# A parallel Buchberger algorithm for multigraded ideals

Emil Sköldberg and Mikael Vejdemo-Johansson and Jason Dusek

###### Abstract.

We demonstrate a method to parallelize the computation of a Gröbner basis for
a homogenous ideal in a multigraded polynomial ring. Our method uses anti-
chains in the lattice $\mathbb{N}^{k}$ to separate mutually independent
S-polynomials for reduction.

##  1\. Introduction

In this paper we present a way of parallelizing the Buchberger algorithm for
computing Gröbner bases in the special case of multihomogeneous ideals in the
polynomial algebra over a field. We describe our algorithm as well as our
implementation of it. We also present experimental results on the efficiency
of our algorithm, using the ideal of commuting matrices as illustration.

###  1.1. Motivation

Most algorithms in commutative algebra and algebraic geometry at some stage
involve computing a Gröbner basis for an ideal or module. This ubiquity
together with the exponential complexity of the Buchberger algorithm for
computing Gröbner bases of homogeneous ideals explains the large interest in
improvements of the basic algorithm.

###  1.2. Prior Work

Several approaches have been tried in the literature. Some authors, such as
Chakrabarti–Yelick [1] and Vidal [2] have constructed general algorithms for
distributed memory and shared memory machines respectively. Reasonable
speedups were achieved on small numbers of processors. Another approach has
been using factorization of polynomials; all generated S-polynomials are
factorized on a master node, and the reductions of its factors are carried out
on the slave nodes. Work by Siegl [3], Bradford [4], Gräbe and Lassner [5]. In
a paper by Leykin [6] a coarse grained parallelism was studied that was
implemented both in the commutative and non-commutative case.

Good surveys of the various approaches can be found in papers by Mityunin and
Pankratiev [7] and Amrhein, Bündgen and Küchlin [8]. Mityunin and Pankratiev
also give a theoretical analysis of and improvements to algorithms known at
that time.

Finally, an approach by Reeves [9] parallelizes on the compiler level for
modular coefficient fields.

###  1.3. Our approach

Our approach restricts the class of Gröbner bases treated to homogenous
multigraded Gröbner bases. While certainly not general enough to handle all
interesting cases, the multigraded case covers several interesting examples.
For these ideals we describe a coarsely grained parallelization of the
Buchberger algorithm with promising results.

######  Example 1.1.

Set $R=k[x_{1},\dots,x_{n^{2}},y_{1},\dots,y_{n^{2}}]$ where $k$ is a field.
Let $X$ and $Y$ be square $n\times n$-matrices with entries the variables
$x_{1},\dots,x_{n^{2}}$ and $y_{1},\dots,y_{n^{2}}$ respectively. Then the
entries of the matrix

$I_{n}=XY-YX$

form $n^{2}$ polynomials generating the ideal $I_{n}$ $R$.

The computation of a Gröbner basis for $I_{1}$ and $I_{2}$ is trivial and may
be carried out on a blackboard. A Gröbner basis for $I_{3}$ is a matter of a
few minutes on most modern computer systems, and already the computation of a
Gröbner basis for $I_{4}$ is expensive using the standard reverse
lexicographic term order in $R$; the Macaulay2 system [10] several hours are
needed to obtain a Gröbner basis with 563 elements. However, using clever
product orders, Hreinsdóttir has been able to find bases with 293 and 51
elements [11, 12]. As far as we are aware of, a Gröbner basis for $I_{5}$ is
not known.

By assigning multidegrees $(1,0)$ to all the variables $x_{1},\dots,x_{n^{2}}$
and $(0,1)$ to all the variables $y_{1},\dots,y_{n^{2}}$, the ideal $I_{n}$
becomes multigraded over $\mathbb{N}\times\mathbb{N}$, and thus approachable
with our methods.

######  Example 1.2.

While this paper presents only the approach to multigraded ideals in a
polynomial ring, an extension to free multigraded modules over multigraded
rings is easily envisioned, and will be dealt with in later work.

Gröbner bases for such free modules would be instrumental in computing
invariants from applied algebraic topology such as the _rank invariant_ as
well as more involved normal forms for higher dimensional persistence
modules.[13]

##  2\. Partially ordered monoids

We shall recall some definitions and basic facts about partially ordered sets
that will be of fundamental use in the remainder of this paper.

A partially ordered set is a set equipped with a binary, reflective, symmetric
and transitive _order_ operation $\leq$. Two objects $a,b$ such that either
$a\leq b$ or $b\leq a$ are called _comparable_ , and two objects for which
neither $a\leq b$ nor $b\leq a$ are called _incomparable_. A subset $A$ of a
partially ordered set in which all objects are mutually incomparable is called
an _antichain_. An element $p$ is _minimal_ if there are no distinct $q$ with
$q\leq p$, _maximal_ if there are no distinct $q$ with $p\leq q$, _smallest_
if all other $q$ fulfill $p\leq q$ and _largest_ if all other $q$ fulfill
$q\leq p$.

There is a partially ordered set structure on $\mathbb{N}^{d}$ in which
$(a_{1},\dots,a_{d})\leq(b_{1},\dots,b_{d})$ iff $a_{i}\leq b_{i}$ for all
$i$. This structure is compatible with the monoid structure on
$\mathbb{N}^{d}$ in the sense that if $\bar{a}\leq\bar{b}$, then
$\bar{c}*\bar{a}\leq\bar{c}*\bar{b}$ for
$\bar{a},\bar{b},\bar{c}\in\mathbb{N}^{d}$. If a monoid has a partial order
compatible with the multiplication in this manner, we call it a _partially
ordered monoid_.

In a partially ordered set $P$, we say that a subset $Q$ is an _ideal_ if it
is downward closed, or in other words if for any $p\in P,q\in Q$ such that
$p\leq q$, then $p\in Q$. It is called a _filter_ if it is upward closed, or
if for $p\in P,q\in Q$ such that $q\leq p$, then $p\in Q$.

An element $p$ is maximal in an ideal if any element $q$ such that $p\leq q$
is not a member of the ideal. Minimal elements of filters are defined
equivalently. An ideal (filter) is _generated_ by its maximal (minimal)
elements in the sense that the membership condition of the ideal (filter) is
equivalent to being larger than (smaller than) at least one generator.
Generators of an ideal or filter form an antichain. Indeed, if these were not
an antichain, two of them would be comparable, and then one of these two would
not be maximal (minimal). An ideal (filter) is _finitely generated_ if it has
finitely many generators, and it is _principal_ if it has exactly one
generator.

There is a partially ordered monoid structure on $\mathbb{N}^{d}$, given by
$(p_{1},\dots,p_{d})\leq(q_{1},\dots,q_{d})$ if $p_{i}\leq q_{i}$ for all $i$,
and by
$(p_{1},\dots,p_{d})*(q_{1},\dots,q_{d})=(p_{1}+q_{1},\dots,p_{d}+q_{d})$.
This structure will be the main partially ordered monoid in us in this paper.

##  3\. Multigraded rings and the grading lattice

A polynomial ring $R=\mathbb{k}[x_{1},\dots,x_{r}]$ over a field $\mathbb{k}$
is said to be _multigraded_ over $P$ if each variable $x_{j}$ carries a
_degree_ $|x_{j}|\in P$ for some partially ordered monoid $P$. We expect of
the partial order on $P$ that if $p,q\in P$ then $p\leq p*q$ and $q\leq p*q$.
The degree extends from variables to entire monomials by requiring
$|mn|=|m|*|n|$ for monomials $m,n$; and from thence a multigrading of the
entire ring $R$ follows by decomposing $R=\bigoplus_{p\in P}R_{p}$ where
$R_{p}$ is the set of all _homogenous_ polynomials in $R$ of degree $p$, i.e.
polynomials with all monomials of degree $p$. A homogenous polynomial of
degree $(n_{1},\dots,n_{d})$ is said to be of _total degree_
$n_{1}+\dots+n_{d}$. We note that for the $\mathbb{N}^{d}$-grading on $R$, the
only monomial with degree $(0,0,\dots,0)$ is $1$, and thus the smallest degree
is assigned both to the identity of the grading monoid and to the identity of
the ring.

