# The strong CP problem, general covariance, and horizons

Giorgio Torrieri, Henrique Truran  Universidade Estadual de Campinas -
Instituto de Física “Gleb Wataghin”
Rua Sérgio Buarque de Holanda, 777
CEP 13083-859 - Campinas SP
<EMAIL_ADDRESS>

###### Abstract

We discuss the strong CP problem in the context of quantum field theory in the
presence of horizons. We argue that general covariance places constraints on
the topological structure of the theory. In particular, as in QCD, it means
that different topological sectors of the theory can only sum incoherently,
because the degrees of freedom beyond the horizon must be traced over for
general covariance to apply. This might lead to a solution of the so-called
strong CP problem without extra observable dynamics.

##  I Introduction to the strong CP problem

The strong CP problem strongcp ; qft ; shuryak ; hooft ; witten1979 ;
forkel_instantons ; adam is the only fundamental naturalness problem connected
to QCD. It arises generically in Yang-Mills theories as a consequence of the
existence of an internal direction in the “color-symmetry” gauge group $G$,
where the vector potential $A_{\mu}\rightarrow A_{\mu}^{a}$ is allowed to
twist in topologically non-trivial ways. The non-trivial twisting of $A_{\mu}$
means that finite-action field configurations are separated into topologically
separated equivalence classes, labeled by how they twist in gauge space over
the asymptotic region $A_{\mu}(r\rightarrow\infty)$ in Euclidean space
coleman_instantons ; forkel_instantons . The label for these equivalence
classes is an integer $n[A]\in\mathbb{Z}$, called the winding number of
$A_{\mu}$, given by

$n[A]=\frac{1}{32\pi^{2}}\int
d^{4}xF_{\mu\nu}^{a}\tilde{F}_{\mu\nu}^{a},\quad\tilde{F}_{\mu\nu}^{a}\equiv\frac{1}{2}\epsilon_{\mu\nu\lambda\sigma}F_{\lambda\sigma},$
(1)

and all classical solutions of the Euclidean equations of motion of the theory
with finite action fall within one of these winding sectors.

Classically, the winding number cannot be changed due to energy conservation.
Quantum mechanically, the existence of tunneling solutions of the classical
Euclidean equations of motion (called instantons) that interpolate between two
given equivalence classes $n\rightarrow n+1$, having the form (up to a
parameter $\rho$)

$A_{\mu}^{a}=2\frac{x_{\nu}}{x^{2}}\left(\frac{\eta^{a}_{\mu\nu}\rho^{2}}{x^{2}+\rho^{2}}\right)$
(2)

where $a$ is the gauge index and $\epsilon_{\mu\nu a}$ mixes internal and
spacetime degrees of freedom.

This means the winding number is only conserved perturbatively. Field strength
fluctuations

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}]$ (3)

corresponding to field configurations of the type of equation (2) occur
locally, suppressed non-perturbatively by the field’s action content $\Gamma$

$\Gamma\sim\exp\left[-\frac{1}{4g^{2}}\int
d^{4}x\,F_{\mu\nu}^{a}F_{\mu\nu}^{a}\right],$ (4)

and change the field’s winding number by one unit.

Note that winding numbers are global properties of given field configurations
defined over all space. Since the topological charge density
$F_{\mu\nu}\tilde{F}_{\mu\nu}$ can be written in terms of a total derivative
$\partial_{\mu}K_{\mu}$ (the so-called Chern-Simons current), they are only
“observable” (not in practice but with QFT sources, see footnote 4) by probing
the celestial sphere at at infinity

$n\sim\int
d^{4}xF_{\mu\nu}\tilde{F}^{\mu\nu}\sim\lim_{r\rightarrow\infty}\oint
d^{3}S\,\epsilon_{\mu\nu\rho\sigma}n_{\mu}\mathrm{Tr}\left(A_{\nu}\partial_{\rho}A_{\sigma}+\frac{2}{3}A_{\nu}A_{\rho}A_{\sigma}\right).$
(5)

Instantons, however, are localised, and dominate at a scale in momentum space
(parametrized by $\rho$ in equation (2)) where the theory becomes non-
perturbative.

Instantons break the degeneracy of the classical vacuum solutions with
different winding numbers, and their presence means that the Yang-Mills vacuum
will be in general a superposition of the different topological sectors
weighted by expansion coefficients $c_{n}$. This coefficients are fixed to a
phase $c_{n}=\exp(in\theta)$ by the homogeneity of Minkowski space qft , where
the arbitrary constant $\theta$ parametrizes the coherent superposition of
states with topological winding number $n$

$Z[\theta]=\sum_{n}e^{in\theta}Z_{n}\iff\left|\theta\right>=\mathcal{N}\sum_{n}e^{in\theta}\left|n\right>.$
(6)

This superposition of winding sectors in the vacuum is equivalent, via
imposing the first relation in Eq. 5 as a constraint in the partition
function, to an additional gauge-invariant adam term in the Euclidean
effective Lagrangian

$\mathcal{L}_{\text{eff}}\equiv\ln
Z=\mathcal{L}_{\text{YM}}+\frac{i\theta}{16\pi^{2}}\mathrm{Tr}(F_{\mu\nu}\tilde{F}_{\mu\nu}),$
(7)

which breaks the CP symmetry of the original Yang-Mills Lagrangian. In QCD,
for example, this CP symmetry breaking gives rise to observable effects, such
as $\theta$-dependent electric dipole moments for neutral particles.
Experiments, however, have constrained this CP-violating parameter to the
small bound of $\left|\theta\right|\leq 10^{-9}$ (the experimental limit of
the neutron’s dipole moment is $10^{-18}$ e.m.), beyond naturalness, and the
explanation for such a small value is called the strong CP problem.

A variety of solutions strongcp were proposed for this problem, most involving
beyond the standard model dynamics coupling to $\theta$ (“axions”) or extra
symmetries that fix $\theta$ to zero. Phenomenologically, no signature
suggestive of such models has so far been detected.

In this work, we will try to “go back to the basics” of quantum field theory
and reflect on the nature of the quantum vacuum. In the following three
sections, we will make a series related arguments that link the strong CP
problem to the interplay between quantum field theory and general coordinate
transformations. Section III argues that a theory with a non-zero $\theta$
will likely lose general covariance of any observables depending on $\theta$.
Section IV will show that the “toy model” often used to describe $\theta$
vacua together with the tracing over of trans-horizon degrees of freedom leads
to $\theta=0$ asymptotic states independently of the “real” $\theta$ value
111We note that the question of coherence in the appearance of instantons was
already discussed in khleb . Where our approaches differ is that we consider
the definition of observables characterising the topology of space, while
khleb concentrates on the dynamics.

##  II Quantum field theory at different scales

At first sight the claim at the end of the last section appears counter-
intuitive. We are expected to believe that something as “global” as the
topology of the universe plays an important role in local physics, the
effective Lagrangian of QCD222There is a lot of discussion about weather
General relativity is ”Machian”, although Mach’s principle played a big role
in constructing it. However the Gibbons-Hawking boundary term looks
conceptually like a quantum field version of Mach’s principle gibbons ;
gibbons2 . To justify and also sharpen this claim, we need to discuss a little
bit more the sensitivity of quantum field theory observables to different
scales.

As in quantum mechanics, in quantum field theory all information about the
quantum state is encoded in the generating functional. In terms of fields
$\phi$ and source functions $J(x)$ (See reference gtholo ; jacobson ), it is

$Z=\int[\mathcal{D}\phi]\exp\left[\int\sqrt{-g}d^{4}x\left[\mathcal{L}(\phi)+J(x)\phi\right]\right].$
(8)

For fields of any given spin, both $\phi$ and $J$ acquire the Lorentz and
internal symmetry properties necessary for $J(x)\phi$ to be a Lorentz scalar.

Unlike quantum mechanics, quantum field theory qft has a continuous infinity
of degrees of freedom. This introduces additional mathematical issues, since
the vanishing of the inner product of orthogonal states does not commute with
“the infinity of degrees of freedom”. Key theorems behind quantum mechanics,
such as the Stone Von Neumann theorem, stop being valid. While this introduces
additional phenomenological features (“broken symmetries”, fundamental for the
standard model of modern physics, are impossible in Quantum Mechanics since
the Stone-Von Neumann theorem forces the vacuum to have the same symmetries as
the Hamiltonian), mathematically the construction of observables can be beset
by ambiguities, such as divergences.

The main way physicists deal with these issues is in the language of the
renormalization group. Observables in quantum mechanics are not “states” but
time ordered field correlators of operators $\hat{O}$,
$\langle\hat{O}(x_{1})\hat{O}(x_{2})\rangle$, measured at a scale
$Q\sim\left|x_{1}-x_{2}\right|^{-1}$. Thus, unlike in quantum mechanics, even
fundamental parameters of the Lagrangian of the theory become ambiguous,
related to an observable and sensitive to $Q$. Masses $m$ can be thought of as
the infrared limit of the field 2-particle correlator, the coupling constant
$\lambda$ to the 4-particle correlator and so on.

$m\sim\left(\lim_{Q\rightarrow
0}\left\langle\phi(x_{1})\phi(x_{2})\right\rangle\right)^{-1}\phantom{AA},\phantom{AA}\lambda(Q)\sim\lim_{Q\rightarrow
0,|x_{1,2}-x_{3,4}|\rightarrow\infty}\frac{1}{Q^{2}}\left\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\right\rangle$
(9)

the scale dependence of the action can be represented by a functional integro-
differential equation polch in terms of the tangent vectors of field
redefinitions in scale space, $\xi$.

$\Delta S_{k}[\varphi,\xi]=\frac{1}{2}\int d^{d}x~{}\xi^{a}{\cal
R}^{k}_{ab}(\varphi)\xi^{b}\,\phantom{AA},\phantom{AA}k\partial_{k}\Gamma_{k}[\varphi,\xi]=\frac{1}{2}{\rm
Tr}\left(\frac{k\partial_{k}{\cal
R}_{k}[\varphi]}{\Gamma_{k}^{(0,2)}[\varphi,\xi]+{\cal
R}_{k}[\varphi]}\right),$ (10)

$\text{with}~{}\lim_{k\to 0}{\cal
R}^{k}_{ab}[\varphi]=0~{}~{}\text{and}~{}\lim_{k\rightarrow\infty}{\cal
R}^{k}_{ab}[\varphi]=\infty$. This procedure is convergent in a Euclideanized
space, but, provided quantum field theory makes sense mathematically, it
should also exist, albeit being incalculable with current technology, in
Minkowski and more general curved spaces. The issue with calculation in such
spaces is the generic appearance of strongly oscillatory integrals turok ,
where scale separation fails.

