $H=+\frac{1}{2}(q^{2}+p^{2})$}\\\
\hbox{indefinite}&-ix\psi(x)&\hbox{odd}&d\psi/dx&\hbox{even}&\hbox{$\psi_{0}(q)\propto
e^{-q^{2}/2}$ and $H=-\frac{1}{2}(q^{2}+p^{2})$}\end{array}$ Table 1:
Coordinate representations of a pair of canonical variables $[q,p]=i$, and the
associated ground-state wave functions for the positive-energy harmonic
oscillator.

##  4 4 derivatives want Dirac-Pauli

As discussed in the previous section, and as summarized in table 1, quantum
mechanics has two faces: a canonical coordinate can be represented

  * i)

in the usual way with positive norm;

  * ii)

in the Dirac-Pauli way, with negative norm, eq. (45).

As we now show, theories with 4-derivatives want this latter quantization
choice (that, in the gravitational case, corresponds to a renormalizable
theory with positive energy).

A single 4-derivative real coordinate $q(t)$ contains two degrees of freedom.
The Ostrogradski procedure (section 2) rewrites the theory as a Hamiltonian
system of two canonical coordinates, $q_{1}=q$ and $q_{2}=\lambda\dot{q}$. The
key new feature that arises in 4-derivative theories is that $\dot{q}$ becomes
an extra canonical coordinate. In the classical theory $q_{2}$ is just an
auxiliary variable, and $\lambda$ is an irrelevant constant: Ostrogradski used
$\lambda=1$.

In the quantum theory, $q_{1}$ and $q_{2}$ are operators that allow to define
the basis $|q_{1},q_{2}\rangle$. We now show that the usual quantization must
be used for $q_{1}$ and that the Dirac-Pauli quantization must be chosen for
$q_{2}$, which is equivalent to (and more transparent than) fixing an
imaginary $\lambda$ and using the canonical representation.

As usual, the operator $q_{1}=q$ is invariant under time-reversal $t\to-t$,
and thereby it can follow the usual $T$-even representation. On the other
hand, the operator $\dot{q}$ is $T$-odd, because of the time derivative: the
time-inversion operator $T$ transforms it as $T\dot{q}T^{-1}=-\dot{q}$. This
is the novel key feature.

Taking into account that $T$ is anti-unitary, one can equivalently define a
usual $T$-even coordinate $q_{2}=\lambda\dot{q}_{1}$ by choosing an imaginary
$\lambda$.333Alternative routes lead to the same conclusion. For example, one
can use the $T$-even $\ddot{q}$ (instead of $\dot{q}$) as second canonical
coordinate. In the Ostrogradski formalism $\ddot{q}=-p_{2}$ is a momentum. So
again one gets a canonical coordinate with unusual $T$-parity (normally a
momentum is $T$-odd). In general, the invariance of the commutation relation
$[q_{2},p_{2}]=i$ under the anti-unitary time-inversion implies that $q_{2}$
and $p_{2}$ have opposite $T$-parities. One can switch $q_{2}\leftrightarrow
p_{2}$ in order to restore their usual $T$-parities: but their commutator
changes sign, implying again negative norm quantization. This is indeed what
happens in the auxiliary variable formalism, used in various forms in the
literature as an alternative to the canonical Ostrogradski formalism (see e.g.
[22, 29, 24]). This formalism is convenient when dealing with quantum field
theory instead of quantum mechanics with a finite number of degrees of
freedom. In order to facilitate the contact, we summarize the auxiliary
variable formalism below. Restarting from the Lagrangian in eq. (2), we add
zero as a perfect square containing an auxiliary variable $a$:
$\hbox{\sevenrsfs L}=\hbox{\sevenrsfs
L}+\frac{1}{2}\bigg{[}\ddot{q}+(\omega_{1}^{2}+\omega_{2}^{2})\frac{q}{2}-\frac{a}{2}\bigg{]}^{2}.$
(47) Expanding the square cancels both the second-order and the fourth-order
kinetic terms for $q$ leaving  $\hbox{\sevenrsfs
L}=-\frac{a\ddot{q}}{2}+(\omega_{1}^{2}-\omega_{2}^{2})^{2}\frac{q^{2}}{8}-(\omega_{1}^{2}+\omega_{2}^{2})\frac{aq}{4}+\frac{a^{2}}{8}-V(q).$
(48) Going to the free theory $V=0$, we can diagonalize the kinetic and mass
term through the field redefinition
$\left\\{\begin{array}[]{l}a=\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}(\tilde{q}_{2}-\tilde{q}_{1})\cr
q=(\tilde{q}_{2}+\tilde{q}_{1})/{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}\end{array}\right.\qquad\hbox{i.e.}\qquad\tilde{q}_{1,2}=\frac{q}{2}\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}\mp\frac{a}{2\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}$
(49) obtaining two decoupled oscillators  $\hbox{\sevenrsfs
L}=\frac{\dot{\tilde{q}}_{2}^{2}-\omega_{2}^{2}\tilde{q}_{2}^{2}}{2}-\frac{\dot{\tilde{q}}_{1}^{2}-\omega_{1}^{2}\tilde{q}_{1}^{2}}{2}.$
(50) From its classical solution,
$a=2\ddot{q}+(\omega_{1}^{2}+\omega_{2}^{2})q$, we see that $a$ roughly
corresponds to the Ostrogradski $p_{2}$. Furthermore, inserting such classical
solution in eq. (49) one recovers the formalism used in [22]
$\tilde{q}_{2}=\frac{\ddot{q}+\omega_{1}^{2}q}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}},\qquad\tilde{q}_{1}=-\frac{\ddot{q}+\omega_{2}^{2}q}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}.$
(51) However, it is simpler to forget the $\lambda$ factors and just declare
that the self-adjoint operator $\dot{q}$ is $T$-odd and thereby it follows the
$T$-odd Pauli-Dirac representation. Then, the Ostrogradski Hamiltonian of eq.
(6) is $T$-even. The states satisfy $T|q,\dot{q}\rangle=|q,\dot{q}\rangle$
since $\dot{q}$ has imaginary eigenvalues and since $T$ is anti-unitary.