We write $\operatorname{lm}p$, $\operatorname{lt}p$, $\operatorname{lc}p$ for
the leading monomial, leading term and leading coefficient of $p$.

######  Proposition 3.1.

Suppose $p$ and $q$ are homogenous. If $|p|\not\leq|q|$ then
$\operatorname{lm}p$ does not divide $\operatorname{lm}q$.

###### Proof.

If $\operatorname{lm}p|\operatorname{lm}q$ then
$\operatorname{lm}q=c\operatorname{lm}p$, and thus
$|\operatorname{lm}q|=|c|*|\operatorname{lm}p|$, and thus, since $|1|\leq|c|$,
by our requirement for a partially ordered monoid,
$\operatorname{lm}p\leq\operatorname{lm}q$. ∎

######  Proposition 3.2.

Reduction in the Buchberger algorithm of a given multidegree for a homogenous
generating set depends only on its principal ideal in the partial order of
degrees.

###### Proof.

We recall that the reduction of a polynomial $p$ with respect to polynomials
$q_{1},\dots,q_{k}$ is given by computing

$p^{\prime}=p-\frac{\gcd(\operatorname{lm}p,\operatorname{lm}q_{j})}{\operatorname{lm}p}q_{j}$

for a polynomial $q_{j}$ such that
$\operatorname{lm}q_{j}|\operatorname{lm}p$. We note that by Proposition 3.1,
this implies $|\operatorname{lm}q_{j}|\leq|\operatorname{lm}p|$ and thus
$|q_{j}|\leq|p|$. ∎

We note that Proposition 3.2 implies that if two S-polynomials are
incomparable to each other, then their reductions against a common generating
set are completely independent of each other. Furthermore, since
$|p^{\prime}|=|p|$, in the notation of the proof of Proposition 3.2, a
reduction of an incomparable S-polynomial can never have an effect on the
future reductions of any given S-polynomial.

Hence, once S-polynomials have been generated, their actual reductions may be
computed independently across antichains in the partial order of multidegrees,
and each S-polynomial only has to be reduced against the part of the Gröbner
basis that resides below it in degree.

##  4\. Algorithms

The arguments from Section 3 lead us to an approach to parallelization in
which we partition the S-polynomials generated by their degrees, pick out a
minimal antichain, and generate one computational task for each degree in the
antichain.

One good source for minimal antichains, that is guaranteed to produce an
antichain, though most often will produce more tasks than are actually
populated by S-polynomials is to consider the minimal total degree for an
unreduced S-polynomial, and produce as tasks the antichain of degrees with the
same total degree.

Another, very slightly more computationally intense method is to take all
minimal occupied degrees. These, too, form an antichain by minimality, and are
guaranteed to only yield as many tasks as have content.

Either of these suggestions leads to a master-slave distributed algorithm as
described in pseudocode in Algorithms 1 and 2.

The resulting master node algorithm can be seen in Algorithm 1, and the
simpler slave node algorithm in Algorithm 2.

loop

if have waiting degrees and waiting slaves then

nextdeg $\leftarrow$ pop(waiting degrees)

nextslave $\leftarrow$ pop(waiting slaves)

send nextdeg to nextslave

else if all slaves are working then

wait for message from slave newslave

push(newslave, waiting slaves)

else if no waiting degrees and some working slaves then

wait for message from slave newslave

push(newslave, waiting slaves)

else if no waiting degrees and no working slaves then

generate new antichain of degrees

if no such antichain available then

finish up

end if

else

continue

end if

end loop

Algorithm 1 Master algorithm for a distributed Gröbner basis computation

loop

receive message msg from master

if msg = finish then

return

else if msg = new degree $d$ then

reduce all S-polynomials in degree d and append to Gröbner basis

compute new S-polynomials based on new basis elements

send finished degree to master

end if

end loop

Algorithm 2 Slave algorithm for a distributed Gröbner basis computation

##  5\. Experiments

We have implemented the master-slave system described in Section 4 in Sage
[14], using MPI for Python [15, 16] for distributive computing infrastructure
and SQLAlchemy [17] interfacing with a MySQL database [18] for an abstraction
of a common storage for serialized python objects.

In order to test our implementation, we have used a computational server
running 8 Intel Xeon processors at 2.83 GHz, with a 5M cache, and a total RAM
available of 16G.

We have run test with the Gröbner basis problem $I_{3}$, and recorded total
wallclock timings, as well as specific timings for the S-polynomial generation
and reduction steps. The problem was run for each possible number of allocated
core (1 to 7 slave processors), and the server was the entire time otherwise
un-utilized.

Figure 1. Timings (seconds) for $I_{3}$ on an 8-core computational server.
Timing runs were made with between 2 and 8 active processors, and the total
wallclock times (top left), SQL interaction times (top right, the S-polynomial
reduction times (bottom left), and the S-polynomial generation times (bottom
right) were measured.

Figure 2. Logarithms of maximal, minimal and average timings (seconds) for
reduction (left) and generation (right) in the $I_{3}$ computations.

As can be seen in Figure 1, parallelization decreases the wall-clock timings
radically compared to single-core execution (2 processors, with the slave
processor doing all work essentially serially). However, the subsequent
decrease in computational times is less dramatic.

Looking into specific aspects of the computation, we can see that while the
average computational times decrease radically with the number of available
processors, the maximal computation time behaves much worse. With the
reduction step, maximal computation times still decrease, mostly, with the
number of available processors. The S-polynomial generation step however
displays almost constant maximal generation times along the computation.

Furthermore, compared to the time needed for the algebraic computations, the
relatively slow, database engine mediated storage and recovery times are
almost completely negligible.

These trends are even more clear when we concentrate on only the maximal,
minimal and average computation times, as in Figure 2. We see a proportional
decrease in average computation times, and a radical drop-off in minimal
computation times, which certainly sounds promising. The global behaviour,
however, is dictated by the maximal thread execution times, which are rather
disappointingly behaving throughout.

##  6\. Conclusions and Future Work

In conclusion, we have demonstrated that while the parallel computation of
Gröbner bases in general is a problem haunted by the ghost of data dependency,
the lattice structure in an appropriate choice of multigrading will allow for
easy control of dependencies. Specifically, picking out antichains in the
multigrading lattice gives a demonstrable parallelizability, that saturates
the kind of computing equipment that is easily accessible by researchers of
today.

Furthermore, we have developed our methods publically
accessible,111http://code.google.com/p/dph-mg-grobner, the code used for
Section 5 can be found in the sage subdirectory and released it under the very
liberal BSD license. Hence, with the ease of access to our code and to the
Sage computing system, we try to set the barrier to build further on our work
as low as we possibly can.

However, the techniques we have developed here are somewhat sensitive to the
distribution of workload over the grading lattice: if certain degrees are
disproportionately densely populated, then the computational burden of an
entire Gröbner basis is dictated by the essentially _serial_ computation of
the highly populated degrees. As such, we suspect these methods to work at
their very best in combination with other parallelization techniques.

The Gröbner basis implementation used was a rather naïve one, and we fully
expect speed-ups from sophisticated algorithms to combine cleanly with the
constructions we use. This is something we expect to examine in future
continuation of this project.

There are many places to go from here. We are ourselves interested in
investigating many avenues for the further application of the basic ideas
presented here:

  * •

Adaptation to state-of-the-art Gröbner basis techniques for single processors.
Improve the handling of each separate degree, potentially subdividing work
even finer.

  * •

Multigraded free modules, and Gröbner bases of these; opening up for the use
of these methods in computational and applied topology, as a computational
back bone for multigraded persistence.

  * •

Multigraded free resolutions; opening up for the application of these methods
in parallelizing computations in homological algebra.

  * •

Adaptation to non-commutative cases; in particular to use for ideals in and
modules over quiver algebras.