Provided a stable vacuum is well-defined, if one calls the scale of the
detector interaction $Q\sim\mu$ and chooses a scale $\Lambda\gg\mu$, one can
evolve Eq. 10 from $\Lambda$ to $\mu$ and separate the Lagrangian into a
finite number of renormalizable terms for which $\Lambda$ can be absorbed into
unobservable divergences, and a series in $Q/\Lambda$. In this regard, the
similarity of the equations 10 to those of generally covariant motion in
curved space are not coincidental, they represent the “scheme independence” of
the regularization of the “diverging” terms of the procedure (see schafer for
a good discussion of this). The standard model has been known to arise out of
just such a construction, and only physics reducible to renormalizable terms
has been seen so far.

This construction has, however, as yet not been univocally and generally
extended to fully cover infrared physics, for which $Q$ is much smaller than
$\mu$, the inverse of the “detector size”, with in a Minkowski picture, where
this infrared limit could exhibit a non-trivial horizon and causal structure.
Just like evolving the “topology of the universe” has local dynamical
consequences, evolving from a scale $k\ll\mu$ to a scale $k\sim\mu$ does
depend on the initial conditions.

This $k\ll\mu$ scale would cover the properties of “the vacuum”, only a
partially rigorously defined object qft , covering a continuous infinity of
degrees of freedom and entangling every point on a time-slice with every other
one. While renormalization physics is insensitive to vacuum properties 333The
Nobel prize of 1999 was awarded for this realization there is no equivalent to
universality and scale separation w.r.t. $\mu$ vs. the vacuum characteristic
scale. Infrared degrees of freedom can produce arbitrarily large effects at
order $\mu$ in the effective theory, and vice-versa. For instance, it has long
been known that the vacuum energy of a vacuum with the boundary is very
sensitive to the shape of that boundary, to the effect that it even changes
sign casimir . As a result, even at semiclassical level, if the cosmological
constant is assumed to be “background dependent” it becomes dynamical markanen
; choi , reflecting dissipation from boundary properties. It’s dependence on
intermediate scales can be very different from naive dimensional analysis
visser . Indeed, the “moving mirror picture” of horizon radiation mirror makes
this apparent, since there are no geometry modifications, the vacuum is
modified by dynamic boundary conditions. There, it is clear that non-inertial
boundary conditions mean ultraviolet and infrared degrees of freedom get
mixed.

While the cosmological constant is a super-renormalizable operator, and
dimensionally $\theta$ is superficially a marginal coupling, topological terms
are also known to depend on infrared wavelenghts renotop , the relationship
with vacuum energy made explicit in color counting witten1979 . In case of the
$\theta$ vacuum this UV-IR mixing can be seen in the instanton liquid model by
calculating the energy expectation value, as was done in coleman_instantons

$<0|H|0>_{\theta}\sim\int_{0}\frac{d\rho}{\rho^{5}}F\left(\rho\Lambda\right)$
(11)

Where $F(...)$ can qualitatively depend on both the UV and IR limit non-
trivially. Physically, this mixing of scales is generated by the fact that
these configurations are dominated by high mixing numbers of ground state
quanta. In the moving mirror picture mentioned earlier mirror , if one were to
construct topologically non-trivial configurations in such a setup, the
relation between Minkowski and mirror boundary conditions would map “soft
quanta” carrying topological information into harder ones which, because of
asymptotic freedom, are insensitive to the presence of instantons in the
vacuum.

Since effective field theory techniques rely on a $Q\ll\mu$ expansion, these
subtleties are generally incorporated in the $\mathcal{O}\left(1\right)$ terms
of the effective Lagrangian. For instance, the calculation of the
$\theta$-dependence of the vacuum energy in ulf has no trace of the integral
in Eq. 11 and appears to only be set by dimensionful parameters of the order
of $\Lambda_{QCD}$. A similar divergence appears in the Higgs mechanism, were
in Minkowski space it is renormalized by putting the infrared scale and the
ultraviolet scale as reciprocals polyakov . This is justified if no intrinsic
scale other than the electroweak symmetry breaking appears in observables, but
would change if, for example, such a theory was formulated on a torus, in
analogy with torus .

The arguments above show that, while one must be careful with infrared scales,
effective field theories can be a useful guide because they can indicate which
terms are compatible with symmetries at the infrared. For collider
experiments, the infrared vacuum structure is often irrelevant blum since
basic invariance and causality can be used to link correlators to scattering
experiments in a straight-forward fashion independent of scales $\ll\mu$, via
the LSZ reduction formula. If the vacuum is stable and there exist stable
particles, by definition IR corrections cannot detectably shift their mass and
coupling constants.

However, beyond the scattering problem things are more complicated: In QED and
linearized gravity the fact that emission of quanta of the order $Q\ll\mu$
does not effect observables at $Q\sim\mu$ is related to the symmetries of the
theory. The full consequences of this have been the subject of a lot of
theoretical development bms , and the topology of the space in question (the
BMS group in asymptotically flat space) is crucial to this. An infinitesimal,
arbitrary small graviton mass, which could reduce the observable cosmological
constant, generally violates these symmetries in a way that also spoils the
theory consistency in the ultraviolet massgrav .

This hierarchy of scales is complemented by the scale separation necessary for
both the Unruh effect and instantons to be properly defined. Acceleration and
force are by their nature semiclassical concepts, since they consider momentum
a differentiable number rather than an operator. This means that
acceleration’s lifetime $T$ needs to be long-lived compared to its scale $a$,
and also smaller than the fundamental scale of the detector, for example it’s
mass $M$

$1/T\ll a\ll M\underbrace{\Rightarrow}_{instanton}\mu\ll a\ll\rho$ (12)

in this context, the instanton’s bulk of the action is in the peak, of size
$\sim\rho$ (Eq. 2), yet the topology information will be in the tail,
dominated by a diverging occupation number of quanta of characteristic
frequency $\ll 1/\rho$. One can therefore have an acceleration small enough
w.r.t. the instanton size, long enough to maintain the semiclassical
approximation, where the semiclassical expansion will break down at the tail.
The first hierarchy in Eq. 12 is therefore far more dubious in its
applicability than the second.

We therefore turn to the other ingredient necessary to construct effective
field theories: Unbroken symmetries in the infrared and in the semiclassical
approximation. The obvious symmetry mentioned previously is the equivalence
principle, built out of general covariance of observables. The above issues
make it obvious that these constructions are difficult to extend to general
covariance in non-linear gravity, necessary for general relativity gibbons ;
gibbons2 ; hawkingano ; gtholo . It has long been known that at one loop in
the effective theory correlators do not transform covariantly (for a
particular example, see the light-bending calculations at donog ). It has also
been known that quantum vacua with different topologies fall into unitarily
inequivalent representations (the “black hole information paradox” is a direct
consequence of this).

At its root the problem is, since general coordinate transformations change
the causal structure of the theory, the vacuum undergoes non-unitary
transformations. Even if one concentrates on particular geometries,
quantization becomes very sensitive to topology. While theories with timelike
Killing vectors (such as static black holes and anti-de Sitter spaces) are
relatively straightforward and edge state techniques can be used in 1D systems
to quantize the boundary separately from the bulk blomm ; blomm2 , in other
geometries, such as de Sitter spacetimes, fundamental observables which are
manifestly unitary in Minkowski space, such as the S-matrix sdesitter , become
ill-defined. No systematic way as of yet exists to treat quantum fields in
such a spacetime, but it can be argued gtholo ; jacobson that if general
covariance is to be fundamental one must give up unitarity and treat the
degrees of freedom behind the horizon in the language of open quantum systems.
In this case, the vacuum’s “state purity” is ill-defined, since it is frame
dependant, but observables could acquire general covariance via the boundary
term. Instead of unitarity, a fluctuation-dissipation theorem would constrain
the partition function zubarev .

Several lines of evidence point to the fact that such a construction might be
achievable. The soft graviton theorems mentioned earlier weinberg relate the
independence of infrared physics to Lorentz invariance. To first order, it has
been known for a while that correlators along Rindler paths laflamme ; matsas
transform covariantly once boundary terms are added. For a correlator
$\left\langle\psi(x^{\mu}_{1}(\tau_{1})\psi(x_{2}^{\mu}(\tau_{2})X\right\rangle$,where
$x_{1,2}^{\mu},\tau_{1,2}$ are along a Rindler trajectory and $X$ is either a
photon in QED sudarsky or a neutrino in Fermi theory vanzella one can
calculate the Minkowski correlator in terms of interactions with a classical
field and the Rindler correlator in terms of interactions with the Unruh bath.
The “interpretation” of the calculation will be different but the calculated
matrix elements will be the same.

It is therefore worth thinking about the form a “generally covariant”
interacting quantum field theory would have, and in particular if, in analogy
with renormalization requirements, some terms in the Lagrangian would be
forbidden by imposing background independence as an infrared symmetry.

In this regard, the $\theta$ term looks suspicious. The vacuum energy density
depends on the boundary and on $\theta$ separately. The effective $\theta$,
$\sim\frac{\delta\ln\mathcal{Z}}{\delta F\tilde{F}}$ will also depend on the
boundary and on the temperature. Hence, there is no a priori reason for the
effective $\theta$ calculated from a Rindler boundary to be the same as the
effective $\theta$ at finite temperature, something already clear from group
theory arguments calixto in a generic scenario with degenerate vacua
constructed from conformal zero modes (semiclassical instantons are similar in
this respect). But local Lorentz invariance forces this equivalence axiom ,
leading to non-zero $\theta$ being forbidden.

This insight can be sharpened in lower-dimensional theories, where the non-
trivial interplay of asymptotic global symmetry properties of the theory and
topological terms is well-known dunne . Appendix A.1 describes in detail a
quantum particle on a ring with a magnetic field in thermal equilibrium. The
different geometric and thermodynamic temperature in this system can be mocked
up by a device that changes the magnetic field based on the system’s heat
capacity. While such a setup can be engineered to add no entropy (no
microstates are measured, and the adiabatic limit is maintained) in this case,
the thermodynamic temperature is different from the geometric temperature
(given by the time periodicity) by a generally non-perturbative factor, eq.
81. Note that the difference is not necessarily connected to a scale
separation, since the two dimensionful scales combine non-perturbatively.

Section A.2 summarizes an equivalent 1+1 dimensional topological theory. As
shown there, while in such a theory the $\theta$ term is set as a boundary
condition, a source “communicating” with a boundary could modify the
temperature from the Unruh value by arbitrary amounts. Dynamics spontaneously
creating such topological terms in 3+1 theory, therefore, could potentially
mean that the geometric temperature of an accelerating observer would be
different from the temperature the observer would measure from the background.
A dynamical symmetry breaking would be equivalent to assuming an effective
$\left\langle J_{L}\right\rangle$ in Eq. 94, coinciding with that of Eq. 5. As
the next section III will argue, unless $\theta=0$ it is impossible to make
such a term truly background independent. Section IV will further show, using
the often-used periodic potential analogy, how the IR tracing out could wash
out an arbitrary $\theta$ to an effective $\theta=0$.