The strange extra factor of $i$ has been justified from first principles. A
posteriori, it was not so strange. After all, it is well known that the self-
adjoint spatial gradient is $i\vec{\nabla}$ rather than $\vec{\nabla}$. In a
relativistic theory, one could have guessed that similarly the self-adjoint
time derivative is $i\partial/\partial t$ rather than $\partial/\partial t$.
Loosely speaking, while from a classical perspective $\dot{q}$ was the natural
auxiliary variable, from a quantum perspective the natural extra coordinate
operator is $i\dot{q}$.444The possibility of converting a non-normalizable
wave-function $\psi_{0}\propto e^{z^{2}/2}$ into a normalizable one by
restricting $z=x+iy$ to the imaginary axis, rather than along the real axis,
was presented as an ad hoc recipe to get something sensible in earlier works
by Bender and Mannheim [25]. At the technical level, their approach differs
from ours because they added an $i$ factor to the variable $q$, rather than to
$\dot{q}$. Our approach follows from general considerations, and has the
advantage that $q,p$ and thereby the Hamiltonian are self-adjoint (although
their matrix representations look anti-hermitian), such that the
generalization to an interacting theory will be immediate (section 6.1).

Using the Heisenberg representation, one has $q(t)=U^{\dagger}(t)q(0)U(t)$ and
$\dot{q}=-i[q,H]=U^{\dagger}(t)\dot{q}(0)U(t)$ with unitary $U$, so $q(t)$
keeps real eigenvalues and $\dot{q}(t)$ keeps imaginary eigenvalues at any $t$
(these statements are not contradictory, given that $q(t)$ also depends on
$p_{1}(0)$ and $p_{2}(0)$).

### The frequency eigenstates

We conclude this section by computing what the Dirac-Pauli representation
adopted for $q_{2}=\dot{q}$ implies for the frequency eigenstates. We restart
from the Hamiltonian eq. (6) and bring it in diagonal form

$H=-\frac{1}{2}(\tilde{p}^{2}_{1}\tilde{\lambda}^{2}+\omega_{1}^{2}\frac{\tilde{q}_{1}^{2}}{\tilde{\lambda}^{2}})+\frac{1}{2}(\tilde{p}^{2}_{2}+\omega_{2}^{2}\tilde{q}^{2}_{2})$
(52)

through the canonical transformation

$q_{1}=\frac{\tilde{q}_{2}-\tilde{\lambda}\tilde{p}_{1}/\omega_{1}}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}},\qquad{q_{2}\over\lambda}=\frac{\tilde{p}_{2}-\omega_{1}\tilde{q}_{1}/\tilde{\lambda}}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}},\qquad
p_{1}=\omega_{1}\frac{\omega_{1}\tilde{p}_{2}-\omega_{2}^{2}\tilde{q}_{1}/\tilde{\lambda}}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}},\qquad
p_{2}\lambda=\frac{\omega_{2}^{2}\tilde{q}_{2}-\omega_{1}\tilde{\lambda}\tilde{p}_{1}}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}.$
(53)

which satisfies
$q_{1}p_{1}-q_{2}p_{2}=\tilde{p}_{2}\tilde{q}_{2}-\tilde{p}_{1}\tilde{q}_{1}$.
For the sake of generality, we here allow for generic factors $\lambda$ and
$\tilde{\lambda}$. The non-vanishing commutators,
$[\tilde{q}_{1},\tilde{p}_{1}]=i$ and $[\tilde{q}_{2},\tilde{p}_{2}]=i$, can
be rewritten in terms of
$\tilde{a}_{2}=\sqrt{\omega_{2}/2}(\tilde{q}_{2}+i\tilde{p}_{2}/\omega_{2})$
and of
$\tilde{a}_{1}=\sqrt{\omega_{1}/2}(\tilde{q}_{1}/\tilde{\lambda}-i\tilde{\lambda}\tilde{p}_{1}/\omega_{1})$
reproducing the Hamiltonian of eq. (14) and the commutators of eq. (13). The
ground-state wave function is easily computed imposing
$\langle\tilde{q}_{1},\tilde{q}_{2}|\tilde{a}_{1,2}|0\rangle=0$ finding

$\psi_{0}(\tilde{q}_{1},\tilde{q}_{2})\propto\exp\bigg{[}-\omega_{2}\frac{\tilde{q}^{2}_{2}}{2}+\omega_{1}\frac{\tilde{q}^{2}_{1}}{2\tilde{\lambda}^{2}}\bigg{]}.$
(54)

For $\tilde{\lambda}=1$ it is not normalizable [26]. It is normalizable if
instead $|{\rm Im}\,\tilde{\lambda}|>|{\rm Re}\,\tilde{\lambda}|$.

The Dirac-Pauli representation for $q_{2},p_{2}$ corresponds to imaginary
$\lambda$. Imposing that $q_{2},p_{1}$ are $T$-odd and that $q_{1},p_{2}$ are
$T$-even (i.e. that $q_{2}$ and $p_{2}$ have the unusual $T$ parity) implies
that $\tilde{q}_{1},\tilde{p}_{2}$ are $T$-odd and that
$\tilde{q}_{2},\tilde{p}_{1}$ are $T$-even (i.e. that the canonical
coordinates of the negative norm mode $\tilde{q}_{1}$ and $\tilde{p}_{1}$ have
the unusual $T$ parity). This is obtained for imaginary $\tilde{\lambda}$.

As a check, let us connect the $q_{1},q_{2}$ basis with the
$\tilde{q}_{1},\tilde{q}_{2}$ basis for generic $\lambda$ and
$\tilde{\lambda}$. It is convenient to start from the $T$-odd basis
$\tilde{q}_{1},\tilde{p}_{2}$, in which the ground state wave function is

$\psi_{0}(\tilde{q}_{1},\tilde{p}_{2})\propto\exp\bigg{[}-\frac{\tilde{p}^{2}_{2}}{2\omega_{2}}+\omega_{1}\frac{\tilde{q}^{2}_{1}}{2\tilde{\lambda}^{2}}\bigg{]}.$
(55)

Next, the transition to the $T$-odd variables $p_{1},q_{2}$ is simply

$\langle
p_{1},q_{2}|\tilde{q}_{1},\tilde{p}_{2}\rangle\propto\delta\bigg{(}\frac{\tilde{q}_{1}}{\tilde{\lambda}}-\frac{p_{1}-q_{2}\omega_{1}^{2}/\lambda}{\omega_{1}\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}\bigg{)}\delta\bigg{(}\tilde{p}_{2}-\frac{p_{1}-q_{2}\omega_{2}^{2}/\lambda}{\sqrt{\omega_{1}^{2}-\omega_{2}^{2}}}\bigg{)}.$
(56)

Inserting the change of variables dictated by the $\delta$ functions into
$\psi_{0}(\tilde{q}_{1},\tilde{p}_{2})$ one obtains