  * •

Building on work by Dotsenko and Khoroshkin, and by Dotsenko and Vejdemo-
Johansson, there is scope to apply this parallelization to the computation of
Gröbner bases for operads. [19, 20]

## References

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# Quantum torsion and a Hartle-Hawking “beam”

João Magueijo<EMAIL_ADDRESS>Theoretical Physics Group, The
Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ,
United Kingdom  Tom Zlosnik<EMAIL_ADDRESS>CEICO, Institute of Physics of the
Czech Academy of Sciences, Na Slovance 1999/2, 182 21, Prague

###### Abstract

In the Einstein-Cartan framework the torsion-free conditions arise within the
Hamiltonian treatment as second-class constraints. The standard strategy is to
solve these constraints, eliminating the torsion from the classical theory,
before quantization. Here we advocate leaving the torsion inside the other
constraints before quantization, leading at first to wave functions that can
be called “kinematical” with regards to the torsion, but not the other
constraints. The torsion-free condition can then be imposed as a condition
upon the physical wave packets one constructs, satisfying the usual
uncertainty relations, and so with room for quantum fluctuations in the
torsion. This alternative strategy has the surprising effect of clarifying the
sense in which the wave functions solving an explicitly real theory are
“delta-function normalizable”. Such solutions with zero (or any fixed)
torsion, should be interpreted as plane waves in torsion space. Properly
constructed wave packets are therefore normalizable in the standard sense.
Given that they are canonical duals, this statement applies equally well to
the Chern-Simons state (connection representation) and the Hartle-Hawking wave
function (metric representation). We show how, when torsion is taken into
account, the Hartle-Hawking wave function is replaced by a Gauss-Airy
function, with finite norm, which we call the Hartle-Hawking beam. The Chern-
Simons state, instead, becomes a packet with a Gaussian probability
distribution in connection space. We conclude the paper with two sections
explaining how to generalize these results beyond minisuperspace.

##  I Introduction

Classical general relativity may be variously formulated giving primacy to the
metric (the second order formalism), or allowing the metric and the connection
to be initially independent variables (the first order formalism). The latter
may be motivated in many different ways (e.g. Kibble ; Hehl ), but
historically has its roots in the Einstein-Cartan formalism. In the minimal
theories, in the absence of spinors, the first and second order formulations
are classically equivalent, since one of the Einstein-Cartan equations of
motion relates the torsion to the spin current (which we will assume vanishes
throughout this paper). A question arises concerning the quantum mechanics of
the two formulations: are they equivalent? In the Einstein-Cartan formalism
zero-torsion appears classically as an equation of motion (and not a built-in
fact, as is the case in the metric formalism, with a Christoffel connection).
Quantum mechanics probes the phase space off-shell, i.e. away from the
classical solutions, so it could feel the non-zero torsion possibility of
Einstein-Cartan theory. Could quantum torsion fluctuations lead to an
inequivalent quantum theory?

A closer inspection reveals that the issue is technically non-trivial. In the
canonical framework (used in most quantization schemes), the torsion-free
condition appears as part of the secondary constraints, which turn out to be
second class (e.g. notimegauge ; realK ). Quantization using the usual Poisson
bracket is therefore inconsistent. One approach to second class constraints is
to employ the Dirac bracket in lieu of the Poisson bracket Diracbk . Another
approach is, if at all possible, to solve the constraints classically,
inserting the resulting equations of motion into the action before quantizing.
This is usually the route taken regarding the torsion. But is this right? Is
this the only path to quantization? It certainly imposes a freeze on torsion
quantum fluctuations. This, we shall argue in this paper, is the origin of
some apparent infinite norms found in the usual approach.

We have in mind the fact that some well-known solutions in quantum gravity are
not normalizeable in the conventional sense. This can be seen both in the
metric and connection representation, since the norm is independent of the
representation. In the connection representation, the matter concerns the
Chern-Simons wave function (also called the Kodama state Jackiw (1983); Witten
(2003); Kodama (1990); Smolin (2002); Freidel and Smolin (2004)). Here, we
must distinguish two types of normalization problems. If we take the state
that solves the Lorentzian complex Hamiltonian equation, then is suffers from
a number of acknowledged pathologies (e.g. Witten (2003)): CPT violating
properties (and consequent impossibility of a positive energy property), lack
of gauge invariance under large gauge transformations, and, indeed, non-
normalizability with respect to the standard inner product. However, if one
employs an explicitly real theory (phase space variables and action) these
problems disappear, as shown in realK ; MSS . Then, the “real” Chern-Simons
state is a pure phase in the Lorentzian signature (just like in the Euclidean
signature), and so it is “delta-function” normalizable Randono0 ; Randono1 ;
Randono2 . It is in this last format that we want to examine the
normalizability of the Chern-Simons state.

But foremost, we want to highlight the fact that this is nothing but the dual
of a seemingly unrelated problem. The Hartle-Hawking (HH) wave function is the
Fourier dual of the Chern-Simons state reduced to mini-superspace, assuming
real variables throughout Magueijo (2020). Its normalizability issues are
well-known. We may take the view that one needs a higher dimensional space to
measure and probability, from a Klein-Gordon current, as suggested by Vilenkin
Vilenkin (1988, 1994). But we may also take the problem face value and wonder
what it means. Flipping the question to the Fourier dual, the issue is the
same as the one for the Chern-Simons state, and the question becomes: what
does it mean to say that the Hartle-Hawking/Chern-Simons wave functions are
“delta-function” normalizable? With respect to what?

“Quantum torsion” provides an answer, as we show in this paper. We start by
reviewing the mini-superspace structures behind the equivalence of the Chern-
Simons and Hartle-Hawking states: complementary variables, Fourier transform
and integration measure (Section II). We also explain how the torsion is
usually forced to be zero by means of second class constraints. We then show
how solutions to the quantum constraints may be obtained by first ignoring the
torsion conditions (Section III). This leads to torsionful versions of the
Hartle-Hawking and Chern-Simons states. Computing the norms of these states,
at once we see why the result is infinite if the torsion is fixed (at zero or
otherwise), and proportional to a delta-function in terms of the torsion
(Section IV).

With this important piece of information in hand we may now build wave packets
in torsion space, which are normalizable in the conventional sense (Section
V). We may also build coherent states centred at zero. These are the physical
states with regards to the torsion conditions. We build these states
explicitly in both representations (Section VI). The uniform probability
extended over an infinite domain, implied by the unbridled Chern-Simons state,
is replaced by a Gaussian distribution. The Hartle-Hawking wave function is
replaced by a regularized Gauss-Airy function: the Hartle-Hawking beam.
Suddenly it all makes sense.

We close with two sections explaining how to extend our construction beyond
mini-superspace. They do not impact on our main results, but may be an
important first step towards the phenomenology of quantum torsion.

##  II Review of mini-superspace structures

The Einstein-Cartan action reduced to homogeneity and isotropy is Magueijo
(2020); Magueijo and Zlosnik (2019):

$S=6\kappa V_{c}\int
dt\bigg{(}a^{2}\dot{b}+Na\bigg{(}b^{2}+k-c^{2}-\frac{\Lambda}{3}a^{2}\bigg{)}\bigg{)}.$
(1)

where $\kappa=1/(16\pi G_{N})$, $k=0,\pm 1$ is the normalized spatial
curvature, and $\Lambda$ is the cosmological constant. Here $a$ is the
expansion factor (the only metric variable), and $b$ and $c$ are components of
the connection, respectively the off-shell version of the Hubble parameter
(since $b\approx\dot{a}$ if there is not torsion; see below), and the parity-
violating Cartan spiral staircase Lazar and Hehl (2010); Magueijo and Zlosnik
(2019). The Lagrange multiplier $N$ is the lapse function.

Hence, the Poisson bracket is:

$\\{b,a^{2}\\}=\frac{1}{6\kappa V_{c}},$ (2)

inducing the mini-superspace commutator:

$\left[\hat{b},\hat{a^{2}}\right]=\frac{il_{P}^{2}}{3V_{c}},$ (3)

where $l_{P}=\sqrt{8\pi G_{N}\hbar}$ is the reduced Planck length. Given that
it will appear recurrently, to simplify the notation we define:

$\mathfrak{h}=\frac{l_{P}^{2}}{3V_{c}}.$ (4)

We may also see $\mathfrak{h}$ as an “effective” Planck constant Barrowhbar ,
but we do not need to accept this interpretation.