Summarizing, a “background-independent quantum field theory”’s requirement
would be the separation of local physics (at scale $\mu$) from scales
sensitive to the topology of space. This is very different from saying the
latter are relevant to observable physics (indeed, as ulf shows explicitly,
they are not). Analogously to counter-terms in Wilsonian renormalization
enforcing independence of detectable physics from its UV completion, infrared
constraints enforce independence of local physics from the topological
features of the chosen coordinate system. $\theta=0$ would be such a
constraint.

##  III The topological term from the partition function

As can be seen in the previous section, the question of the general covariance
of quantum field theories is a subtle one. It is straight-forward to write an
action invariant under general coordinate transformations, since
$\sqrt{-g}d^{4}x$ is generally covariant and $\mathcal{L}$ can be made so by
the use of covariant derivatives. Indeed, a $\theta$ term can also arise
within an effective theory extension of general relativity deser .

The issue is the role of the integration measure $\mathcal{D}\phi$. What
configurations are counted if the coordinate system contains singularities, or
the spacetime is divided into casually disconnected regions? We note that, as
shown in hawkingano , Hawking radiation can be thought of as an anomaly, i.e.
a tension between the fundamental symmetry of the theory and the integration
measure – in this case provoked by the boundary structure of the Schwarzschild
spacetime at the horizon.

At the partition function level, it has long been known gibbons ; gibbons2
that the partition function for general coordinate systems with causal
horizons will necessitate of a surface term, which for a time-like Killing
horizon can be modeled by a thermal bath. This is the essence of the Unruh
effect, and, in 1+1D, can be rigorously connected to topological terms (see
appendix and blomm ).

In gtholo , it was proposed that perhaps promoting the partition function with
a source to a dynamical object would generate a generally covariant quantum
theory. A general non-inertial transformation will alter both bulk and
boundary, but there is a possibility that, with Lagrangians describing both,
the total partition function will be invariant under general coordinate
transformations. In gtholo we argued that imposing this must lead to treating
all quantum states as open, since such general coordinate transformations
necessarily break unitarity.

This extension, however, is unobservable in experiments done so far as it
concerns situations with strong classical accelerations. Such an approach, as
we argued, could be used to write down an effective quantum field theory
covariant under general coordinate transformations. Not quantum gravity, of
course, but a field theory respecting the symmetries of gravity at quantum
level.

In this spirit, let us try to define $\theta$ as a running parameter in the
spirit of Eq. 9. We immediately see that Eq. 7 is valid in flat Euclidean
space, but any infrared subtlety, like the presence of a horizon, would change
the asymptotic shape of the instanton and hence the $k\ll\mu$ initial
condition of equation 10. Put it differently, while of course
$F_{\mu\nu}\tilde{F}^{\mu\nu}$ is a generally covariant scalar the equation 5
does not transform covariantly and hence the constraint leading to Eq.7 is not
generally covariant.

We can however circumvent this problem remembering the effective action can be
defined also in terms of a source in Eq. 8. The winding number will be related
to the infrared limit of a probe such as a “loop” of “color sources”
$J_{a}^{\mu}$ placed at infinity444A technical note: of course it is
impossible even in principle to put a probe at infinity. However, one can
define such a measurement in terms of Bayesian limits. Here, by “a probe at
infinity” we mean a probe placed at a sequence of larger and larger distances.
The chance of the winding number not being inferred correctly by the
measurement goes down with distance in a calculable way. Such a sequence can
be used to define something like Eq. 5, while $\theta$ will be related to the
entanglement of such winding numbers

$n\propto\lim_{r_{0}\rightarrow\infty}\mathrm{Tr}\oint
dS_{d-1}.J^{a}\delta(r-r_{0})A_{a}\phantom{AA},\phantom{AA}\theta\propto-i\left\langle
n\right\rangle$ (13)

however, in Minkowski space $S_{d-1}$ can be oriented in either a time-like or
a space-like direction. In the first case, we will be projecting on $\theta$,
as it commutes with the Hamiltonian.As a particular case, if instead of an
isotropic $dS_{d-1}$ we consider a “long thin sausage”, our observable
coincides with the dipole moment of a neutral particle antiparticle
combination. This measurement is in fact ”asymptotic”, since observing the
neutron for any amount of finite time there is a finite probability that
topological fluctuations will give us an effective non-zero dipole moment (of
course this is irrelevant on microscopic scales).

In the second, spacelike loop, we will be sensitive to the topological number
$|n>$ of our system, rather than $\theta$. This ambiguity is unique for
“topological terms”, and suggests that general topological terms are not
generally covariant. Indeed, it is obvious that the boundary integral has the
same structure of equation (5); A function of $A_{\mu}^{a}$ integrated over
the surface $dS_{d-1}$ which in turn is the edge of a particular geometry. A
non-zero $\theta$ term means that different $n$’s are coherently entangled,
and the degree of this coherence would be modified by the boundary term.
Hence, the entanglement entropy, and all of its derivatives ($\theta$ is
proportional to the first derivative, the topological susceptibility to the
second), would be modified by the boundary term in a way that is very
sensitive to the geometry in question. In fact, calculating Eq. 5 using the
WKB approximation parikh and a path crossing the horizon can easily be seen to
be boundary-dependent.

This suggestion has been put on a firmer footing within etesi , following an
unsuccessful attempt by the same author etesi2 to argue, in a manner similar
to blackholes , that Yang-Mills instantons are incompatible with a
Schwarzschild geometry. As it turns out etesi this is not quite correct.
However, unlike in Minkowski the instanton in Euclidean Schwarzschild space
cannot be gauge transformed into a smooth temporal gauge; By looking at a
Rindler patch of this space, it is clear that a Minkowski and a Rindler
observer, or a Schwarzschild vs a Lemaitre observer, will see a different
instanton content and a different $n$. As $n$ is a scalar, the only way to
preserve an undetermined $n$ (required by local quantum mechanics) that
transforms covariantly is to make sure the summation over $\left|n\right>$ was
incoherent even in the freely falling frame, where geometry is closest to
Minkowski. This corresponds to the case of $\theta=0$. Note that this argument
only applies to the average topological value $\left\langle
F\tilde{F}\right\rangle$. Phenomenologically useful fluctuations
$\sim\left\langle F\tilde{F}F\tilde{F}\right\rangle$ lattice are not affected
because of the time-ordered nature of the product in the fluctuation
witten1979 , which isolates the local (instanton peak, always in causal
contact with the observer) over the global asymptotic state (possibly affected
by the horizon). The analogy here is the cosmological constant (of which
$F\tilde{F}$ is the “topological QCD part, witten1979 ) and local
gravitational physics 555This analogy can be seen via equation (11) of
witten1979 , where the spectral structure of the 2-point correlator of
$F\tilde{F}$ is examined. In Euclidean space at $k\rightarrow 0$ the 2-point
function is determined by the 1-point function, but in the equivalent
situation in a local Minkowski theory causality changes the conclusion. The
discussion about time orderings in witten1979 makes it clear that the
analytical IR behavior of Euclidean and Minkowski signatures can be very
different.. Background independence requires that Infrared physics could get
corrections from quantum fluctuations and decoherence markanen ; choi , but
the same symmetry requires that locally physics is unaffected. This can happen
if all local physics is sensitive to the second derivative and the first
derivative is zero, as indeed seems to be the case.

Let us explore this argument in more detail, but concentrating on Rindler
patches. For static and quasi-static spacetimes this is equivalent to filling
the manifold with a bath of quanta with the temperature $T_{h}\sim 1/R_{h}$,
where $R_{h}$ is the horizon scale (the Schwarzschild radius for black holes,
the Hubble radius for dS space, the acceleration for Rindler space and so on).
Thermality appears as a consequence of the symmetries of the quasi-static
accelerated spacetime witten ; calixto , and hence can be thought of as as
embedded in the effective action

$S_{eff}\sim\ln\left[\int\mathcal{D}\phi\exp\underbrace{\int\sqrt{-g}d^{4}x}_{\rightarrow\oint
dt\int d^{3}x\sim
T\sum_{n}\delta\left(t-\frac{n}{T}\right)}\left[\mathcal{L}(\phi)+J(x)\phi\right]\right].$
(14)

As shown in axiom , for axiomatic field theory (defined in terms of
correlators), Lorentz invariance implies this relation is exact.

As further shown in matsas ; laflamme and references therein, one can derive
the Unruh effect by tracing over degrees of freedom beyond the horizon.

Figure 1:  The asymptotic horizons of Minkowski and Rindler observers, on
which winding numbers are measured (middle and right panels). As shown in
laflamme , a rigorous dictionary between the two partition functions can be
established by complexifying coordinates and a choice of contours, shown on
the left

The two pictures are complementary since, at least for Rindler patches horibe
; truran , tracing over in Minkowski space can be achieved by complexifying
the action and appropriately choosing contours constructed to respect the
periodicity of the time coordinate. This way we can write, up to a
normalization factor, the generating functional for a field as

$Z[J]\sim\int\mathcal{D}_{\mathcal{C}}\phi\exp\left\\{i\int_{\mathcal{C}}\mathop{\sqrt{-g}d^{4}x}\left(\mathcal{L}[\phi,\partial\phi]+J\phi\right)\right\\},$
(15)

where $\mathcal{C}$ corresponds to a contour choice in complex coordinate
space – analogous to the Schwinger-Keldysh and thermofield-dynamical
formalisms of usual finite-temperature quantum field theory. An extension of
Rindler to Minkowski spacetime can be realized, in Rindler null-coordinates
$l_{\pm}=\tau\pm\xi$, by the horizontal patches
$\mathcal{C}_{1},\mathcal{C}_{2}$ of the contour shown in figure (1), defined
by

$\mathcal{C}_{1}\equiv l_{\pm},\quad\mathcal{C}_{2}\equiv
l_{\pm}-\frac{i\beta}{2},\quad l_{\pm}\in\mathbb{R}.$ (16)

In this way, all four wedges $W_{\pm,F,P}$ of Minkowski space correspond to
different combinations of horizontal sections of the Rindler null-coordinate
contours. The fields associated with the horizontal sections
$l_{\pm}\in\mathcal{C}_{2}$ (with non-zero imaginary part $\Im\mathcal{C}\neq
0$) correspond to the causally inaccessible regions of spacetime $W_{-,F,P}$,
and act as invisible fields from the point of view of an accelerated observer.
In other words: The extension of the fields from Rindler to Minkowski
spacetimes acts as a purification of the Hawking-Unruh thermal state.