$\psi_{0}(p_{1},q_{2})\propto\exp\bigg{[}-\frac{p_{1}^{2}+2\omega_{1}\omega_{2}p_{1}q_{2}/\lambda-\omega_{1}\omega_{2}(\omega_{1}^{2}+\omega_{1}\omega_{2}+\omega_{2}^{2})(q_{2}/\lambda)^{2}}{2\omega_{1}\omega_{2}(\omega_{1}+\omega_{2})}\bigg{]}$
(57)

where $q_{2}$ and $p_{1}$ are both complex and linked by ${\rm
Re}\,p_{1}=\omega_{1}^{2}{\rm Re}\,(q_{2}/\lambda)$ and ${\rm
Im}\,p_{1}=-\omega_{2}^{2}{\rm Im}\,(q_{2}/\lambda)$. $\psi_{0}$ can be
trivially analytically continued to real $p_{1},q_{2}$. For $\lambda=\pm i$ it
remains a bounded Gaussian. Finally, one performs the Fourier transform from
$p_{1}$ to $q_{1}$, obtaining from $\psi_{0}(p_{1},q_{2})$ the ground state
wave function $\psi_{0}(q_{1},q_{2})$, which agrees with eq. (16). The same
equality holds for excited states, that can be computed acting with creation
operators on the ground state.

In the limit $\omega_{1}=\omega_{2}$ one gets the critical situation described
in section 3.2.

##  5 Path-integral quantization

We now present the path-integral quantization of the same 4-derivative theory.

###  5.1 Path-integral for generic norm

Our generic-norm formalism makes easy to write down the equivalent path-
integral formalism, an issue already considered in [21]. Inserting $1=\int
dq\,|^{q}\rangle\langle_{q}|$ at intermediate times $t_{m}=t_{i}+m\,dt$ one
has

$\langle^{q_{f},t_{f}}|_{q_{i},t_{i}}\rangle=\prod_{m}\int
dq_{m}\langle^{q_{m+1},t_{m+1}}|_{q_{m},t_{m}}\rangle.$ (58)

Each step $\langle^{q_{m+1},t_{m+1}}|_{q_{m},t_{m}}\rangle$ can be evaluated
as

$\langle^{q_{m+1}}|e^{-iHdt}|_{q_{m}}\rangle=\int
dp_{m}\,\langle^{q_{m+1}}|^{p_{m}}\rangle\langle_{p_{m}}|e^{-iHdt}|_{q_{m}}\rangle=\int\frac{dp_{m}}{2\pi}e^{i[p_{m}(q_{m+1}-q_{m})-H_{\rm
cl}dt]}$ (59)

having defined

$H_{\rm cl}\equiv\frac{\langle_{p}|H|_{q}\rangle}{\langle_{p}|_{q}\rangle}$
(60)

and used $\langle^{q}|^{p}\rangle=e^{ipq}/\sqrt{2\pi}$ and
$\langle_{p}|_{q}\rangle=e^{-ipq}/\sqrt{2\pi}$. The final result is the path
integral

$\langle^{q_{f},t_{f}}|_{q_{i},t_{i}}\rangle=\int{Dq\,Dp}~{}e^{i\int
dt[p\dot{q}-H_{\rm cl}]}\qquad\hbox{where}\qquad Dq\,Dp=\lim_{dt\to
0}\prod_{m}\frac{dq_{m}dp_{m}}{2\pi}$ (61)

and with boundary conditions $q(t_{i})=q_{i}$, $q(t_{f})=q_{f}$.

###  5.2 Path-integral for 4 derivative quantum theories

Applying the generic path-integral of eq. (61) to the 4-derivative oscillator
in the canonical Ostrogradski formalism, one gets the transition amplitude

$\langle^{q_{1f},q_{2f},t_{f}}|_{q_{1i},q_{2i},t_{i}}\rangle\propto\int
Dq_{1}Dp_{1}Dq_{2}Dp_{2}\,\exp\bigg{[}{i\int
dt\left[p_{1}\dot{q}_{1}+p_{2}\dot{q}_{2}-H_{\rm
cl}+J_{1}q_{1}+J_{2}q_{2}\right]}\bigg{]}$ (62)

where for generality we added currents $J_{1,2}$ (such that acting with
functional derivatives with respect to them one can form more general matrix
elements of time-ordered operators; $J_{1}$ is $T$-even and $J_{2}$ is
$T$-odd). The Pauli-Dirac representation for $\dot{q}$ manifests in two ways:

  * 1)

A propagator with an unusual $-$ in its external state.

Rewriting the transition amplitude in the usual positive-norm formalism, it
acquires an usual $-$ sign, becoming
$\langle{q_{f},-\dot{q}_{f},t_{f}}|{q_{i},\dot{q}_{i},t_{i}}\rangle$. In the
limit $t_{f}\to t_{i}$ one has
$\langle^{q_{f},\dot{q}_{f}}|_{q_{i},\dot{q}_{i}}\rangle=\delta(q_{f}-q_{i})\delta(\dot{q}_{f}-\dot{q}_{i})$,
so that the unusual $-$ sign is equivalent to the Dirac-Pauli negative
norm.555 In the limit $dt=t_{f}-t_{i}\to 0$ the classical action becomes
$S_{\rm
cl}\simeq\frac{6}{dt^{3}}\bigg{(}q_{f}-q_{i}-dt\frac{\dot{q}_{i}+\dot{q}_{f}}{2}\bigg{)}^{2}+\frac{(\dot{q}_{f}-\dot{q}_{i})^{2}}{2\,dt}+\cdots$
(63) which is minimal for a motion with constant speed
$(\dot{q}_{i}+\dot{q}_{f})/2$. The classical action satisfies
$S(q_{f},\dot{q}_{f},t_{f};q_{i},\dot{q}_{i},t_{i})=-S(q_{i},\dot{q}_{i},t_{i};q_{f},\dot{q}_{f},t_{f})$
as well as $S_{\rm
cl}(q_{f},\dot{q}_{f},t_{f};q_{i},\dot{q}_{i},t_{i})=-S_{\rm
cl}(q_{f},-\dot{q}_{f},t_{i};q_{f},-\dot{q}_{i},t_{i})$. Furthermore, the
$T$-odd nature of $\dot{q}$ is hardwired in the path-integral, as a
geometrical feature. For each trajectory $q(t)$ with boundary conditions
$q(t_{i,f})=q_{i,f}$ and $\dot{q}(t_{i,f})=\dot{q}_{i,f}$ the time-inverted
trajectory has the same action and the following boundary conditions:

$q_{i}\to q_{f},\qquad q_{f}\to
q_{i},\qquad\dot{q}_{i}\to-\dot{q}_{f},\qquad\dot{q}_{f}\to-\dot{q}_{i}.$ (64)

Thereby the propagator given by the path-integral satisfies the identity

$\langle q_{f},-\dot{q}_{f},t_{f}|q_{i},\dot{q}_{i},t_{i}\rangle=\langle
q_{i},\dot{q}_{i},t_{f}|q_{f},-\dot{q}_{f},t_{i}\rangle$ (65)

which is equivalent to the operator identity
$\langle\psi_{f}|\psi_{i}\rangle=\langle T\psi_{i}|T\psi_{f}\rangle$ given
that $T|q,\dot{q},t\rangle=|q,\dot{q},-t\rangle$.