The canonical structure and commutator imply the Fourier transform and its
inverse:

$\displaystyle\psi_{a^{2}}(a^{2})$ $\displaystyle=$
$\displaystyle\int\frac{db}{\sqrt{2\pi\mathfrak{h}}}e^{-\frac{i}{\mathfrak{h}}a^{2}b}\psi_{b}(b),$
$\displaystyle\psi_{b}(b)$ $\displaystyle=$
$\displaystyle\int\frac{da^{2}}{\sqrt{2\pi\mathfrak{h}}}e^{\frac{i}{\mathfrak{h}}a^{2}b}\psi_{a}(a^{2}).$
(5)

for changing between duals (where we have used a symmetric definition for FT
and its inverse). The associated inner product between states and integration
measure are the trivial:

$\displaystyle\langle\psi|\phi\rangle$ $\displaystyle=$ $\displaystyle\int
db\,\psi^{\star}_{b}(b)\phi_{b}(b)$ (6) $\displaystyle=$ $\displaystyle\int
da^{2}\,\psi^{\star}_{a^{2}}(a^{2})\phi_{a^{2}}(a^{2}).$ (7)

All integrations will be understood to be over the whole real line here and
from now on; we will mention alternatives at the end of the paper. The Fourier
transform can then be seen as an insertion of a partition of identity with:

$\displaystyle\psi_{a^{2}}(a^{2})$ $\displaystyle=$ $\displaystyle\langle
a^{2}|\psi\rangle$ (8) $\displaystyle\psi_{b}(b)$ $\displaystyle=$
$\displaystyle\langle b|\psi\rangle$ (9)

and

$\langle
b|a^{2}\rangle=\frac{e^{\frac{i}{\mathfrak{h}}a^{2}b}}{\sqrt{2\pi\mathfrak{h}}}$
(10)

As explained in Magueijo (2020) (ignoring torsion), the solution to the
Hamiltonian constraint in the $a^{2}$ representation is the Hartle-Hawking
wave function, whereas in the $b$ representation it is the Chern-Simons state.
The two are the Fourier transform of each other (with the definitions of
measure and Fourier transform induced by the canonical structure, which we
have just presented). The theory’s Hamiltonian has the form $H=6\kappa
V_{c}Na{\cal H}$ with:

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle-b^{2}-k+c^{2}+\frac{\Lambda}{3}a^{2}.$ (11)

But what about the conditions forcing the torsion to vanish classically?

In the canonical framework we should distinguish between two different types
of torsion. One type of torsion is that contained in $b$, the mini-superspace
reduction of the extrinsic curvature (the space-time components of the spin-
connection Alexander et al. (2019b); Magueijo and Zlosnik (2019), also denoted
$K^{i}$ in the Ashtekar formalism realK ). Specifically:

$b=\dot{a}/N-T$ (12)

the first term containing the torsion-free component, the second the torsion.
The Hamilton equation for $a$ reads:

$\dot{a}=Nb$ (13)

so that it fixes the torsion $T$ to zero on-shell, as part of the time
evolution.

A quite different type of torsion is that contained in $c$, the mini-
superspace version of the torsion contained in the purely spatial components
of the spin-connection (denoted $\Gamma^{i}$ in the Ashtekar formalism realK
). In mini-superspace (and more generally in the time gauge) $c$ (the torsion
in $\Gamma^{i}$) does not have a momentum. This can be phrased by adding to
the action a term corresponding to the Legendre transform between $c$ and its
conjugate momentum $p_{c}$, plus a constraint forcing $p_{c}$ to be zero:

$S\rightarrow S+\int dt(\dot{c}p_{c}-\lambda p_{c})$ (14)

Then,

$\\{c,p_{c}\\}=1$ (15)

but since:

$\\{p_{c},{\cal H}\\}=2c$ (16)

we get the secondary constraint:

$c\approx 0.$ (17)

However because of (15), the two new constraints ($c\approx 0$ and
$p_{c}\approx 0$) are second class constraints.

The usual argument is that second class constraints should be solved
classically, before quantization, because they cannot be imposed consistently
at the quantum mechanical level111Certainly we cannot impose
${\hat{c}}\psi={\hat{p}_{c}}\psi=0$, in whatever rep. Note that if we imposed
only $p_{c}\approx 0$ that would be fine, and only state that the wave
function could not depend on $c$, forcing a uniform distribution in $c$. Would
this be inconsistent with “observation”?. Thus one sets $c=0$ in the
Hamiltonian equation, quantizes, and gets on with it. But is this the only way
to quantize? Certainly not (we recall Gupta Bleuler quantization). Here we
propose an alternative, which happens to explain what is meant by delta-
function normalization of the Chern-Simons and Hartle-Hawking wave functions.

##  III Kinematical torsionful wave functions

Let us solve the quantum Hamiltonian constraint, leaving the $c$ there, to be
seen as a parameter, just like the (torsion-free) spatial curvature $k=0,\pm
1$. As in Magueijo (2020) we find that with suitable ordering, the solution to
the Hamiltonian constraint in the $b$ representation is the torsionful version
of the Chern-Simons state in mini-superspace:

$\psi_{CS}(b,c)={\cal
N}\exp{\bigg{(}i\frac{3}{\Lambda\mathfrak{h}}\left(\frac{b^{3}}{3}+(k-c^{2})b\right)\bigg{)}}.$
(18)

In the $a^{2}$ representation, we find a straightforward modification of the
Wheeler-DeWit equation, leading (with simple adaptations from Magueijo (2020))
to the torsionful version of the Hartle-Hawking wave function222Note that
there is a typo in the definition of $z$ in Magueijo (2020). :

$\psi_{HH}(b,c)={\cal N}^{\prime}{\rm Ai}(-z),$ (19)

with

$z=-\left(\frac{3}{\Lambda\mathfrak{h}}\right)^{2/3}\left(k-c^{2}-\frac{\Lambda
a^{2}}{3}\right).$ (20)

Using standard results in the theory of Airy functions we have:

${\cal N}^{\prime}={\cal
N}\sqrt{\frac{2\pi}{\mathfrak{h}}}\left(\frac{\Lambda\mathfrak{h}}{3}\right)^{1/3}.$
(21)

As in Magueijo (2020) we may argue that there is only one state (once the
matter of the domains of variation is fixed), taking the form of the Chern-
Simons or Hartle-Hawking wave functions depending on the chosen
representation. The only novelty here is that the state is index by the
torsion $c$. We denote this state $|\Psi_{HC}(c)\rangle$ and in a
representation-free notation we may write:

$\hat{\cal H}(a^{2},b;c)|\Psi_{HC}(c)\rangle=0.$ (22)

The explicit representations result from:

$\displaystyle\langle b|\Psi_{HC}(c)\rangle$ $\displaystyle=$
$\displaystyle\psi_{CS}(b,c)$ (23) $\displaystyle\langle
a^{2}|\Psi_{HC}(c)\rangle$ $\displaystyle=$ $\displaystyle\psi_{HH}(a^{2},c)$
(24)

and proving that $\psi_{CS}(b,c)$ and $\psi_{HH}(a^{2},c)$ are the Fourier
transform of the other is then a particular case of the insertion of a
partition of identity as defined above.

These wave functions may be called “kinematical” with regards to the torsion,
because we have not yet imposed the conditions which we know they must
satisfy, in some sense, regarding the torsion. Before proposing a prescription
for these conditions (Section V), we show how the kinematical wave functions
clarify the important issue of the normalizability of the Chern-Simons state
and the Hartle-Hawking wave function.

##  IV Normalizability of the Chern-Simons and Hartle-Hawking states

We start by examining the sense in which the “real” Chern-Simons state realK ,
which is a pure phase in Lorentzian signature (just like its Euclidean
cousin), is “delta-function” normalizable. The state certainly resembles a
wave extending over an infinite domain (in $b$, in mini-superspace; or in
$a^{2}$ for its dual). Such waves, spread over an infinite domain, are
regularizable by confining them into a finite box, but this would be
equivalent to truncating the domain of $b$ (or $a^{2}$) and discretizing
$a^{2}$ (or $b$), clearly a very crude expedient.