Extending such a calculation to a topologically non-trivial Yang-Mills theory
is a formidable project. However, the arguments in etesi make it clear that
the fact that winding numbers can be both in the observed and the hidden
patches will generally change the topological structure of the resulting
partition function. A physical reason is that for real positive quark masses,
the partition function in equation (6) admits a quasi-probabilistic
interpretation

$Z=\sum_{n}Z_{n}e^{in\theta},\quad P_{n}\sim\frac{Z_{n}}{Z(\theta=0)},$ (17)

with $P_{n}$ is the “probability” to “measure” a winding number $n$ ulf . Note
that ulf this is a Wigner Quasi-probability wigner rather than a probability,
since for generic $\theta$ it might not be real-valued. However, it expresses
the quantum uncertainty of winding numbers when $\theta$ is fixed, and it
obeys Wigner’s quasi-probability axioms. The winding number, in this picture,
is measurable via the probe in equation (5).

In the quasi-probabilistic interpretation motivated in equation (6), $Z_{n}$
and the topological susceptibility $\partial^{2}\ln Z/\partial^{2}\theta$ will
transform between Minkowski and Rindler space with a factor representing the
ratio of visible to invisible fields. By Bayes’s theorem

$Z_{n}={Z(\theta=0)}\sum_{\mu}\frac{Z_{\mu}}{Z(\theta=0)}\times
P\left(m-n\in\mathcal{C}_{2}\right),$ (18)

and the $P(...)$ term will be directly proportional to the proportion of the
given Rindler time-slice covering each section of Rindler space. This is not
unity, and will depend on the proper time of the Rindler observer. It will
also be an observable, measurable in a Gedankenexperiment by repeated
applications of the operator defined in equation (5).

This lack of covariance has a root in two issues: the winding number is not a
conserved quantum number, and hence is expected to change with the
Hamiltonian. However, “time” is not just a coordinate but also defines the
order in which the co-moving observer makes observations (“collapses the
wavefunction”, or, rather, samples correlators). If one sequentially “observes
it”, over time intervals of the order of an instanton size, one expects it to
change by one unit. However, this change in Hamiltonians will also affect the
partition function according to

$Z=\left<0\right|\exp\left[i\int dt\hat{H}\right]\left|0\right>,$ (19)

and so the procedure

$\ \int dt\rightarrow\frac{1}{N}\sum_{n}\int_{nT/N}^{(n+1)T/N}dt$

would break down. If the winding number is observed along a Minkowski vs. a
Rindler trajectory, the degree of quantum coherence, and $P(...)$ will vary.
The only way to make equation (18) generally covariant appears to be for
$P\left(m-n\in\mathcal{C}_{2}\right)$ to be a unity operator in the winding
numbers basis, which is equivalent to assuming $\theta=0$.

Note that, since we are describing the vacuum rather than correlators,
axiomatic field theory results regarding the equivalence of thermal and
accelerated dynamics axiom need not apply. In fact, symmetry arguments can be
used to understand how non-perturbative vacuum degeneracies change the Unruh
state w.r.t. a thermal state calixto 666note that the $\theta$ state there is
not a QCD one but a generic zero mode condensate. It has however similarities
with the QCD $\theta$ vacuum. The result of calixto and the qualitative
discussion here would imply the accelerated effective $\theta$ is not the same
of the effective $\theta$ at finite temperature.

To get a physical feeling of what is going on here, consider the case of
accelerated photon “Bremsstrahlung” emission, or the famous $p\rightarrow
ne^{+}\nu$ decay examined in matsas . As matsas makes the case, the
“interpretation” in inertial and comoving frames is different (in one case the
decay is a quantum reaction to a semiclassical field, in the other it is
interaction with the Unruh bath) the decay matrix elements calculated in both
cases will be the same.

The equivalent Gedankenexperiment would have us compare the EDM measured on an
accellerated neutron, interpreted by a Minkowski observer (who sees the effect
of accelleration) and the comoving observer (who sees a finite temperature
$\theta$). Because the causal structure of the two observers are different, so
the EDM operator sampled by Eq. 13 will contain a different combination of
winding numbers and $\theta$. Hence, a Minkowski observer and a comoving
observer will see different $Z_{n}$, and hence different effective $\theta$’s
and topological susceptibilities. The only way general covariance is to be a
fundamental principle, $\theta$ must be equal to zero, so the sum of different
topological sectors appears incoherent in all frames.

This looks quite an abstract argument, but it has been known for some time in
the context of lower dimensional Chern-Simons theories, known as the “finite
temperature puzzle” (see section 5.4 of dunne and references therein), whereat
finite temperature the breaking of Lorentz invariance introduces the
appearance of violation of large Gauge invariance order by order. In dunne ,
the breaking of Lorentz invariance is physical, because the system is prepared
at finite temperature. However, such a “finite temperature” state could never
describe the ground state from the vantage point of a reference frame.

The above discussion can also be incorporated into the effective theory of
constituent quarks consti (to which the $\sigma$-models sigma1 are related via
the NJL model) because in such local models “local” and “large” Gauge
transformations are separated, and the former handled perturbatively by adding
a Wilson line $U_{\infty}$ at infinity to each color charge. This way, in the
perturbative limit, topological transformations completely decouple from local
transformations and only the latter are relevant for Feynman diagram
expansion. The problem is that once a horizon exists, $U_{\infty}$ will span
both “visible” and “invisible” fields. Hence, no effective theory,
perturbative or otherwise, can be made from visible fields alone and
topological and local transformations are inseparable. Naively we can
speculate that since (as argued in consti ) local color-charged objects are
forbidden by confinement, this means the $\theta$ angle should also go to
zero. Note that the association between confinement and $\theta$ can be made
using renormalization group arguments rggroup . Of course, as argued in
section 4.1 of polch ; schafer , there is a case for relating these arguments,
with renormalization scheme independence taking the role of general
covariance. If one wants, the symmetry of general covariance and the arguments
in section II provide a theoretical justification for the infrared fixed point
to be of the form of rggroup .

To summarize this section, we made a heuristic and speculative argument that
the only way to restore general covariance is for $\theta$ to be equal to
zero. Because the argument was based on visible and invisible fields, the crux
is that only for $\theta=0$ the coefficient $c_{m}=e^{im\theta}$ does not
change between a coherent and an incoherent sum. In the next section we shall
show, using a toy model from condensed matter physics solved in the 80’s
caldeirabook ; caldeira2 , that explicitly tracing over degrees of freedom
beyond the horizon will generally confirm this conclusion.

##  IV The Periodic potential wells picture of $\theta$

Since it is fashionable to illustrate the $\theta$ problem using the periodic
potential well shuryak ; forkel_instantons ; coleman_instantons , let us try
to get some additional understanding using such a toy model. The models are
physically completely different, in the sense that the periodic potential well
is “engineered” to have a $\theta$-like parameter characterizing the
eigenstates of the Hamiltonian and topological terms have no kinetic modes.
Nevertheless, we think that a dictionary between these problems is useful
enough to extend it to an open quantum system.

Figure 2:  A schematic illustration of the periodic well analogy of
instantons, and the role that horizon radiation plays in them. The QCD
solution is represented by a periodic potential, whose wavefunction is
represented by a tunneling process (corresponding to a localized field
configuration, the instanton) and a dominant “peak” (corresponding to the
asymptotic field configuration, delocalized). Horizon radiation decoheres the
IR peak without altering the localized tunneling. Top image from
forkel_instantons

Let us consider a system consisting of a quantum mechanical particle moving
through potential wells satisfying the periodicity condition $V(x\pm a)=V(x)$,
shown by the black lines in Figure (2). The wells, in this example, represent
topological configurations (winding numbers) of a non-abelian gauge theory at
the horizon. The Hamiltonian for this system is given by the kinetic and
potential terms

$\hat{H}_{S}=\frac{\hat{p}^{2}}{2M}+V(\hat{x}).$ (20)

The solution for these kinds of systems is well known sakurai , and given in
terms of the energy eigenstates

$\left|\theta\right>=\sum_{n\in\mathbb{Z}}\exp(in\theta)\left|n\right>,$ (21)

where $\left|n\right>$ is state of the particle localized at the $n$the site
(or winding number, in the Yang-Mills correspondence), and $\theta$ is a
parameter that labels the simultaneous eigenstates of the Hamiltonian and the
$a$-translation operator $T^{\dagger}(a)V(x)T(a)=V(x+a)$. The associated
density matrix for such $\theta$ state is given by

$\hat{\rho}=\sum_{m,n\in\mathbb{Z}}\exp(i(m-n)\theta)\left|m\right>\left<n\right|.$
(22)

If the potential barriers are tall with respect to the energy of the system,
we can use the tight-binding approximation and obtain an approximation to the
$\theta$-state energy

$E(\theta)=E_{0}-2\Delta\cos\theta$ (23)

from the energy of the localized states
$E_{0}=\left<n\right|\hat{H}\left|n\right>$ and the splitting energy between
adjacent sites $\Delta=|\left<n\pm 1\right|\hat{H}\left|n\right>|$. The
ground-state wavefunction $\psi_{\theta}(x)=\left<x|\theta\right>$ takes the
Bloch form

$\psi_{\theta}(x)=e^{ikx}u_{k}(x),\quad k=\theta/a,$ (24)

where $u_{k}$ a periodic function with period $a$ and $k=\theta/a$ is the
“Bloch momentum” associated with the periodic potential.

We now need to add to the toy model of the $\theta$ vacuum a toy model of the
thermal bath due to the horizon, and the interactions of the system with the
bath. Following caldeirabook ; caldeira2 , let us add a “bath” consisting of
an infinite number harmonic oscillators interacting with the particle in the
periodic potential as

$\hat{H}=\hat{H}_{S}+\hat{H}_{B}$ (25)

where $\hat{H}_{B}$ is the Hamiltonian for the simple harmonic oscillators of
the thermal bath together with a linear interaction term between the bath and
the particle, given by

$\hat{H}_{B}=\sum_{n\in\mathbb{Z}}\left(\frac{1}{2m_{n}}\hat{p}_{n}^{2}+\frac{1}{2}m_{n}\omega_{n}^{2}\left(\hat{q}_{n}+\frac{C_{n}\hat{x}}{m_{n}\omega_{n}^{2}}\right)^{2}\right).$
(26)

For convenience, we define the pure interacting Hamiltonian by
$\hat{H}_{I}=\sum_{n\in\mathbb{Z}}C_{n}\hat{q}_{n}\hat{x}$.