  * 2)

An unusual classical Hamiltonian.

Inserting the Ostrogradski Hamiltonian of eq. (6) in the generic path integral
of eq. (61) one gets the following classical Hamiltonian:

$H_{\rm
cl}=\frac{\langle_{p_{1},p_{2}}|H|_{q_{1},q_{2}}\rangle}{\langle_{p_{1},p_{2}}|_{q_{1},q_{2}}\rangle}=ip_{1}q_{2}+\frac{p_{2}^{2}}{2}+\frac{\omega_{1}^{2}+\omega_{2}^{2}}{2}q_{2}^{2}+\frac{\omega_{1}^{2}\omega_{2}^{2}}{2}q_{1}^{2}+V(q_{1}).$
(66)

This is the same as eq. (6) with $\lambda=-i$. $H_{\rm cl}$ can be complex
because $q_{2},p_{2}$, in the Dirac-Pauli representation, have complex
eigenvalues.666The classical Hamiltonian is real if one instead uses the
equivalent oscillator basis
$\tilde{q}_{1},\tilde{q}_{2},\tilde{p_{1}},\tilde{p}_{2}$ of eq. (53). Thanks
to the unusual $i$, it is invariant under time-reversal.

Let us now try to evaluate the path-integral. As usual, one can perform the
Gaussian $Dp_{1}Dp_{2}$ path integrals. The $Dp_{1}$ path-integral formally
gives the Dirac delta function $\delta(q_{2}-\lambda\dot{q}_{1})$, allowing to
eliminate the $Dq_{2}$ path-integral, leaving

$\langle^{q_{f},\dot{q}_{f},t_{f}}|_{q_{i},\dot{q}_{i},t_{i}}\rangle\propto\int
Dq\,\exp\bigg{[}{i\int
dt\left[\mathscr{L}(q)+J_{1}q+J_{2}\lambda\dot{q}\right]}\bigg{]},$ (67)

where $\mathscr{L}$ coincides with the original 4-derivative Lagrangian. By
partial integration, the source term for $\dot{q}$ can be transformed into a
source for $q$ or for $\ddot{q}$ (like in the auxiliary-field method). This
computation however has three problems:

  1. 1.

the $Dp_{1}$ path-integral is, in general, divergent. Thereby the subsequent
result is only formal.

  2. 2.

the $\delta(q_{2}-\lambda\dot{q}_{1})$ always vanishes if $q_{1}$ and $q_{2}$
are real. Thereby the $Dq_{2}$ path-integral is only formal.

  3. 3.

Once interactions are turned on, the Lagrangian admits classical run-away
solutions, reflected in the path-integral.

Given that the theory is well defined in the operator formalism, somehow this
path integral must have a sense.

###  5.3 Euclidean Path-integral for 4 derivative quantum theories

A sensible path-integral is found by restarting from eq. (62) and continuing
it to Euclidean time, $it=t_{E}$, such that $dq/dt=i\,dq/dt_{E}$ i.e.
$\dot{q}=iq^{\prime}$. One gets the Euclidean path integral

$\langle^{q_{1f},q_{2f},t_{Ef}}|_{q_{1i},q_{2i},t_{Ei}}\rangle\propto\int
Dq_{1}Dq_{2}Dp_{1}Dp_{2}~{}\exp\bigg{[}\int
dt_{E}(ip_{1}q^{\prime}_{1}+ip_{2}q^{\prime}_{2}-H_{\rm
cl}+J_{1}q_{1}+J_{2}q_{2})\bigg{]}.$ (68)

Now the $Dp_{1}$ integral is convergent and gives
$\delta(q_{2}-q^{\prime}_{1})$, such that the $Dq_{2}$ path integral just
fixes $q_{2}=q^{\prime}_{1}$. Next, the remaining terms in $H_{\rm cl}$ are a
sum of positive squares so all other integrals are convergent. Performing them
one finds the Lagrangian Euclidean path-integral:

$\displaystyle\langle^{q_{f},q^{\prime}_{f},t_{Ef}}|_{q_{i},q^{\prime}_{i},t_{Ei}}\rangle\propto\int
Dq\,\exp\bigg{[}-\int
dt_{E}\left[\mathscr{L}_{E}(q)+J_{1}q+J_{2}q^{\prime}\right]\bigg{]}$ (69)

where the classical Euclidean Lagrangian corresponding to eq. (2) is

$\mathscr{L}_{E}=\frac{1}{2}\bigg{(}\frac{d^{2}q}{dt_{E}^{2}}\bigg{)}^{2}+\frac{\omega_{1}^{2}+\omega_{2}^{2}}{2}\bigg{(}\frac{dq}{dt_{E}}\bigg{)}^{2}+\frac{\omega_{1}^{2}\omega_{2}^{2}}{2}q^{2}+V(q).$
(70)

Let us now check the result. The classical free solution is

$q(t_{E})=a_{1}e^{-\omega_{1}t_{E}}+a_{2}e^{-\omega_{2}t_{E}}+b_{1}e^{\omega_{1}t_{E}}+b_{2}e^{\omega_{2}t_{E}}.$
(71)

It already contains run-away exponentials, characteristic of any Euclidean
theory. Interactions compatible with the positivity of the action lead to an
equally good path-integral. By imposing the boundary conditions
$q=q^{\prime}=0$ at $t_{Ei}=-\infty$ and evaluating the classical action, one
finds the normalizable ground-state wave function

$\langle
q,q^{\prime},t_{E}=0|0,0,t_{E}=-\infty\rangle\propto\exp\bigg{[}-\frac{q^{2}\omega_{1}\omega_{2}+q^{\prime
2}}{2}(\omega_{1}+\omega_{2})+qq^{\prime}\omega_{1}\omega_{2}\bigg{]}.$ (72)