A more palatable alternative consists of taking the kinematical torsionful
wave functions and evaluate their inner product for generic torsion values.
Using (6) and (18) we find, in the $b$ representation:

$\displaystyle\langle\Psi_{HC}(c)|\Psi_{HC}(c^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\int
db\,\psi^{\star}_{CS}(b,c)\psi_{CS}(b,c^{\prime})$ (25) $\displaystyle=$
$\displaystyle{\cal N}^{2}\int
db\,\exp{\bigg{(}i\frac{3}{\Lambda\mathfrak{h}}\left((c^{2}-c^{\prime
2})b\right)\bigg{)}}$ $\displaystyle=$ $\displaystyle
2\pi\frac{\Lambda\mathfrak{h}}{3}{\cal N}^{2}\delta(c^{2}-c^{\prime 2})$

(note how the cubic term in $b$ in the phase cancels, leaving the right
factors for a delta function). By choosing:

${\cal N}=\sqrt{\frac{3}{2\pi\Lambda\mathfrak{h}}}$ (26)

we therefore have:

$\langle\Psi_{HC}(c)|\Psi_{HC}(c^{\prime})\rangle=\delta(c^{2}-c^{\prime 2})$
(27)

giving a clear implementation of the statement that the real Chern-Simons wave
functions are “delta-normalizable”.

We now understand why even the non-pathological real Chern-Simons state, which
is a pure phase, has infinite norm:

$\int db\,|\psi_{CS}(b,c)|^{2}={\cal N}\int db=\infty$ (28)

for any torsion $c$, and in particular for $c=0$. It is because

$\int
db\,|\psi_{CS}(b,c)|^{2}=\langle\Psi_{HC}(c)|\Psi_{HC}(c)\rangle=\delta(0)=\infty.$

In particular if we force the torsion to be zero by construction (as in the
usual approach to the second class constraints associated with it), we have:

$\langle\Psi_{HC}(c\equiv 0)|\Psi_{HC}(c\equiv 0)\rangle=\infty.$ (29)

By building wave packets in $c$ (possibly centred at $c=0$, but with a width),
as we shall do in the next Section, the norm becomes finite, in perfect
analogy with delta-normalizable plane waves and their wave packets.

Having gone this far, we note that norms do not depend on the representation,
so we can translate everything we said so far into the metric representation,
and see what it implies for the Hartle-Hawking wave function. It is known that
the (torsion-free) Hartle-Hawking state is not normalizable:

$\int da^{2}\,|\psi_{HH}|^{2}={\cal N}^{\prime 2}\int da^{2}\,{\rm
Ai}^{2}(z)=\infty,$ (30)

a matter which has led to several probability interpretations(see Vilenkin
(1994, 1988) and alsoHalliwell  and references therein). Without discussing
their merits, we note that this non-normalizability is just an expression of
(29) in the metric representation, and results from fixing the torsion to zero
too abruptly and rigidly, just as was found in the connection representation.
And likewise, the problem can be removed by introducing the torsionful Hartle-
Hawking wave functions and note that these are already delta-function
normalized in the sense of (27) (since the norm does not depend on the
representation). Finite norm versions of the the Hartle-Hawking wave function
can then be found constructing wave packets in the torsion out of these delta-
normalized torsionful waves.

Just to make sure, we double check our statement with a direct calculation.
Bearing in mind the identity:

$\int dz{\rm Ai}(z+x){\rm Ai}(z+y)=\delta(x-y),$ (31)

we can evaluate (27) in the $a^{2}$ representation as:

$\displaystyle\langle\Psi_{HC}(c)|\Psi_{HC}(c^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\int
da^{2}\psi^{\star}_{HH}(a^{2},c)\psi_{HH}(a^{2},c^{\prime})$ (32)
$\displaystyle=$
$\displaystyle\frac{3}{\Lambda}\left(\frac{\Lambda\mathfrak{h}}{3}\right)^{4/3}{\cal
N}^{\prime 2}\delta(c^{2}-c^{\prime 2}).$

Hence we recover (27) if:

${\cal
N}^{\prime}=\sqrt{\frac{\Lambda}{3}}\left(\frac{3}{\Lambda\mathfrak{h}}\right)^{2/3}$
(33)

which could have been found to have this value directly from (21) and (26).

##  V Wave packets and the torsion condition

We can now consider general normalizable solutions to the Hamiltonian
constraint. The $|\Psi_{HC}(c)\rangle$ can be seen as monochromatic waves (and
so have infinite norm), but by linearly superposing them into wave packets one
arrives at normalizable solutions, since they are delta-function normalized.
Such packets solve the Hamiltonian constraint because the Hamiltonian is
linear and does not contain $p_{c}$ (so, multiplying each
$|\Psi_{HC}(c)\rangle$ by an amplitude, $A(c)$, dependent on $c$, still
produces a solution).

Given the form of the delta-function normalization (cf. Eq. (27)), the most
convenient measure with which to label the superposition is $dc^{2}$. Thus, we
arrive at:

$|\phi\rangle=\int dc^{2}A(c^{2})|\Psi_{HC}(c)\rangle.$ (34)

The normalization condition $\langle\phi|\phi\rangle=1$ becomes the statement:

$\int dc^{2}|A(c^{2})|^{2}=1,$ (35)

so, any suitably chosen $A(c^{2})$ leads to well defined probabilities. These
packets may be written in the $a^{2}$ or $b$ representation, applying to them
(8) or (9), i.e.:

$\displaystyle\phi(a^{2})$ $\displaystyle=$ $\displaystyle\int
dc^{2}A(c^{2})\psi_{HH}(a^{2},c)$ (36) $\displaystyle\phi(b)$ $\displaystyle=$
$\displaystyle\int dc^{2}A(c^{2})\psi_{CS}(b,c).$ (37)

The amplitude $A(c^{2})$ is the same in both representations.

We are finally ready to construct states compliant with a quantum version of
the torsion-free condition. They should be coherent states centred at $c=0$.
Given that $c$ does not have a momentum there is some arbitrariness in the
construction. Specifically, in order to fit the measure $dc^{2}$, we could
have replaced (14) by333Note that this choice implies that the term $c^{2}$ in
the Hamiltonian density is now linear in the configuration variable which may
alternatively be regarded as a term proportional to the momentum _of_
$p_{c^{2}}$. Models containing such contributions can be candidates for
emergent clocks in quantum gravity Gielen:2020abd  but we will not explore
this further in the present paper.:

$S\rightarrow S+\int dt(\dot{c^{2}}p_{c^{2}}-\lambda p_{c^{2}})$ (38)

suggesting we build coherent states from:

$[\hat{c}^{2},\hat{p}_{c^{2}}]=i\hbar\implies\hat{p}_{c^{2}}=-i\hbar\frac{\partial}{\partial
c^{2}}.$ (39)

As with the case for free-particles (lacking a potential term in the
Hamiltonian), the fact that $p_{c}$ does not appear in the Hamiltonian means
there is a free length scale, $\ell$, required to define dimensionless
quadratures and “annihilation” operators (see freecoh , for a very pedagogical
review). Thus, coherent states will be eigenstates of:

$\hat{Z}=\frac{\ell^{2}}{\sqrt{2}}\hat{c^{2}}+i\frac{1}{\hbar\ell^{2}\sqrt{2}}\hat{p}_{c^{2}}=\frac{\ell^{2}}{\sqrt{2}}c^{2}+\frac{1}{\ell^{2}\sqrt{2}}\frac{\partial}{\partial
c^{2}}.$ (40)

The scale $\ell$ can be the Planck length, Lambda’s length scale
$|\Lambda|^{-1/2}$, or any function thereof (we could also appeal to $V_{c}$).
In particular, we can investigate:

$\ell=\left(l_{P}^{n}\Lambda^{-m/2}\right)^{\frac{1}{n+m}}.$ (41)

The case $n=2$ and $m=1$ is studied in RandoMeso ; Barrowhbar  and leads to
interesting mesoscopic implications. The choice $n=1$ and $m=2$ leads to
effects on the scale of $10^{2}$ Hz, of possible interest for gravity wave
detection. The fact that $\ell$ is not fixed is not surprising and we will
comment on it at the end of this section, once coherent states are
constructed.