In the interaction picture, the equation of motion for the reduced density
matrix of the particle is given by

$\frac{d\hat{\rho}_{S}}{dt}=-i\operatorname{Tr}_{B}\left[\hat{H}_{I}(t),\hat{\rho}_{S+B}(t)\right].$
(27)

When the couplings $C_{n}$ are equal to zero one recovers equation (21) for
the ground state of the particle, since $\hat{\rho}_{S}$ and $\hat{\rho}_{B}$
decouple and evolve, respectively, under $\hat{H}_{S}$ and $\hat{H}_{B}$
separately. If $C_{n}$ are not equal to zero, the toy model can only be solved
in very particular cases and under certain approximations. Analytical
solutions can be found for simple potential profiles, such as the double well
potential caldeirabook , and the (biased) periodic potential caldeira2 . These
solutions hinges on the fact that the interaction of the Brownian particle
couples to the thermal bath frequencies via the spectral density function

$J(\omega)=\frac{\pi}{2}\sum_{n}\frac{C_{n}^{2}}{m_{n}\omega_{n}}\delta(\omega-\omega_{n}),\quad\omega>0.$
(28)

Exact solutions can be found by limiting the particle’s response to the
particular case of a thermal bath with ohmic profile and a high-frequency
cutoff $\Lambda>0$, given by

$J(\omega)=\eta\omega,\quad 0<\omega<\Lambda.$ (29)

This means that for these kinds of toy models, the dissipative dynamics
induced by environment interactions mostly involves the low-frequency modes of
the environment. In the Yang-Mills correspondence, that means topological
information (associated with infrared degrees of freedom777It is also worth
noting that in usual quantum field theory in flat spacetime, renormalization
group counter-terms are constrained to preserve Lorentz symmetry. It is clear
that if this can be extended to general covariance while preserving local
Lorentz symmetry, it is the IR structure of the theory that must be
constrained gtholo .) decoheres, while localized field configurations, such as
instantons, would remain intact.

In the particular case of a double-well potential caldeirabook , for example,
we can map the high-frequency dynamics of the model to an effective two-level
system. The degrees of freedom in this case are the symmetric and anti-
symmetric combinations of the damped harmonic oscillator states centered at
the bottom of each potential well

$\Psi_{\pm}(x,\\{q_{n}\\})=\frac{1}{\sqrt{2}}[\Psi_{R}(x,\\{q_{n}\\})\pm\Psi_{L}(x,\\{q_{n}\\})].$
(30)

Generally one can separate the associated density matrix into a coherent and
an incoherent part, where the coherent part can be rotated as the projection
part within a certain direction $\hat{S}_{z}$ for the two-level system above
caldeirabook

$\hat{\rho}_{S}=\sum_{n}A_{n}\left|n\right>\left<n\right|+P(t)\sum_{m\neq
n}B_{mn}\left|m\right>\left<n\right|.$ (31)

Since $H_{I}$ does not commute with $H_{S}$, the equation of motion for the
reduced density matrix of the system becomes an initial value problem, with
all time dependence can put into $P(t)$ (which could be a matrix in the
“winding number” basis). It can be shown then that $P(t)$ obeys the damped
harmonic oscillator equation

$\ddot{P}+T^{-1}\dot{P}+\Delta^{2}P=0,$ (32)

with $T^{-1}$ and $\Delta$ functions of the original parameters.

The exact form of the parameters can be an involved calculation, but their
dependence on the length scales of the problem is universal. As shown in
caldeirabook , while $\Delta$ is dominated by short-range physics (in our
context this means instantons equation (2), hence $\Delta^{-1}\sim\rho$), the
damping time-scale $T$ depends on the softest scale connected to the size of
the reservoir (in our context, this is the horizon radius). The latter can be
removed from the system by “adiabatic renormalization”, where all dimensionful
parameters are presented as a ratio of the cutoff frequency. In the limit we
want to reach, where the cutoff frequency dependence is very small, one needs
$\alpha\rightarrow 0$ (defined in chapter 9 of caldeirabook ). In fact, if one
compares the scales in Eq. 12 to the running of Eq. 32 one sees that the
applicability of the Unruh EFT in an instanton context is equivalent to the
theory being near the $\alpha\rightarrow 1$ infrared fixed point. This means
in a theory having a generally covariant infrared limit the effective
$P$,$\sim\theta$ will decay to zero on a time-scale set and inversely
proportional to the lowest frequency. In a scenario where the cosmological
constant is dynamical, such as inflation theory weinbergcosmo this would have
happened very soon in the inflation era.

To compare our models to a real quantum field theory vacuum, either Yang-Mills
or its effective theory implementation (either constituent quarks consti or
mesons where the $\theta$ term appears as a parity-odd rotation in isospin
space sigma1 ) we need to be a bit more rigorous. The path integral formalism
can be connected to density matrix language via entangle , where the
similarity between QCD and the periodic potential setup is more clear.

The density matrix is related to the partition function via

$\langle
x|\,\rho\,|x^{\prime}\rangle=\frac{1}{Z}\int\mathcal{D}\phi(t=t_{0},x)<\phi|\Psi><\Psi|\phi>$
(33) $\
=\frac{1}{Z}\int^{\tau=\infty}_{\tau=-\infty}\int\left[\mathcal{D}\phi,\mathcal{D}y(\tau)\,\mathcal{D}y^{\prime}(\tau)\right]\,e^{-iS(\phi
y,y^{\prime})}\cdot\delta\left[y(0^{+})-x^{\prime}\right]\,\delta\left[y^{\prime}(0^{-})-x\right]\
,$

where $|\Psi>$ is some Eigenstate basis and $\psi$ are field configurations to
be integrated over.

Let us now consider expand $\Psi$ in terms of $\Psi_{n}$, the wave functional
corresponding to equation (17). In both the QCD case and the periodic
potential case,

$\hat{\rho}\underbrace{\rightarrow\int\mathcal{D}\phi}_{\lim_{x\rightarrow
0^{+}-O^{-}}\mathcal{D}\phi(t=t_{0},x)}<\phi|\Psi_{n}><\Psi_{m}|\phi>\exp\left[\theta(n-m)\right]$
(34)

where $\Psi_{n}$ are Eigenstates of the Hamiltonian with, additionally,
boundary condition set by the winding number (For the periodic potential, it
is a fixed number of turns around $N$ wells, for the QCD case it is a given
winding number).

Now, the previous section has argued that background independence implies
independence from horizon terms. In this section, the tracing over the horizon
terms was argued to be equivalent to tracing over the “infrared” degrees of
freedom $k\sim(0^{+}-0^{-})^{-1}$ in equation (33).

Our mechanism adds to the system (S) a thermal bath (B) and in both cases

$\hat{\rho}_{QCD}=\mathrm{Tr}_{B}\hat{\rho}_{S+B},\quad\hat{\rho}_{S+B}\sim
c_{kk^{\prime}}\left|k\right>\left<k^{\prime}\right|,\quad
k_{B}\sim(0^{+}-0^{-})^{-1}\ll k_{S},$ (35)

where $k$ refers to momentum and $c_{kk^{\prime}}$ encode all structure of the
vacuum. The arguments of the previous section make it clear that for
background independence to be achieved, the tracing of the bath degrees of
freedom must make no difference to the effective Lagrangian.

Let us, as above, refer to $\Psi_{m}$ as physical states in flat space of
winding number $m$ and $\Phi_{m}$ as $\Psi_{m}$ with the infrared limit
decohered (all dependence of $k_{B}$ removed, and $k_{S}$ $\sim\rho^{-1}$ of
equation (2) unchanged). While $\Psi_{m}$ and $\Psi_{n}$ are not Eigenstates
of the Hamiltonian, they do form an orthogonal set. In contrast, the
decoherence of $k_{bath}$ means that

$\left<\Phi_{m}|\Phi_{n}\right>\sim Tr_{k}c_{k\sim k_{Bath}}\neq 0$ (36)

with the fact that instanton states have diverging infrared Fourier
coefficients and infinite occupation numbers ensuring a finite overlap even if
the ultraviolet part of the instanton is unchanged.

Thus, the decohered density matrix $\hat{\rho}_{D}$, will not be diagonal in
the $\left|n\right>$ basis. Shifting them to a diagonal basis
$\left|\Psi^{\prime}\right>$ will involve a generally complex rotation in
phase space, whose $j$th eigenvalue can be represented as complex numbers
$r_{j}e^{i\alpha_{j}}$. Putting these together we get, from equation (34),
(35) and (36),

$\hat{\rho}\propto\sum_{n,m,j}e^{i(\theta
n+\sum_{j}\alpha_{j})}\mathcal{D}\phi\left<\phi|\Psi_{n}^{\prime}\right>\left<\Psi_{n}^{\prime}|\phi\right>$
(37)

$\left|\Psi_{n}^{\prime}\right>$ and $\left|\Psi_{n}\right>$ are
distinguishable only via the IR part of the partition function and an
unobservable renormalization parameter given by $r_{j}$. The phase, however,
was rotated by an “infinite” number of angles $\alpha_{j}$, and hence can be
safely assumed to decohere.

Considering the density matrix to evolve dynamically, we would conclude the
infrared form of the dynamics of this decoherence will be controlled by a
damped equation of the type equation (32), with the damping time of the order
of the horizon parameter (the cosmological constant in a de-Sitter space,
acceleration for Rindler space and so on). This “coincidence” can actually be
seen from the form of equation (8) and the argument, made in zubarev , that
general covariance of the time parameter requires correlated fluctuation
(“many outcomes for an initial condition”) and dissipation (“one outcome for
many initial conditions”) that transform covariantly. Topological quantum
fluctuations that fix the $\theta$ term are “the softest scale”, set at the
horizon scale. They must be matched by an equally soft dissipation, making
such “slow” dynamics is unobservable. This implies $\theta$ damping is
irrelevant for any measurement $k\sim k_{Bath}$.

Finally, the interpretation of decoherence as the effect of unmeasured degrees
of freedom can connect this section to the previous section III. Because the
horizon hides the information allowing the winding number to be
instantaneously measured, the density matrix in the basis of $\left|n\right>$
becomes

$e^{in\theta}\left|n\right>\left<m\right|\delta_{mn}\rightarrow\sum_{m^{\prime},n^{\prime}}e^{in\theta}P(n^{\prime},n)P(m^{\prime},m)\left|n^{\prime}\right>\left<m^{\prime}\right|\delta_{m^{\prime}n^{\prime}},$
(38)

where $P(n^{\prime},n)$ represent semi-classical probabilities of winding
numbers disappearing behind a horizon. Rotating bases will add complex
coefficients, parametrized by the $r_{j}e^{i\alpha_{j}}$ of the previous
equation equation (37). The $r_{j}$’s reflect the normalization of the field
strength in response to the horizon changing the configuration space. It
should be unobservable for frequencies higher than the horizon radius. The
phases $\alpha_{j}$ however remains and rotates $\theta$ chaotically for each
momentum mode, equivalent to decoherence (see a good discussion of “coherent”
and “chaotic” sources here gyul ).