This agrees with the ground-state wave-function $\psi_{0}(q_{1},q_{2})$ in eq.
(16), that was computed in the Dirac-Pauli formalism in Minkowski space, after
identifying $q=q_{1}$ and $q^{\prime}=q_{2}$. In other words,
$q^{\prime}=dq/dt_{E}=-idq/dt$ coincides with $q_{2}$, as computed for
$\lambda=-i$. The novel feature introduced by 4-derivatives is that
$q^{\prime}$ must not be continued into an imaginary $-i\dot{q}$ (which would
give divergent wave functions), because it already describes the $T$-odd
variable $q_{2}$, which contains the Dirac-Pauli $i$ factor of eq. (45).777
Hawking and Hertog [22] found a non-normalizable Minkowskian wave-function
because they expressed eq. (72) in terms of $\dot{q}$, which is not the
canonical coordinate $q_{2}$ appropriate for 4-derivative theories. Their
proposal of integrating out $\dot{q}$ in the Euclidean before performing the
analytic continuation to the Minkowskian is not necessary: the Minkowskian
wave functions are normalizable if the appropriate analytic continuation is
performed.  The final result is that the Minkowskian theory is an unusual
analytic continuation of the Euclidean theory.

##  6 Interactions, Quantum Field Theory, probability

Summarising, we so far considered a 4-derivative harmonic oscillator. One
might think that we achieved nothing [26]. After all, a classical 4-derivative
harmonic oscillator has no run-away problems, see eq. (9), given that it
splits into two decoupled oscillators, one with negative energy and one with
positive energy. The classical trouble starts when they interact. In this
section we will explain that we have achieved instead something useful in an
interacting quantum field theory.

###  6.1 Adding interactions

The quantum formalism was so far developed for the harmonic oscillator (which
corresponds to the modes of a free 4-derivative quantum field theory), finding
that the quantum theory has a positive energy spectrum and no run-away
behaviours. Adding interactions, the quantum interacting inherits all these
good properties, as long as interactions are perturbative and as long as the
interacting Hamiltonian $H$ remains self-adjoint.

The second issue was the main obstacle to past attempts of adding ad-hoc
unusual $i$ factors in order to make the quantum free theory consistent [25]
(normalizable wave functions and unitary evolution with negative norm and
positive energy eigenvalues): adding extra complex factors can render
interactions complex, ruining the theory [26].

In our approach the only extra $i$ factor arose from a principled reason:
$\dot{q}$ is a $T$-odd coordinate that follows the negative-norm Dirac-Pauli
representation. This satisfies all the properties of quantum mechanics, as
generalised to negative norms: $\dot{q}$ itself is self-adjoint, like $q$ and
$\ddot{q}$. Thereby any interaction which is a real function of them is self-
adjoint. Our procedure immediately generalizes to the interacting case (in
agravity [8] all interactions are dictated by general covariance).

The perturbativity assumption means that, as long as the energy spectrum of
the free oscillator gets slightly distorted by interactions, the energy
eigenvalues will remain real and bounded from below (strongly interacting
theories could also be good; however they seem not needed for the physical
application to agravity [8]).

One might worry that, even if all energy eigenvalues are positive, the theory
possesses negative-norm states with $\langle\psi|H|\psi\rangle<0$. Eq. (29)
shows how transition amplitudes can be computed trough perturbation theory: we
see that the energy eigenvalues are the quantity that enters into conservation
of energy. Thereby a theory where all eigenvalues of $H$ (of $H_{0}$ in the
perturbative expansion) are positive is consistent. As usual, perturbative
computations can be systematised in terms of the propagator. By expressing
$q=q_{1}$ in terms of the annihilation and creation operators $a_{i}$,
$a_{i}^{\dagger}$ through eq.s (11) and (12) and using the commutation
relations $[\tilde{a}_{i},\tilde{a}_{i}^{\dagger}]=s_{i}$ we find the
propagator

$\halign to=0.0pt{\@eqnsel\hskip\@centering$\displaystyle{#}$&\global\@eqcnt\@ne\hskip 2\arraycolsep\hfil${#}$\hfil& \global\@eqcnt\tw@\hskip 2\arraycolsep$\displaystyle{#}$\hfil&\llap{#}\cr 0.0pt plus 1000.0pt$\displaystyle{\langle 0 | T q(t)q (t')|0\rangle&10.0pt\hfil${=&10.0pt$\displaystyle{\langle 0 | \theta(t-t') q(t)q(t')+\theta(t'-t)q(t')q(t) |0\rangle\\\ &\hbox to0.0pt{\hss=&\frac{1}{\omega_{1}^{2}-\omega_{2}^{2}}\sum_{i}\frac{s_{i}}{2\omega_{i}}[e^{i\omega_{i}(t-t^{\prime})}\theta(t^{\prime}-t)+e^{i\omega_{i}(t^{\prime}-t)}\theta(t-t^{\prime})]\\\&=&i\frac{1}{\omega_{1}^{2}-\omega_{2}^{2}}\int\frac{dE}{2\pi}\sum_{i}\frac{s_{i}\,e^{-iE(t-t^{\prime})}}{E^{2}-\omega_{i}^{2}+i\epsilon}\\\&=&\int\frac{dE}{2\pi}\frac{-i\,e^{-iE(t-t^{\prime})}}{(E^{2}-\omega_{1}^{2}+i\epsilon)(E^{2}-\omega_{2}^{2}+i\epsilon)}.&(73)\cr}}}}}$

where $\epsilon$ is a small positive quantity and we used $s_{1}=-1$ and
$s_{2}=1$ in the last step.

One might worry that, using the Heisenberg picture, operators satisfy the time
evolution equation $\dot{A}=-i[A,H]$, which looks dangerously similar to the
classical equation of motion, as given by Poisson parentheses, which has run-
away solutions. However, the quantum solutions are equal to the classical
solutions only in a free theory. In general operators are not numbers, and the
difference (in particular, the Pauli-Dirac representation) manifests when non-
linear interactions are present. As well known, the Heisenberg equations are
in general solved by $A(t)=U^{\dagger}(t)A(0)U(t)$. So, all good properties of
negative norm states found in the Schrödinger picture remain valid in the
Heisenberg picture, given that they are equivalent.