The torsion-free condition may be implemented by constructing coherent states
centred at the origin, i.e. by solving $\hat{Z}A(c^{2})=0$, with $\hat{Z}$ as
in (40). These have the form:

$A(c^{2})=\frac{\exp{\left(-\frac{c^{4}}{4\sigma^{2}_{c}}\right)}}{(2\pi\sigma_{c}^{2})^{1/4}},$
(42)

the denominator chosen so as to enforce (35). Here we use the (slightly
misleading) notation:

$\sigma_{c}^{2}=\langle c^{4}\rangle=2/\ell^{4}$ (43)

to avoid the heavy alternative $\sigma^{2}_{c^{2}}$. The amplitude (42) will
be used for the rest fo this paper.

We close by noting that we could have built coherent states centred around any
point $\\{c_{0}^{2},p_{c^{2}_{0}}\\}$. For example, for the quasi-Euler theory
Alexander et al. (2019a, b); Magueijo and Zlosnik (2019) we should set
$p_{c^{2}_{0}}=0$, but $c_{0}^{2}$ could be any constant. We could also have
considered squeezed coherent states. The more general expression is:

$A(c^{2})=\frac{1}{(2\pi\zeta^{2}\sigma_{c}^{2})^{1/4}}\exp{\left(-\frac{(c^{2}-c_{0}^{2})^{2}}{4\zeta^{2}\sigma^{2}_{c}}+\frac{i}{\hbar}p_{c^{2}_{0}}c^{2}\right)},$
(44)

where $\zeta$ is the squeezing parameter.

The uncertainty in the length scale $\ell$ (and so in $\sigma_{c}$) is
therefore an uncertainty in the definition of squeezing, since only
$\sigma_{c}\zeta$ appears in the states. Its origin lies in the fact that the
conjugate momentum to $c^{2}$ does not appear anywhere, so it can be defined
in a number of ways. We can saturate the Heisenberg bound imposed by the
complementarity of $c^{2}$ and its momentum, but have no way to define
“balanced” uncertainties between conjugates for an coherent state (or how
unbalanced they are for a squeezed state). Note that a similar situation
happens for a free particle freecoh , amounting to the introduction of a
length scale to define dimensionless quadratures.

##  VI The Hartle-Hawking beam

To complete our construction we evaluate explicit forms for the wave packets.
It is easier to put it all together in the $b$ representation. Inserting (42)
into (37) (recalling (18)) leads to:

$\displaystyle\phi(b)$ $\displaystyle=$ $\displaystyle{\cal
N}(8\pi\sigma_{c}^{2})^{1/4}\exp{\left[-\frac{9b^{2}\sigma_{c}^{2}}{\Lambda^{2}\mathfrak{h}^{2}}+i\frac{9}{\Lambda\mathfrak{h}}\left(b^{3}+3kb\right)\right]}$
(45) $\displaystyle=$
$\displaystyle(8\pi\sigma_{c}^{2})^{1/4}\exp{\left[-\frac{9b^{2}\sigma_{c}^{2}}{\Lambda^{2}\mathfrak{h}^{2}}\right]}\psi_{CS}(b,0).$

The last expression relates the infinite norm zero-torsion Chern-Simons state
to the finite norm wave packet built around zero-torsion. We see that it is
dressed by a Gaussian, which regularizes it. Indeed the associated probability
density is:

$P(b)=|\phi(b)|^{2}={\cal
N}^{2}\sqrt{8\pi\sigma_{c}^{2}}\exp{\left[-\frac{18b^{2}\sigma_{c}^{2}}{\Lambda^{2}\mathfrak{h}^{2}}\right]}.$
(46)

This is just a Gaussian distribution in $b$, with variance:

$\sigma_{b}=\frac{\Lambda\mathfrak{h}}{6\sigma_{c}}.$ (47)

By using (26) we can check that the distribution is properly normalized. To
illustrate the effect of different values of $\sigma_{c}$ we can introduce the
following dimensionless variables:

$\displaystyle\tilde{b}$
$\displaystyle=\frac{3}{\beta\Lambda}\frac{b}{\mathfrak{h}}$ (48)
$\displaystyle\tilde{\sigma}_{c}$ $\displaystyle=\beta\sigma_{c}$ (49)

where

$\displaystyle\beta$
$\displaystyle=\bigg{(}\frac{3}{\Lambda\mathfrak{h}}\bigg{)}^{2/3}$ (50)

such that

$\displaystyle P(\tilde{b})$
$\displaystyle=\sqrt{\frac{2}{\pi}}\tilde{\sigma}_{c}e^{-2\tilde{\sigma}_{c}^{2}\tilde{b}^{2}}$
(51)

satisfies $\int P(\tilde{b})\tilde{b}$ = 1.

Figure 1: Comparison of $P(\tilde{b})$ for various values of
$\tilde{\sigma}_{c}$.

In Fig. 1 we have plotted this distribution. As $\tilde{\sigma}_{c}$ goes to
zero, the probability approaches a uniform distribution with infinitesimal
density. The higher the fluctuations in the torsion $c$, the more peaked the
distribution of $b$ around zero. Quantum fluctuations in the torsion seem to
be acting as a “filter” suppressing high quantum spacetime curvature.

In the $a^{2}$ representation we must perform the integral (36), with (42).
This is best evaluated appealing to the concept of Airy transform Airybook ,
which can be written as:

$\displaystyle\phi_{\alpha}(y)$
$\displaystyle=\frac{1}{\alpha}\int^{\infty}_{-\infty}f(x){\rm
Ai}\bigg{(}\frac{y-x}{\alpha}\bigg{)}dx\quad(\alpha>0).$ (52)

It is a standard result that the Airy transform of the Gaussian function:

$\displaystyle{\cal G}(x)$ $\displaystyle=\frac{1}{\sqrt{\pi}}e^{-x^{2}}$ (53)

is:

$\displaystyle\phi_{({\cal G})\alpha}(x)$
$\displaystyle=\frac{1}{|\alpha|}e^{\frac{1}{4\alpha^{3}}(x+\frac{1}{24\alpha^{3}})}{\rm
Ai}\bigg{(}\frac{x}{\alpha}+\frac{1}{16\alpha^{4}}\bigg{)}.$ (54)

With this result in hand we find, after some algebra, that our wave-packet can
be written in the form

$\displaystyle\phi(\tilde{a}^{2})$ $\displaystyle=\tilde{\cal
N}e^{\tilde{\sigma}_{c}^{2}\big{(}\tilde{k}-\tilde{a}^{2}+\frac{2}{3}\tilde{\sigma}_{c}^{4}\big{)}}\mathrm{Ai}\bigg{(}\tilde{k}-\tilde{a}^{2}+\tilde{\sigma}_{c}^{4}\bigg{)}$
(55)

where we have defined the following dimensionless variables:

$\displaystyle\tilde{a}^{2}$ $\displaystyle=$ $\displaystyle\beta\frac{\Lambda
a^{2}}{3}$ $\displaystyle\tilde{k}$ $\displaystyle=$ $\displaystyle\beta k$
$\displaystyle\tilde{\sigma}_{c}$ $\displaystyle=$
$\displaystyle\beta\sigma_{c}$ $\displaystyle\tilde{\cal N}$ $\displaystyle=$
$\displaystyle(8\pi\sigma_{c}^{2})^{1/4}{\cal N}^{\prime}.$

This can be more suggestively written as:

$\phi(a^{2})=Ce^{-\lambda_{a}a^{2}}\psi_{HH}(a^{2},c_{0}^{2})$ (56)

which we call the Hartle-Hawking beam, and illustrate in Fig. 2 for various
values of $\tilde{\sigma}_{c}$.