In conclusion, we note that a similar mechanism to what we suggest was
proposed in the 80’s as an origin of spontaneously broken  symmetries broksym
. The success of the Higgs model and chiral symmetry breaking invalidated
further development of this idea, but for anomalously broken symmetries it has
potential.

##  V Discussion

The previous section IV makes it clear that any tracing over of degrees of
freedom outside a causal horizon generally results in the asymptotic
relaxation of a density matrix of a theory with connected topological sectors
to a density matrix where they are connected incoherently. This means that
equation (6) is therefore updated to

$\sum_{m,n}e^{i(m-n)\theta}\left|m\right>\left<n\right|\rightarrow\sum_{n}c_{n}\left|n\right>\left<n\right|\rightarrow\sum_{n}\left|n\right>\left<n\right|,$
(39)

where $c_{n}$ doesn’t depend on $\theta$ and is put to unity via field
strength renormalization. $\theta=0$ in this regard is “special” because
coherent and incoherent summation is indistinguishable.

Note that instantons, as any “UV” degrees of freedom, are preserved (as they
should be if there is any hope of the idea presented in this paper to be
correct, since they are necessary for phenomenology (the $\eta^{\prime}$ mass
issue) shuryak ; hooft ; witten1979 , seen on the lattice lattice and hinted
at in experiment instexp ). Equation (39) only modifies the infrared sector of
the theory. In mathematical terms, instanton fluctuations and topological
susceptibility should therefore not be different from the lattice value
lattice . If a universe is “prepared” with a finite $\theta$ and a horizon,
the timescale for the $\theta$ to relax to its effectively zero value can be
given by the methods of section IV. The asymptotic density matrix picture is
also covariant under general coordinate transformations, at least for non-
inertial transformations having an acceleration smaller than the horizon size,
according to section III.

The mechanism described here could be valid, however, even if general
covariance is broken by quantum effects. For instance, it could arise
dynamically in a cosmological scenario. In a “long” high cosmological constant
phase in the early universe, natural in the slow-roll inflation scenario
weinbergcosmo , the topological sector of the QCD vacuum would have time to
decohere if this phase is also deconfined (either due to temperature of a dS
constant above $T_{c}$). Afterwards, the decohered $\theta=0$ state would be
“locked”, since long wavelength colored perturbations which cause tunneling
would be below the confinement mass gap. Such a dynamical non-perturbative QFT
problem is of course well outside this work’s scope, but qualitatively this
scenario might be implementable.

An obvious drawback of the explanation presented here is its lack of
falsifiability. Unlike with more traditional mechanisms of the resolution of
the $\theta$ problem, we do not predict new particles, and is founded on a
fundamental modification of the quantum field theory vacuum state gtholo that
in turn generally does not produce verifiable predictions. However, one might
be able to experimentally test our model with analogue systems. Provided a
fluid with internal symmetries exhibiting topological properties similar to
Yang-Mills theory is found (for example a fluid with polarization montenegro
), one might be able to put this fluid in Minkowski and de-Sitter
configurations mosna . An effective $\theta$ term could then appear in the
former and disappear in the latter, decohered by Hawking sound. The violation
of unitarity that we believe is implicit in the structure of spacetime would
here arise out of the fluid dynamics limit.

Since the effect suggested here involves integrating out infrared degrees of
freedom, there might be a parallel to non-perturbative renormalization group
approaches, investigated elsewhere rggroup and motivated by the analogy
between renormalization group scheme independence and general covariance polch
; schafer . In both cases, the approach does not rely on new physics, but
rather on making sure the underlying theory’s local physics is insensitive to
unobserved infrared perturbations.

In conclusion, we have heuristically discussed the topology of a quantum field
theory in curved space. We have argued that the presence of causal horizons,
or, equivalently, the expectation of general covariance of the theory at the
quantum level, generally require the asymptotic wave functional of the theory
to have different topological sectors incoherently summed. This is just what
one needs to require the $\theta$ angle of the theory to vanish, in accordance
with observations. While we are very far from proving that this is the correct
explanation of the strong CP problem, it is a suggestion that merits further
development.

GT acknowledges support from FAPESP proc. 2017/06508-7, participation in
FAPESP tematico 2017/05685-2 and CNPQ bolsa de produtividade 301432/2017-1. H.
T. Would like to acknowledge the support from the Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), proc. 141024/2017-8. We want
to thank Gabor Etesi, Amir Caldeira and Sabine Hossenfelder for discussions
and suggestions.

##  Appendix A Toy models

In this section we shall describe toy models which include a boundary term and
the possibility to calculate the density matrix and the various forms of
entropy explicitly. They are limited in that they do not allow us to fully
understand how topological terms are generated by the dynamics, since for such
1+1 theories topology arises out of boundary conditions rather than dynamics,
and is decoupled from the bulk.

However, dynamical coupling between bulk and boundary can be put in by hand in
an adiabatic way which generates no entropy. In this case, it becomes clear
that the “geometric” and “statistical” definitions of entropy do not match,
thereby giving an example of the tension between general covariance and
topological terms.

###  A.1 Quantum particle on a ring with a magnetic field

We consider the quantum mechanical system of a particle in a ring $S^{1}$,
characterized by the Euclidean action

$iS=\int
dt\left(\frac{1}{2}I\dot{\phi}^{2}+\mathcal{B}\dot{\phi}\right)\longrightarrow-
S_{E}=-\int
d\tau\left(\frac{1}{2}I\dot{\phi}^{2}-i\mathcal{B}\dot{\phi}\right)$ (40)

for the angular variable $\phi$. The periodic-boundary conditions
characterizing the system are given by $\phi(\tau+\beta)=\phi(\tau)$ and
$\phi\sim\phi+2\pi n$. This theory then represents a $(0+1)$ dimensional field
theory $\phi:S^{1}\rightarrow S^{1}$, which is a topological theory since
$\pi_{1}(S^{1})=\mathbb{Z}$. The topological charge is represented by how many
times around the circle the particle runs on the course of one period $\beta$
– counted by the $i\mathcal{B}\dot{\phi}$ term on the action.

We use this model as a toy example for the Laflamme procedure of obtaining a
description for the density matrix of a (free) field theory in an accelerated
frame. This procedure is described as follows: The comoving-time of an
accelerated observer is periodic in Euclidean signature, due to the natural
invariance of Minkowski spacetime under boosts with imaginary-boost parameter,
and that means the accelerated observer’s proper-time is represented in
Euclidean signature by the angular variable in a polar-coordinate
representation of spacetime. The presence of causal horizons for the
accelerated observer is reflected on the fact that you can’t cover the circle
with a single coordinate chart: That can be seen from the transformation
$\tau\rightarrow\tau+\pi$ on the circle, which maps the points
$(\tau+\pi,r)\rightarrow(\tau,-r)$. Under reverse Wick rotation
$\tau\rightarrow i\tau$, that transformation maps a Rindler wedge into it’s
causal complement (that means mapping the right wedge into left wedge and vice
versa).

Now we can describe the causally disconnected regions of Minkowski space by
considering a division of the circle into two sections, with the common
boundary (shown in figure (3)) representing the causal horizon structure of
the accelerated frame.

Figure 3: Figure from laflamme , representing the Euclidean section of
spacetime representing an accelerated observer. $\mathcal{M}_{\pm}$ are the
pieces representing the right and left wedges respectively, and
$\mathcal{S}_{k}$ the boundaries associated with the presence of causal
horizons. The circumference of the circle is $\beta$, and
$\tau_{\mathcal{S}_{2}}=\tau_{\mathcal{S}_{1}}+\beta/2$.

We also include the magnetic field term $\mathcal{B}$ and the boundary
regulator described at the end of this section, leading to Eq. 81. These are
represented by the circuit connecting $\mathcal{B}$ to $S_{1,2}$

To obtain a description of physics from the point of view of the accelerated
observer, we integrate out the contributions from the inaccessible region.

In mathematical terms, that corresponds to describing the theory on
$\mathcal{M}_{+}$ by the path-integral summing over the complementary region
$\mathcal{M}_{-}$: Let $\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})$ be the
partition function for the fields on $\mathcal{M}_{\pm}$ satisfying the
boundary conditions $\phi(\mathcal{S}_{1})=\tilde{\phi}_{1}$ and
$\phi(\mathcal{S}_{2})=\tilde{\phi}_{2}$. The path integral we are interested
in evaluating is, then, given by

$\rho(\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime})=\int\mathcal{D}\tilde{\phi}_{2}\Psi_{+}(\tilde{\phi}_{1},\tilde{\phi}_{2})\Psi_{-}(\tilde{\phi}_{1}^{\prime},\tilde{\phi}_{2}),$
(41)

where $\rho$ is the density matrix associated with the states
$\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime}\in\mathcal{M}_{+}$ accessible to
the observer.

In the context of quantum field theory, figure (3) is only a representation of
the true Euclidean section of spacetime, whose spatial dimensions compose the
correct picture in higher dimensions. However, for free theories such as the
scalar field

$S=\frac{1}{2}\int
d^{D+1}x\sqrt{g}\left(g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi+m^{2}\Phi^{2}\right),$
(42)

we can make use of the Fourier transform and study a single mode of the field
laflamme

$S_{E}^{\lambda}=\frac{1}{2}\int
d\tau\left(\dot{\phi}_{\lambda}^{2}+\lambda\phi_{\lambda}^{2}\right),$ (43)

which reduces to a quantum harmonic oscillator in a ring. Given the close
connection with Euclidean-signature quantum mechanics on a ring with the
physics of accelerated observers, we use this analogy to study the topological
system from equation (40) in an accelerated frame.

Our target is therefore calculating the density matrix in equation (41) for
the model described by the action (40) for the particle in the ring. First,
then, we calculate the partition functions

$\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})=\int_{\mathcal{C}[\mathcal{M}_{\pm}]}\mathcal{D}\phi\,e^{-S_{E}}$
(44)

for the $(0+1)$-dimensional field $\phi$ with corresponding boundary
conditions

$\mathcal{C}[\mathcal{M}_{\pm}]\equiv\\{\phi(\mathcal{S}_{1})=\tilde{\phi}_{1}\text{
and }\phi(\mathcal{S}_{2})=\tilde{\phi}_{2}\\}.$ (45)

Being a free theory, the path-integral can be solved by means of the solutions
to the Euclidean equations of motion

$\frac{d^{2}}{d\tau^{2}}\phi_{cl}(\tau)=0$ (46)

that extremizes the action. By virtue of the topological nature of the theory,
the solutions of this classical equation of motion are parametrized by an
integer $n\in\mathbb{Z}$ that counts how many times the particle winds around
the circle for given boundary conditions. Let $\phi_{cl}^{\pm}$ be the
classical solutions to the Euclidean equations of motion (46)

$\begin{split}\phi_{cl}^{+}(\tau)&=\tilde{\phi}_{1}+\frac{2}{\beta}\left(\tilde{\phi}_{2}-\tilde{\phi}_{1}+2\pi
n_{+}\right)\tau\\\
\phi_{cl}^{-}(\tau)&=2\tilde{\phi}_{2}-\tilde{\phi}_{1}-2\pi
n_{-}+\frac{2}{\beta}\left(\tilde{\phi}_{1}-\tilde{\phi}_{2}+2\pi
n_{-}\right)\tau\end{split}$ (47)

satisfying the boundary conditions (45). The integers $n_{\pm}\in\mathbb{Z}$
are the winding numbers of the field configurations, which appear in the
combination $2\pi n_{\pm}$ since $\phi(\tau)$ lives on the circle $S^{1}$. The
field configurations

$\exp\left\\{i\phi_{cl}^{\pm}(\tau)\right\\}\in S^{1}$ (48)

are called instantons, and they interpolate between the classical,
$\tau$-independent, zero-action “vacuum” solutions
$\tilde{\phi}_{cl}^{vac}(\tau)=\tilde{\phi}_{1/2}$, on each
$\mathcal{M}_{\pm}$ manifold.