###  6.2 Extension to quantum field theory

As well known, a single harmonic-oscillator degree of freedom $q(t)$ is the
building block for a field such as $\phi(t,x,y,z)$ or $g_{\mu\nu}(t,x,y,z)$.
The expansion of a field in Fourier modes with given momentum works in the
4-derivative case similarly to the 2-derivative case. As long as, at the end,
we are only interested in $S$-matrix elements, all the detailed structure of
the quantum mechanical theory, such as the wave-functions, gets hidden behind
the commutation relations of eq. (13), which hold separately for each mode.
The usual $i\epsilon$ prescription for the field propagator dictates that
amplitudes can be analytically continued from the Euclidean. Details will be
presented elsewhere.

One would like to claim that quantum field theory inherits all good properties
of quantum mechanics also when negative norms are present. However, while in
quantum mechanics interactions can easily satisfy the condition that avoids
‘tachionic ghosts’ (namely, the interaction strength between two opposite-norm
states must be smaller than their energy difference as discussed in section
3.2), any interesting quantum field theory leads to situations that might
violate this condition. The simplest situation where this occurs is the decay
of a ghost (for example a massive spin 2 graviton at rest), which can be
degenerate with a multi-particle state (for example two photons going in
opposite directions with energy equal to half of the ghost mass), such that
the interaction, no matter how small, can be smaller than the energy
difference. Actually, the ghost is degenerate with an infinite number of
similar states, such that an appropriate limit procedure is needed: in the
positive norm case, entropic considerations allow to interpret this situation
as particle decay. A 4-derivative kinetic term
$\Pi(p)=-(p^{2}-m_{1}^{2})(p^{2}-m_{2}^{2})$ acquires a positive imaginary
part. We will explore if ‘ghost decay’ can be interpreted like in [24].

###  6.3 Ghost does not play dice?

So far we carefully avoided talking about probabilities.

The theory is unitary in a negative-norm space. Thereby the only remaining
difficulty is assigning an interpretation to states that entangle positive
norm components with negative norm components. The Copenhagen interpretation
added an extra ingredient external to the deterministic formalism of quantum
mechanics: the Born postulate, according to which:

> “when an observable corresponding to a self-adjoint operator $A$ is measured
> in a state $|\psi\rangle$, the result is an eigenvalue $A_{n}$ of $A$ with
> probability
>
>
> $P_{n}=\frac{\langle\psi|\Pi_{n}|\psi\rangle}{\langle\psi|\psi\rangle}\qquad\hbox{where}\qquad\Pi_{n}=\frac{|n\rangle\langle
> n|}{\langle n|n\rangle}$ (74)
>
> is the projector over the eigenstate $|n\rangle$ of $A$”.

For positive norms, these $P_{n}$ satisfy the probability rules $0\leq
P_{n}\leq 1$ and $\sum_{n}P_{n}=1$; the average value of $A$ satisfies
$\langle\psi|A|\psi\rangle/\langle\psi|\psi\rangle=\sum_{n}A_{n}P_{n}$.

At the moment we do not have a satisfactory generalisation to indefinite norm.
Even worse, the Born postulate is unsatisfactory by itself, given that it
describes a non-local collapse of the wave-function [33]. In order to make
progress, one needs to operate close to the heart of quantum mechanics. As
well known this presents fatal risks: physicists tend to become philosophers.
We conclude by listing some interpretations of quantum mechanics, equivalent
to the Copenhagen interpretation, which could lead to a satisfactory
indefinitive norm quantum mechanics.

  1. 1.

Feynman clarified the ontological basis of the Born postulate: it agrees with
experiments, so ‘shut up and compute’. All experiments have so far been
performed with positive norm states. The negative norm graviton predicted by
agravity is beyond the reach of present experiments. On the one hand, this is
good because it means that Einstein’s general relativity is recovered at large
distances; on the other hand, however, we do not have experimental guidance.
Lee and Wick proposed that the interpretation issue is bypassed, given that in
quantum field theory we can only observe asymptotic states, which are made of
positive-norm quanta [17]. The Lee-Wick idea may be applied to the
gravitational theory proposed by Stelle [2], as discussed in [5, 34].

  2. 2.

Any self-adjoint Hamiltonian $H$ gives unitary evolution with respect to many
different norms, since each energy eigenstate evolves picking just a phase.
Defining ghost parity ${\cal G}$ to be the metrics in the special basis of
energy eigenstates and $|^{\psi}\rangle={\cal G}^{-1}|_{\psi}\rangle$, a
possible generalization of the Born postulate to generic norm is (see also
[25])

$P_{n}=\langle^{\psi}|\Pi_{n}|_{\psi}\rangle\qquad\hbox{where}\qquad\Pi_{n}=|^{n}\rangle\langle_{n}|.$
(75)

The example of section 3.2 gets converted into normal oscillations with mixing
angle $\sin^{2}2\theta=E_{I}^{2}/E_{R}^{2}$. However,
$\langle_{\psi}|A|_{\psi}\rangle$ is real but does not have a probabilistic
interpretation, while $\langle^{\psi}|A|_{\psi}\rangle$ has a probabilistic
interpretation but can be complex.

  3. 3.

Various authors claim that the Born postulate is just an emergent phenomenon
(somehow like friction) that follows from the fundamental deterministic
equations when applied to complex systems in view of spontaneous decoherence
[35].

  4. 4.

Cramer [36] proposed a “transactional interpretation”, claiming that EPR non-
locality results from a cancellation of advanced and retarded waves, in a
time-symmetric set-up (see also [37]) inspired by the analogous formulation of
classical electro-dynamics proposed by Dirac and Feynman-Wheeler. The
$\langle\psi^{\prime}|\psi\rangle$ amplitude in the Dirac-Pauli coordinate
representation supports the interpretation as being the overlap of a wave
$\psi$ moving forward in time with a wave $\psi^{\prime}$ moving backwards in
time.

We plan to further investigate such issues.

##  7 Conclusions (so far)

We presented the quantization of 4-derivative theories, finding that a unique
structure emerges. We can summarise it as follows.