This beam is a Hartle-Hawking wave function dressed with an exponential that
makes it fall off at large $a^{2}$ faster than the usual (non-normalizable)
power-law. Unlike the equivalent result for the $b$ representation (cf. Eq.
(45)), the state being dressed has an effective torsion, formally given by:

$c_{0}^{2}=-\left(\frac{3}{\Lambda\mathfrak{h}}\right)^{2}\sigma_{c}^{4}.$
(57)

This shifts the zeros of the wave function, but does not affect their spacing.
The phenomenology described in RandoMeso , therefore, is not expected to
change. The suppression exponent, $\lambda_{a}$, is given by:

$\lambda_{a}=\frac{3}{\Lambda\mathfrak{h}}\sigma_{c}^{2}=\frac{54V_{c}^{2}}{\Lambda
l_{P}^{4}\ell^{4}}.$ (58)

Depending on the torsion scale $\ell$ and on the observation volume this could
have meso- and macroscopic observable effects. We will return to this matter
elsewhere.

Finally, we note that for $a^{2}<0$ the dressing exponential blows up, but the
Airy function decays exponentially even faster. The Hartle-Hawking beam is a
perfectly well behaved distribution. For completeness we include the
irrelevant expression for the proportionality constant $C$ in (56):

$C=\exp{\left[\left(\frac{3}{\Lambda\mathfrak{h}}\right)^{2}\sigma_{c}^{2}\left(k+\frac{2}{3}\left(\frac{3}{\Lambda\mathfrak{h}}\right)^{2}\sigma_{c}^{4}\right)\right]}.$
(59)

.

Figure 2: Comparison of $|\phi(\tilde{a})|^{2}$ for various values of
$\tilde{\sigma}_{c}$. For ease of illustration, plots are normalized so that
$\int|\phi|^{2}d\tilde{a}^{2}=1$. Formally, as $\tilde{\sigma}_{c}\rightarrow
0$, the solution $\phi$ approaches proportionality with the Hartle-Hawking
wave function $\psi_{HH}$.

##  VII Beyond mini-superspace

One may wonder if our normalization procedure can extend beyond mini-
superspace. The answer is yes, as we proceed to schematically show, drawing
heavily on realK ; GenHH  (to which we refer the reader for details). In
general, we should apply to our constructions the correspondence:

$\displaystyle a^{2}$ $\displaystyle\rightarrow E^{a}_{i}$ (60) $\displaystyle
b$ $\displaystyle\rightarrow K^{i}_{a}$ (61)

where $E^{a}_{i}$ are the densitized inverse triads, $K^{i}_{a}$ are the
extrinsic curvature 1-forms (the imaginary part of the Ashtekar Self-Dual
connection, $A^{i}_{a}$), with $i$ the $SU(2)$ indices and $a$ the spatial
coordinate indices. The spatial connection $\Gamma^{i}_{a}$ (the real part of
$A^{i}_{a}$) can be split into a torsion-free part $\bar{\Gamma}^{i}$ and a
contorsion $\tilde{\Gamma}^{i}$:

$\Gamma^{i}=\bar{\Gamma}^{i}(E)+\tilde{\Gamma}^{i}$ (62)

In mini-superspace, $k=\pm 1$ provides an example of the effect of the first
term, $c$ of the second. We shall say more about this splitting in the next
Section.

With this dictionary in hand we can reproduce most of the steps in this paper
in a setting beyond mini-superspace. The Fourier transform defined in (5)
becomes the one proposed in GenHH :

$\psi_{E}(E,\Gamma)=\prod_{\vec{x},a,i}\int\frac{d[K^{i}_{a}(\vec{x}))]}{\sqrt{2\pi
l_{P}^{2}}}e^{-\frac{i}{l_{P}^{2}}E^{a}_{i}(\vec{x})K^{i}_{a}(\vec{x})}\psi_{K}(K,\Gamma),$
(63)

where the product include all spatial points $\vec{x}$, discretized or as a
continuum of infinitesimals. The inner product that generalizes (6) is:

$\displaystyle\langle\psi_{1}(\Gamma)|\psi_{2}(\Gamma^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\prod_{\vec{x},a,i}\delta V_{c}\int
d[K^{i}_{a}(\vec{x}))]\psi_{K1}^{\star}(K^{i}_{a}(\vec{x})),\Gamma^{i}_{a}(\vec{x}))$
(64)
$\displaystyle\psi_{K2}(K^{i}_{a}(\vec{x})),\Gamma^{i^{\prime}}_{a}(\vec{x}))$

where $\delta V_{c}$ is the volume element around point $\vec{x}$ introduced
for convenience in the next steps, mimicking our construction in mini-
superspace. Indeed, the generalization of (18) is realK :

$\displaystyle\psi_{CS}$ $\displaystyle=$ $\displaystyle{\cal
N}\exp{\left[-\frac{3i}{\Lambda l_{P}^{2}}\int
K^{i}\,{}^{(3)}\\!R^{i}-\epsilon_{ijk}\frac{K^{i}K^{j}K^{k}}{6}\right]}.$ (65)

This is the general solution to the Hamiltonian constraint in the connection
representation, with the torsion left in. It should be seen as the general
state which is still kinematical with regards to the torsion, but not
otherwise. It has dual metric representations, with formal definition and
examples beyond mini-superspace given in GenHH . It factorizes as:

$\displaystyle\psi_{CS}$ $\displaystyle=$
$\displaystyle\prod_{\vec{x},a,i}{\cal
N}_{ai}(\vec{x})\exp{\left[-\frac{3i\delta V_{c}}{\Lambda
l_{P}^{2}}\epsilon^{abc}\left(K^{i}_{a}\,{}^{(3)}\\!R^{i}_{bc}-\epsilon_{ijk}\frac{K^{i}_{a}K^{j}_{b}K^{k}_{c}}{6}\right)\right]}.$
(66)

As in mini-superspace, when we evaluate the inner product for states with
different $\Gamma^{i}$ (using (64)), the cubic term in $K^{i}$ cancels out,
leaving us only with linear terms that lead to delta-functions. Hence, in
general we have:

$\langle\psi_{CS}(\Gamma)|\psi_{CS}(\Gamma^{\prime})\rangle=\prod_{\vec{x},a,i}\delta({}^{(3)}\\!\tilde{R}^{ia}(\vec{x})-{}^{(3)}\\!\tilde{R}^{ia^{\prime}}(\vec{x}))$
(67)

with ${}^{(3)}\\!\tilde{R}^{ia}=\epsilon^{abc}{}^{(3)}\\!R^{i}_{bc}$ and

${\cal N}_{ai}(\vec{x})=\sqrt{\frac{3}{2\pi\Lambda l_{P}^{2}}}.$ (68)

We can now build wave packets out of these states with generic amplitudes
$A(\Gamma)$.

We see that a general treatment generates delta normalization most naturally
in terms of the dual of the 3-curvature. In mini-superspace this reduces to
the single parameter:

$k_{c}=k-c^{2}$ (69)

Obviously if we fix $k$ this is equivalent to the normalization in $c^{2}$
found before. Nothing changes with regards to the definition of the Hartle-
Hawking beam. But an intriguing possibility is raised. Given that states are
now indexed by $k_{c}$, could states with different $k$ but the same $k_{c}$
be the same? Could there be cross talk between the 3 types of FRW Universe? We
will investigate this possibility further in QuantFlat , with direct reference
to the infamous flatness problem in Big Bang cosmology.

##  VIII Torsion in the Einstein-Cartan theory

But there is more. As already noted in Section II, in the canonical formalism
there are two types of torsion: that contained in $\Gamma^{i}$ ($c$ in mini-
superspace), and that contained in $K^{i}$ (the $T$ in $b=\dot{a}/N-T$). They
are treated very differently classically: the first leads to second class
constraints, the second is set to zero by a Hamilton equation. according to
our procedure, quantum mechanically they are treated even more differently.
And yet, classically one can convert one type of curvature into the other as
we now show.