The angular velocity of the classical solutions take the form

$\dot{\phi}_{cl}^{\pm}(\tau)=\frac{4\pi}{\beta}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right),$
(49)

and the classical action for each instanton solution is given by the
expression

$\begin{split}-S_{E}^{\pm}&=-\int_{\mathcal{C}[\mathcal{M}_{\pm}]}d\tau\left\\{\frac{1}{2}I\left(\frac{4\pi}{\beta}\right)^{2}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right)^{2}-\frac{4\pi
i\mathcal{B}}{\beta}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right)\right\\}\\\
&=-\frac{4\pi^{2}I}{\beta}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right)^{2}+2\pi
i\mathcal{B}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right).\end{split}$
(50)

Now, the partition functions $\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})$
are given by the sum over all possible field configurations $\phi^{\pm}$, and
all such field configurations falls under a particular equivalence class
determined by its winding number $n[\phi^{\pm}]\in\mathbb{Z}$. That means we
must sum over all classical instanton solutions, taken to be the
representatives of each winding-sector, as well as all possible fluctuations

$\delta\phi^{\pm}=\phi-\phi_{cl}^{\pm}$ (51)

around these classical backgrounds. The fluctuating fields $\delta\phi^{\pm}$
are topologically trivial, meaning $n[\delta\phi^{\pm}]=0$, and satisfy the
boundary conditions

$\delta\phi^{\pm}(\mathcal{S}_{i})=0.$ (52)

Since we are expanding around the solutions to the classical equations of
motion, the total action is merely the sum

$S[\phi]=S[\phi_{cl}^{\pm}]+S[\delta\phi^{\pm}],$ (53)

and therefore we obtain the factorization

$\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})=\int{\mathcal{D}\phi}\,e^{-S_{E}}\sim
Z_{\delta\phi^{\pm}}\sum_{n_{\pm}\in\mathbb{Z}}e^{-S_{E}[\phi_{cl}^{\pm}]},$
(54)

with $Z_{\delta\phi^{\pm}}$ the (topologically trivial) partition function for
the fluctuating field $\delta\phi^{\pm}$

$Z_{\delta\phi^{\pm}}=\int_{\delta\phi^{\pm}(\mathcal{S}_{1})=0}^{\delta\phi^{\pm}(\mathcal{S}_{2})=0}\mathcal{D}\delta\phi^{\pm}e^{-S[\delta\phi^{\pm}]}.$
(55)

Since we are interested in the topological properties of the reduced density
matrix, and $Z_{\delta\phi^{\pm}}$ is only part of the normalization of
$\Psi_{\pm}$, we only deal with the instanton sum of equation (54).
Substituting the expression for the classical action of each instanton
solution (50), we get

$\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})\sim\sum_{n_{\pm}}\exp\left\\{-\frac{4\pi^{2}I}{\beta}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right)^{2}+2\pi
i\mathcal{B}\left(n_{\pm}\pm\frac{\tilde{\phi}_{2}-\tilde{\phi}_{1}}{2\pi}\right)\right\\}.$
(56)

Now we are in a position to evaluate the reduced density matrix (41) obtained
by integrating out all $\mathcal{M}_{-}$ fields. An elegant approach is to
make use of the representation of the partition functions $\Psi_{\pm}$ in
terms of the Riemann’s theta function moore

$\mathcal{R}\begin{bmatrix}n_{0}\\\
z_{0}\end{bmatrix}(z,\tau)=\sum_{n\in\mathbb{Z}}\exp\left(i\pi\tau(n+n_{0})^{2}+2\pi
i(n+n_{0})(z+z_{0})\right).$ (57)

Writing $\Delta\tilde{\omega}=(\phi_{2}-\phi_{1})/2\pi$, we get the compact
expression for the wedge partition functions

$\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})\sim\mathcal{R}\begin{bmatrix}\pm\Delta\tilde{\omega}\\\
\mathcal{B}\end{bmatrix}\left(0,\frac{4\pi iI}{\beta}\right).$ (58)

Using the transformation property of the $\mathcal{R}$ function under modular
transformations

$\mathcal{R}\begin{bmatrix}n_{0}\\\
z_{0}\end{bmatrix}\left(z,\tau\right)=(-i\tau)^{-1/2}e^{2\pi
iz_{0}n_{0}}e^{-i\pi z^{2}/\tau}\mathcal{R}\begin{bmatrix}z_{0}\\\
-n_{0}\end{bmatrix}\left(\frac{-z}{\tau},\frac{-1}{\tau}\right),$ (59)

we obtain an alternative and equivalent representation

$\mathcal{R}\begin{bmatrix}\pm\Delta\tilde{\omega}\\\
\mathcal{B}\end{bmatrix}\left(0,\frac{4\pi iI}{\beta}\right)=\left(\frac{4\pi
I}{\beta}\right)^{-1/2}e^{\mp 2\pi
i\mathcal{B}\Delta\tilde{\omega}}\mathcal{R}\begin{bmatrix}\mathcal{B}\\\
\mp\Delta\tilde{\omega}\end{bmatrix}\left(0,-\frac{\beta}{4\pi iI}\right).$
(60)

Expanding this expression in terms of the sum representation of $\mathcal{R}$
we end up with the relationship

$\Psi_{\pm}(\tilde{\phi}_{1},\tilde{\phi}_{2})\sim\sum_{n_{\pm}}\exp\left\\{-\frac{4\pi^{2}I}{\beta}\left(n_{\pm}\pm\Delta\tilde{\omega}\right)^{2}+2\pi
i\mathcal{B}\left(n_{\pm}\pm\Delta\tilde{\omega}\right)\right\\}=\\\
=\left(\frac{4\pi
I}{\beta}\right)^{-1/2}\sum_{n_{\pm}}\exp\left\\{-\frac{\beta}{4I}(n_{\pm}-\mathcal{B})^{2}\pm
2\pi in_{\pm}\Delta\tilde{\omega}\right\\}.$ (61)

Now we can proceed to integrate out $\tilde{\phi}_{2}$ using the new
representation of $\Psi_{\pm}$. In this representation, the reduced density
matrix is given by the expression

$\rho(\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime})\sim\int_{0}^{2\pi}d\tilde{\phi}_{2}\Psi_{+}(\tilde{\phi}_{1},\tilde{\phi}_{2})\Psi_{-}(\tilde{\phi}_{1}^{\prime},\tilde{\phi}_{2})\sim\\\
\sim\left(\frac{\beta}{4\pi
I}\right)\int_{0}^{2\pi}{d\tilde{\phi}_{2}}\,\sum_{n_{\pm}\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{4I}\left(n_{+}-\mathcal{B}\right)^{2}+in_{+}\left(\tilde{\phi}_{2}-\tilde{\phi}_{1}\right)\right\\}\times\\\
\times\exp\left\\{-\frac{\beta}{4I}\left(n_{-}-\mathcal{B}\right)^{2}-in_{-}\left(\tilde{\phi}_{2}-\tilde{\phi}_{1}^{\prime}\right)\right\\},$
(62)

and by using the integral representation of the Kronecker delta

$\delta_{mn}=\frac{1}{2\pi}\int_{0}^{2\pi}d\tilde{\phi}_{2}\,e^{i(m-n)\tilde{\phi}_{2}}$
(63)

we obtain

$\rho(\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime})\sim\left(\frac{\beta}{2I}\right)\sum_{n\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{2I}\left(n-\mathcal{B}\right)^{2}-in\left(\tilde{\phi}_{1}-\tilde{\phi}_{1}^{\prime}\right)\right\\}.$
(64)

Once again we normalizing the density matrix by demanding unity trace

$\operatorname{Tr}\rho=\int_{0}^{2\pi}d\tilde{\phi}\,\rho(\tilde{\phi},\tilde{\phi})=1,$
(65)

and that yields the expression for the matrix elements of the density matrix

$\rho(\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime})=\frac{1}{2\pi
Z}\sum_{n\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{2I}\left(n-\mathcal{B}\right)^{2}-in\left(\tilde{\phi}_{1}-\tilde{\phi}_{1}^{\prime}\right)\right\\},$
(66)
$Z=\sum_{n\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{2I}\left(n-\mathcal{B}\right)^{2}\right\\}.$
(67)

We could have obtained this result more directly from the expression of the
Euclidean propagator

$\langle\tilde{\phi}_{1}^{\prime}|e^{-\beta
H}|\tilde{\phi}_{1}\rangle=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{2I}(n-\mathcal{B})^{2}-in(\tilde{\phi}_{1}-\tilde{\phi}_{1}^{\prime})\right\\}.$
(68)

This propagator can be obtained from the system Hamiltonian

$H=\frac{1}{2I}\left(-i\partial_{\phi}-\mathcal{B}\right)^{2}$ (69)

whose spectrum consists in the eigenvectors
$\\{\left|m\right>,m\in\mathbb{Z}\\}$, given by

$\langle\phi|m\rangle=\Phi_{m}(\phi)=\frac{1}{\sqrt{2\pi}}e^{im\phi}.$ (70)

The procedure of integrating out half of the spacetime circle corresponds to
the process of separating $\beta=\frac{\beta}{2}+\frac{\beta}{2}$ and
introducing a complete set of states in between

$\langle\tilde{\phi}_{1}^{\prime}|e^{-\beta
H}|\tilde{\phi}_{1}\rangle=\int_{0}^{2\pi}d\tilde{\phi}_{2}\,\langle\tilde{\phi}_{1}^{\prime}|e^{-\frac{1}{2}\beta
H}|\tilde{\phi}_{2}\rangle\langle\tilde{\phi}_{2}|e^{-\frac{1}{2}\beta
H}|\tilde{\phi}_{1}\rangle\\\
=\int_{0}^{2\pi}d\tilde{\phi}_{2}\,\Psi_{+}(\tilde{\phi}_{1},\tilde{\phi}_{2})\Psi_{-}(\tilde{\phi}_{1}^{\prime},\tilde{\phi}_{2})\sim\rho(\tilde{\phi}_{1},\tilde{\phi}_{1}^{\prime}).$
(71)