Quantum mechanics has its usual visible face, where a coordinate operator $q$
is represented as $q|x\rangle=x|x\rangle$. But quantum mechanics also has a
hidden face, where $q|x\rangle=ix|x\rangle$, as first pointed out by Dirac and
Pauli. Both $q$ and $p$ of a canonical pair $[q,p]=i$ are self-adjoint in both
representations. The main difference is that the usual representation implies
positive norms and $q$ is naturally even under time reflection $T$, while the
DP representation leads to states with indefinite norm and to a naturally
$T$-odd $q$ (in view of the $i$ factor and of the fact that $T$ is anti-
unitary)

The Ostrogradski formulation of a 4-derivative degree of freedom $q(t)$
(summarised in section 2) employs two canonical coordinates: $q_{1}=q$ and
$q_{2}=\dot{q}$. For the first time we have observed that $q_{1}$, which is
$T$-even, naturally follows the usual representation, while $q_{2}$ which is
$T$-odd, naturally follows the Dirac-Pauli negative-norm quantization. This
leads to a sensible quantum theory with positive energies and normalizable
wave-functions, as discussed in section 4.

In section 3 we presented a new formalism appropriate for generic-norm quantum
mechanics, introducing ‘covariant’ $|_{n}\rangle$ and ‘contro-variant’
$|^{n}\rangle$ basis states. This clarifies why a self-adjoint linear operator
can be represented by a matrix that, in some basis, is not hermitian. A self-
adjoint Hamiltonian leads to unitary time-evolution, in the sense that the
negative norm is preserved. Given that $q$, $\dot{q}$, $\ddot{q}$, …are self-
adjoint, a Hamiltonian which is a generic real function of them is self-
adjoint, leading to sensible interacting quantum theory provided that one
avoids tachyons, an observation that was previously overlooked. The usual
condition that the theory should be free of tachyons is generalised to
negative norm quantum mechanics.

In section 5 we presented the path-integral formulation of negative-norm
quantum mechanics. The classical Hamiltonian becomes complex. Another new
result of this paper is the proof that the normalizable wave functions found
in the operator formalism are recovered from the path-integral after
performing naive manipulations over ill-defined objects and/or analytic
continuations. In particular, the version of the path-integral in Euclidean
time $t_{E}=it$ is well defined, and reproduces the usual wave functions
taking into account that $dq/dt_{E}$ already coincides with the Dirac-Pauli
$\dot{q}$.

The fact that (1) our approach leads to normalizable wave-functions and (2)
these wave-functions can also be deduced from a well-defined Euclidean path-
integral clearly show that the right quantization for $\dot{q}$ is the Dirac-
Pauli one.

Two issues must be addressed before that these results can be used to obtain a
predictive renormalizable quantum theory of gravity: generalisation to quantum
field theory, and generalisation of the Born probabilistic interpretation to
negative norms.

#### Acknowledgments

This work was supported by the ERC grant NEO-NAT, by the Spanish Ministry of
Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN
(CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia Severo
Ochoa” Programme and the grant HEPHACOS-S2009/ESP1473 from the C.A. de Madrid.
We thank all the colleagues who told us that working with ghosts is equivalent
to abandoning physics, and Enrique Álvarez, José R. Espinosa, Antonio
González-Arroyo, Martti Raidal, Hardi Veermae, Sergey Sibiryakov, and Guido
Altarelli for useful comments and encouragement. This was the content of my
last conversation with Guido — the present paper was finalized wondering what
Guido would have said.