To recap, the action for the Einstein-Cartan theory is given in differential
forms notation by:

$\displaystyle S[e,\omega]$
$\displaystyle=\int\epsilon_{IJKL}e^{I}e^{J}R^{KL}(\omega)$ (70)

where $I,J,\dots$ are $SO(1,3)$ indices, $e^{I}=e^{I}_{\mu}dx^{\mu}$ is the
co-tetrad, $\omega^{I}_{\phantom{I}J}=\omega^{I}_{\phantom{I}J\mu}dx^{\mu}$ is
the spin connection and
$R^{IJ}=d\omega^{IJ}+\omega^{I}_{\phantom{I}K}\omega^{KL}$ is the curvature
two-form. Considering the action (70) to be a functional of $e^{I}$ and
$\omega^{IJ}$ as independent fields, variation of $\omega^{IJ}$ yields the
following classical equation of motion

$\displaystyle de^{I}+\omega^{I}_{\phantom{I}J}e^{J}\equiv T^{I}=0$ (71)

where $T^{I}$ is the torsion two-form. Alternatively, one may consider the
following decomposition of $\omega^{IJ}$ at the level of the action:

$\displaystyle\omega^{IJ}$ $\displaystyle=\bar{\omega}^{IJ}(e)+C^{IJ}$ (72)

where $\bar{\omega}^{IJ}(e)$ is the unique solution of equation (71) for
$\omega^{IJ}$ assuming that $e^{I}_{\mu}$ considered as a matrix is invertible
and $C^{IJ}$ is the contorsion one-form. Using this ansatz in (70) and
integrating by parts leads to the action

$\displaystyle S[e,C]$
$\displaystyle=\int\epsilon_{IJKL}e^{I}e^{J}\bigg{(}R^{KL}(e)+C^{K}_{\phantom{K}M}C^{ML}\bigg{)}.$
(73)

The first term is simply the Einstein-Hilbert action of General Relativity.
The action can be written as follows:

$\displaystyle S[e,C]$
$\displaystyle=\int(2\bar{R}+C^{I}_{\phantom{I}KI}C^{KJ}_{\phantom{KJ}J}-C^{I}_{\phantom{I}JK}C^{JK}_{\phantom{JK}I})\sqrt{-g}d^{4}x$
(74)

where $C^{IJK}\equiv e^{K\mu}C^{IJ}_{\mu}$. In passing to the Hamiltonian
formulation of this theory, one can consider a phase space coordinatized by
the spatial metric $q_{ab}(x^{c},t)$ \- where $x^{a}$ are spatial coordinates
- its momentum $\Pi^{ab}(x^{c},t)$ as well as $C^{IJK}(x^{a},t)$ and its
momentum ${\cal P}_{IJK}(x^{a},t)$. Equivalence of the classical Euler-
Lagrange equations and Hamilton’s equations would require the imposition of
primary constraints ${\cal P}_{IJK}\approx 0$ and requiring the preservation
of this constraint in time would impose further that $C^{IJK}\approx 0$, which
is indeed the Euler-Lagrange equation for $C^{IJK}$.

Explicitly, the Hamiltonian density $H$ will be given by a sum of constraints:

$\displaystyle H$ $\displaystyle=N{\cal H}+N^{a}{\cal
H}_{a}+\lambda^{IJK}{\cal P}_{IJK}+\xi_{IJK}C^{IJK}$ (75)

where $N$ and $N^{i}$ are the lapse and shift functions whilst $\lambda^{IJK}$
and $\xi_{IJK}$ are Lagrange multiplier fields and

$\displaystyle{\cal H}$ $\displaystyle=-2\sqrt{q}\bar{\cal
R}+\frac{2}{\sqrt{q}}\bigg{(}\Pi^{ab}\Pi_{ab}-\frac{1}{2}\Pi^{a}_{\phantom{a}a}\Pi^{b}_{\phantom{b}b}\bigg{)}$
$\displaystyle-\sqrt{q}\bigg{(}C^{I}_{\phantom{I}KI}C^{KJ}_{\phantom{KJ}J}-C^{I}_{\phantom{I}JK}C^{JK}_{\phantom{JK}I}\bigg{)}$
(76) $\displaystyle{\cal H}_{i}$
$\displaystyle=-4\sqrt{q}q_{ac}\partial_{b}\bigg{(}\frac{1}{\sqrt{q}}\Pi^{cb}\bigg{)}$
(77)

where $q$ is the determinant of $q_{ab}$.

In this paper we adopted the approach of implementing _only_ the constraints
${\cal H}$ and ${\cal H}_{i}$ on quantum states and then building wave packets
that represent coherent states around zero contorsion. But with an integration
by parts we have now put all the torsion into the last procedure. To
illustrate the point in mini-superspace, this is the action we would start
from:

$S[a,\Pi,T,c]=6\kappa V_{c}\int
dt\bigg{(}a^{2}\dot{\Pi}+Na\bigg{(}\Pi^{2}+k+T^{2}-c^{2}-\frac{\Lambda}{3}a^{2}\bigg{)}\bigg{)}.$
(78)

instead of (1). Thus the second type of torsion $T$ (that beforehand revealed
by the dynamics) is transferred to the spatial curvature. We could also
transfer just some of it or none at all. In fact, the action (78) has a local
(in time) $SO_{C}(1,1)$ (complexified 2d Lorentz group) gauge invariance -
reflected in the invariance of $T^{2}-c^{2}$.

Instead of (69) our states are now indexed by:

$k_{c}=k+T^{2}-c^{2}.$ (79)

Nothing changes in our construction of wave packets, and yet the quantum
theory looks totally different with now two dimensions of torsion space
present in the problem. The million dollar question with regards to the other
phenomenology is whether or not we have broken local Lorentz invariance in our
treatment of the torsion.

##  IX Conclusions

The main achievement of this paper was to find a regular, normalizable version
of the Hartle-Hawking wave function, which we called the Hartle-Hawking beam.
The crucial point is not to impose the torsion-free condition too soon, namely
classically, before quantization. The latter may seem innocuous, but would
amount to freezing quantum fluctuations in the torsion, implying essentially
states of the form (34) with

$A(c^{2})=\delta(c^{2})$ (80)

so that their norm is:

$\int dc^{2}|A(c^{2})|^{2}=\int dx\delta^{2}(x)=\infty.$ (81)

With the benefit of hindsight this should have been obvious. Fixing the
torsion to zero is to consider a wave function with an amplitude in torsion
space that is a delta function. The integral of the square of a delta function
is infinite. By allowing the torsion to fluctuate, even with a state of
minimal uncertainty, such as a coherent state centred at the origin, we remove
this infinity. We called such a wave packet (which is a Gauss-Airy function),
the Hartle-Hawking beam.

We probably would not have found the Hartle-Hawking beam, had we not received
inspiration from its dual version, the real Chern-Simons state. It has been
known for a while that the Euclidean version of that state is delta-function
normalizable, since it is a pure phase. A clear statement of what that means
was missing until this paper. Furthermore, it has been found recently that the
Lorentzian state is more similar to the Euclidean state than thought before,
as long as an explicitly real theory is used realK . Again, the matter relates
to what to do with torsion in a complexified theory. Quite often reality
conditions and torsion-free conditions are mixed and confused. A separation of
the two was required in order to find a non-pathological normalizable version
of the Chern-Simons state. Its Fourier dual is precisely the Hartle-Hawking
beam.

Beyond these achievements, many questions remain to be answered on a
mathematical level. But foremost we can look forward to investigating the
phenomenology of torsion fluctuations, particularly in situations where a
mesoscopic scale is present RandoMeso ; Barrowhbar . A major issue to be
cleared is whether or not our quantum treatment of torsion has broken local
Lorentz invariance. We will return to this matter in the not too distant
future

## Acknowledgements

We would like to thank Stephon Alexander, Gabriel Herczeg and Simone Speziale
for enlightening discussions related to this paper. This work was supported by
the STFC Consolidated Grant ST/L00044X/1 (JM) and by the Grant Agency of the
Czech Republic, GAC̆R grant 20-28525S (TZ).

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