## References

  * [1] M. Ostrogradski, Mem. Ac. St. Petersbourg VI (1850) 385.
  * [2] K.S. Stelle, “Renormalization of Higher Derivative Quantum Gravity”, Phys. Rev. D 16 (1977) 953. Recently, the proof of renormalisability was repeated using modern techniques, see G. Rabuffo, “Local cohomology, master equation and renormalization of higher- derivative and nonlocal quantum gravity”, thesis at Pisa University.
  * [3] E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity”, Nucl. Phys. B (1982) 201.
  * [4] I.G. Avramidi and A.O. Barvinsky, “Asymptotic Freedom In Higher Derivative Quantum Gravity”, Phys. Lett. B 159 (1985) 269. I.G. Avramidi, “Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher derivative quantum gravity” [arXiv:hep-th/9510140].
  * [5] I. Antoniadis and E. T. Tomboulis, “Gauge Invariance and Unitarity in Higher Derivative Quantum Gravity”, Phys. Rev. D 33 (1986) 2756.
  * [6] E. Elizalde, S.D. Odintsov, A. Romeo, “Renormalization group properties of higher derivative quantum gravity with matter in (4-epsilon)-dimensions”, Nucl. Phys. B462 (1995) 315 [arXiv:hep-th/9502131]. G. Narain, R. Anishetty, “Running Couplings in Quantum Theory of Gravity Coupled with Gauge Fields”, JHEP 1310 (2013) 203 [arXiv:1309.0473].
  * [7] I. L. Shapiro and A. G. Zheksenaev, “Gauge dependence in higher derivative quantum gravity and the conformal anomaly problem”, Phys. Lett. B 324 (1994) 286. E.V. Gorbar, I.L. Shapiro, “Renormalization group and decoupling in curved space. 2. The Standard model and beyond”, JHEP 0306 (2003) 004 [arXiv:hep-ph/0303124]. G. de Berredo-Peixoto, I.L. Shapiro, “Higher derivative quantum gravity with Gauss-Bonnet term”, Phys. Rev. D71 (2004) 064005 [arXiv:hep-th/0412249].
  * [8] A. Salvio, A. Strumia, “Agravity”, JHEP 1406 (2014) 080 [arXiv:1403.4226].
  * [9] M.B. Einhorn, D.R.T. Jones, “Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity”, JHEP 1503 (2015) 047 [arXiv:1410.8513].
  * [10] K. Kannike, G. Hütsi, L. Pizza, A. Racioppi, M. Raidal, A. Salvio, A. Strumia, “Dynamically Induced Planck Scale and Inflation”, JHEP 1505 (2015) 065 [arXiv:1502.01334].
  * [11] A. Farzinnia, S. Kouwn, “Classically Scale Invariant Inflation, WIMPzilla, and (A)gravity” [arXiv:1512.05890].
  * [12] R.P. Woodard, “Avoiding dark energy with $1/r$ modifications of gravity”, Lect. Notes Phys. 720 (2006) 403 [arXiv:astro-ph/0601672]. R.P. Woodard, “The Theorem of Ostrogradsky” [arXiv:1506.02210].
  * [13] P.A.M. Dirac, “The physical interpretation of quantum mechanics” (1942).
  * [14] W. Pauli, “On Dirac’s New Method of Field Quantization”, Rev. Mod. Phys. 15 (1943) 175.
  * [15] W. Heisenberg, “Quantum Theory of Fields and Elementary Particles”, Rev. Mod. Phys. 29 (3) 269.
  * [16] A. Pais and G. E. Uhlenbeck, “On Field theories with nonlocalized action”, Phys. Rev. 79 (1950) 145.
  * [17] T. D. Lee and G. C. Wick, “Negative Metric and the Unitarity of the S Matrix”, Nucl. Phys. B 9 (1969) 209. T. D. Lee and G. C. Wick, “Finite Theory of Quantum Electrodynamics”, Phys. Rev. D 2 (1970) 1033.
  * [18] R.E. Cutkosky, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, “A non-analytic $S$ matrix”, Nucl. Phys. B 12 (1969) 281.
  * [19] S. Coleman, “Acausality”, in “Erice 1969, Ettore Majorana School On Subnuclear Phenomena”, New York (1970) 282.
  * [20] R.P. Feynman, “Negative probability” in “Quantum implications: Essays in honor of David Bohm”, edited by B.J. Hiley and F.D. Peat (Routledge and Kegan Paul, London, 1987).
  * [21] D.G. Boulware, D.J. Gross, “Lee-Wick Indefinite Metric Quantization: A Functional Integral Approach”, Nucl. Phys. B 233 (1984) 1.
  * [22] S.W. Hawking, T. Hertog, “Living with ghosts”, Phys. Rev. D65 (2001) 103515 [arXiv:hep-th/0107088]. See also H. Kleinert, J. Math. Phys. 27 (1986) 12.
  * [23] G. ’t Hooft, “Unitarity in the Brout-Englert-Higgs Mechanism for Gravity” [arXiv:0708.3184]. G. ’t Hooft, “A class of elementary particle models without any adjustable real parameters”, Found. Phys. 41 (2011) 1829 [arXiv:1104.4543].
  * [24] B. Grinstein, D. O’Connell, M.B. Wise, “Causality as an emergent macroscopic phenomenon: The Lee-Wick $O(N)$ model”, Phys. Rev. D79 (2008) 105019 [arXiv:0805.2156].
  * [25] P.D. Mannheim, “Solution to the ghost problem in fourth order derivative theories”, Found. Phys. 37 (2006) 532 [arXiv:hep-th/0608154]. P.D. Mannheim, A. Davidson, “Dirac quantization of the Pais-Uhlenbeck fourth order oscillator”, Phys. Rev. A71 (2004) 042110 [arXiv:hep-th/0408104]. C.M. Bender, “Making sense of non-Hermitian Hamiltonians”, Rept. Prog. Phys. 70 (2007) 947 [arXiv:hep-th/0703096]. C.M. Bender, P.D. Mannheim, “No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model”, Phys. Rev. Lett. 100 (2007) 110402 [arXiv:0706.0207]. C.M. Bender, P.D. Mannheim, “Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart”, Phys. Rev. D78 (2008) 025022 [arXiv:0804.4190]. C.M. Bender, P.D. Mannheim, “Giving up the ghost”, J. Phys. A41 (2008) 304018 [arXiv:0807.2607]. P.D. Mannheim, “Cosmological Perturbations in Conformal Gravity”, Phys. Rev. D85 (2011) 124008 [arXiv:1109.4119]. P.D. Mannheim, “Comprehensive Solution to the Cosmological Constant, Zero-Point Energy, and Quantum Gravity Problems”, Gen. Rel. Grav. 43 (2009) 703 [arXiv:0909.0212]. P.D. Mannheim, “Making the Case for Conformal Gravity”, Found. Phys. 42 (2011) 388 [arXiv:1101.2186]. P.D. Mannheim, “Advancing the case for $PT$ Symmetry – the Hamiltonian is always $PT$ Symmetric” [arXiv:1506.08432].
  * [26] A.V. Smilga, “Ghost-free higher-derivative theory”, Phys. Lett. B632 (2005) 433 [arXiv:hep-th/0503213]. A.V. Smilga, “Comments on the dynamics of the Pais-Uhlenbeck oscillator”, SIGMA 5 (2008) 017 [arXiv:0808.0139].
  * [27] K. Andrzejewski, J. Gonera, P. Maslanka, “Euclidean Path Integral and Higher-Derivative Theories”, Prog. Theor. Phys. 125 (2009) 247 [arXiv:0904.3055]. R. Di Criscienzo, S. Zerbini, “Functional Determinants in Higher Derivative Lagrangian Theories”, J. Math. Phys. 50 (2009) 103517 [arXiv:0907.4265] and Erratum-ibid. 51 (2010) 059901. S.V. Ketov, G. Michiaki, T. Yumibayashi, “Quantizing with a higher time derivative” [arXiv:1110.1155].
  * [28] T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, “Towards singularity and ghost free theories of gravity”, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249]
  * [29] T. Chen, E.A. Lim, “Stabilization of Linear Higher Derivative Gravity with Constraints”, JCAP 1405 (2014) 010 [arXiv:1311.3189].
  * [30] K. Andrzejewski, J. Gonera, P. Machalski, K. Bolonek-Lason, “On the triviality of higher derivative theories”, Phys. Lett. B706 (2012) 427 [arXiv:1110.0672].
  * [31] J. Maldacena, “Einstein Gravity from Conformal Gravity” [arXiv:1105.5632].
  * [32] H. Lu, Y. Pang, C. N. Pope, “Conformal Gravity and Extensions of Critical Gravity”, Phys. Rev. D 84 (2011) 064001 [arXiv:1106.4657].
  * [33] A. Einstein, B. Podolsky, N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”, Phys. Rev. 47 (1935) 777. J. Bell, “On the Einstein-Poldolsky-Rosen paradox”, Physics 1 (1964) 195. A. Aspect, “Bell’s inequality test: more ideal than ever”, Nature 398 (1999) 189.
  * [34] B. Hasslacher, E. Mottola, Phys. Lett. B 99 (1981) 221.
  * [35] R. Griffiths, “Consistent histories and the interpretation of quantum mechanics”, J. Stat. Phys 36 (1984) 219. R. Omnes, “Consistent interpretation of quantum mechanics”, Rev. Mod. Phys. 64 (1992) 339. M. Gell-Mann, J.B. Hartle, “Classical equations for quantum systems”, Phys. Rev. D 47 (1993) 3445.
  * [36] J.G. Cramer, “An Overview of the Transactional Interpretation”, J. Theor. Phys. 27 (1988) 227. J.G. Cramer, “The Transactional Interpretation of Quantum Mechanics and Quantum Nonlocality” [arXiv:1503.00039].
  * [37] Y. Aharonov, P. G. Bergmann, J. L. Lebowitz, “Time symmetry in the quantum process of measurement”, Phys. Rev. 134 (1964) B1410.