Following this, we proceed to calculate thermodynamical quantities from the
model. First, we note that we can write the identity in two equivalent ways

$\int_{0}^{2\pi}d\phi|\phi\rangle\langle\phi|=\sum_{m\in\mathbb{Z}}|m\rangle\langle
m|=\mathbb{I}.$ (72)

For convenience, we note that (66) can be rewritten in terms of the
eigenfunctions $\Phi_{m}(\phi)$ of the Hamiltonian

$\begin{split}\langle\phi^{\prime}|\rho|\phi\rangle=\frac{1}{2\pi
Z}\sum_{n\in\mathbb{Z}}\exp\left\\{-\frac{\beta}{2I}\left(n-\mathcal{B}\right)^{2}-in\left(\phi-\phi^{\prime}\right)\right\\}=\frac{1}{Z}\sum_{n\in\mathbb{Z}}e^{-\beta
E_{n}}\Phi_{n}(\phi)\Phi_{n}^{*}(\phi^{\prime}).\end{split}$ (73)

We can also find the matrix elements of the density matrix in the Hamiltonian
eigenstate basis

$\begin{split}\langle m|\rho|n\rangle=\frac{1}{Z}e^{-\beta
E_{m}}\delta_{mn}.\end{split}$ (74)

Note that in the basis of Hamiltonian eigenstates $\rho$ is diagonal, and we
find that

$\rho=\frac{1}{Z}e^{-\beta H},$ (75)

which indeed confirms that $\rho$ represents a finite temperature system of
excitations of $H$. The energy and entropy of the system are obtained by using
the diagonal representation of $\rho$ in the basis of eigenstates of $H$, and
are given by

$E\equiv\langle H\rangle\equiv\operatorname{Tr}\rho
H=\frac{1}{Z}\sum_{m\in\mathbb{Z}}E_{m}e^{-\beta E_{m}}$ (76)

in this picture we can calculate the Von Neumann entropy and recover the usual
thermodynamic relation

$\begin{split}S\equiv-\langle\ln\rho\rangle\equiv-\operatorname{Tr}\rho\ln\rho=\beta\langle
H\rangle+\ln Z.\end{split}$ (77)

Differentiating the entropy with respect to the internal energy we get the
expected relation

$\frac{\partial S}{\partial E}=\beta,$ (78)

Confirming that indeed $\beta^{-1}=T$ represents the temperature of the
system. Taking the entropy to be an explicit function of both $\beta$ and
$\mathcal{B}$ we obtain the usual thermodynamical expression

$dS=\left(\frac{\partial
S}{\partial\beta}\right)_{\\!\\!\mathcal{B}}d\beta+\left(\frac{dS}{d\mathcal{B}}\right)_{\\!\\!\beta}d\mathcal{B}=\beta\,dE+\left(\frac{\partial
S}{\partial\mathcal{B}}\right)_{\\!\\!E}d\mathcal{B}.$ (79)

Observing equations (66), it would appear that that the $\mathcal{B}$-term on
the density matrix doesn’t depend on the temperature of the field $\beta\sim
T^{-1}$. That means the topological sector, in this particular case, does not
seem to be sensitive to the Hawking-Unruh temperature associated with the
Euclidean-periodicity in proper-time. In this toy-model, the magnetic field
$\mathcal{B}$ takes the role of the theta-parameter in other scenarios, such
as Chern-Simons theory and the CP-violating $\theta QCD_{4}$ scenario.

On the other hand, in real 3+1 field theory $\mathcal{B}$ is determined
dynamically from the theory. While we cannot reproduce this effect explicitly,
we can “mock it up” and study its consequences in the following way: let us
hook up a system prepared as in Fig. 3, a quantum particle on a circle in
equilibrium, to an adiabatic device measuring heat capacity. This device in
turn regulates a magnetic solenoid generating the field $\mathcal{B}$. This
device is adiabatic and does not measure any microstates, so, unlike
“Maxwell’s demon type setups”, its added entropy content is negligible.

What it does, however, is bring the system away from the thermodynamic limit,
so the thermodynamic entropy is not automatically the Legendre transform of
the energy. Instead, it is defined by

$\beta^{\prime}=\left.\frac{dS}{dE}\right|_{\hat{n}}$ (80)

where $\hat{n}$ is the direction in $\mathcal{B},E$ space where two systems
hooked up in parallel stop.

Considering the tangent vector $\hat{n}$ as the derivative of a curve
$\gamma(\lambda)=(E(\lambda),\mathcal{B}(\lambda))$ at $\lambda=0$, we get
$\beta^{\prime}=\gamma^{\prime}(0)\cdot\nabla S$. Parameterizing the curve
with the internal energy we get that the equilibrium configuration of such a
system is related to the usual equilibrium as

$\beta^{\prime}(E,\mathcal{B})\equiv\hat{n}\cdot\nabla
S=\beta+\left(\frac{\partial
S}{\partial\mathcal{B}}\right)_{\\!\\!E}\frac{d\mathcal{B}}{dE}.$ (81)

The moral of the story here is that if there is coupling between bulk and
topological degrees of freedom, one could expect that geometric and
thermodynamic temperatures are different. While this was a quantum problem
without a “real” Unruh effect, in the next section we shall generalize this
reasoning to a one dimensional quantum field theory.

###  A.2 Maxwell theory in (1+1) dimensions

Let us now consider the 1+1D Maxwell theory examined in adam ; blomm ; blomm2
. This is a quantum field theory, where the Unruh effect is possible. At first
site, the equivalence of Eq. 2.28 of blomm for the topological configuration
with Eq. 49 of calixto , together with the discussion following Eq. 49, would
seal our case that a non-trivial topological charge will spoil the Unruh
effect. However, this is not the case, since the theory examined in blomm has
no propagating degrees of freedom.

In fact, this theory can be mapped on the problem examined in the previous
section, since pure electrodynamics in $(1+1)$ dimensions is a topological
theory, since $\pi_{1}(U(1))=\mathbb{Z}$. This means pure electrodynamics
admits a topological $\theta$ term, similar in nature to the QCD case, given
by

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{e\theta}{4\pi}\epsilon_{\mu\nu}F^{\mu\nu}\equiv\frac{1}{2}\mathcal{E}^{2}+\frac{e\theta}{2\pi}\mathcal{E},$
(82)

where $\mathcal{E}\equiv F_{01}$. Since the Lagrangian only depends on
$\mathcal{E}$, we can treat it like the canonical field and find its equation
of motion

$\mathcal{E}=-\frac{e\theta}{2\pi}.$ (83)

Since the field is just a constant $\mathbb{C}$-number, the theory is trivial.
That means there is only one physical state, the vacuum state
$\left|\theta\right>$, satisfying

$\mathcal{E}\left|\theta\right>=-\frac{e\theta}{2\pi}\left|\theta\right>,\quad\mathcal{H}\left|\theta\right>=\frac{1}{2}\left(\frac{e\theta}{2\pi}\right)^{2}\left|\theta\right>\phantom{AA},\phantom{AA}\dot{F}=0$
(84)

In other words, $\theta$ is the simultaneous eigenstate of the Hamiltonian
density and the electric field, and $\theta$ can be interpreted as a constant
background electric field $\mathcal{E}=\frac{e\theta}{2\pi}$. In terms of
$\mathcal{E}$, that means the Hamiltonian density takes the form

$\mathcal{H}\left|\mathcal{E}\right>=\frac{\mathcal{E}^{2}}{2}\left|\mathcal{E}\right>.$
(85)

Furthermore, as shown in blomm2 , this theory on a disk can be mapped exactly
to the particle on a ring problem of the previous section. From the
electromagnetic duality in $(1+1)$ dimensions, the Euclidean partition
function of the theory in Eq. (82) is related to the particle in a ring by

$\begin{split}Z&=\int\mathcal{D}A\exp\left\\{-\int
d^{2}x\left(\frac{1}{4}F_{\mu\nu}F_{\mu\nu}-i\frac{e\theta}{2\pi}\epsilon_{\mu\nu}F_{\mu\nu}\right)\right\\}\\\
&=\int\mathcal{D}\mathcal{E}\delta(\dot{\mathcal{E}})\exp\left\\{-\int_{0}^{\beta}d\tau\,\left(\frac{a}{2}\mathcal{E}^{2}-i\frac{e\theta}{2\pi}\mathcal{E}\right)\right\\}\\\
&\equiv\int\mathcal{D}\phi\exp\left\\{-\int_{0}^{\beta}d\tau\left(\frac{1}{2a}\dot{\phi}^{2}-i\frac{e\theta}{2\pi}\dot{\phi}\right)\right\\},\end{split}$
(86)

where $a$ is defined in relation to the area enclosed by the ring
$a=\beta^{-1}\int\sqrt{g}$. Thus, the correspondence becomes

$\mathcal{E}\rightarrow\dot{\phi},\quad a\rightarrow
I^{-1},\quad\text{and}\quad\frac{e\theta}{2\pi}\rightarrow\mathcal{B}.$ (87)

As seen in entangle the partition function requires asymptotic states as well
as the action. However, we can use this equivalence as a starting point to
calculate exactly the vacua in inequivalent frames.

Let us consider such a field prepared to be in a Minkowski vacuum state. By
means of equation (4.24) of blomm , defining the action of the Wilson line on
the wedge-decomposition of a state as

$\mathcal{W}_{e}\left|\Psi\right>=\mathcal{W}_{e}\frac{1}{Z}\sum_{\mathcal{E}}e^{-A\frac{\mathcal{E}^{2}}{2}}\left|\mathcal{E}\right>_{L}\otimes\left|\mathcal{E}\right>_{R}=\frac{1}{Z}\sum_{\mathcal{E}}e^{-A\frac{\mathcal{E}^{2}}{2}}\left|\mathcal{E}+e\right>_{L}\otimes\left|\mathcal{E}+e\right>_{R},$
(88)

we can reconstruct the Minkowskian wavefunctional for $\mathcal{E}=0$, given
in equation (4.30) of blomm , using the Wilson line for an electric field
$\mathcal{E}=\mathcal{E}_{0}$ as

$\mathcal{W}_{\mathcal{E}_{0}}\left|\Psi\right>=\mathcal{W}_{\mathcal{E}_{0}}\frac{1}{Z}\sum_{\mathcal{E}}e^{-\frac{\pi
R^{2}}{2}\frac{\mathcal{E}^{2}}{2}}\left|\mathcal{E}\right>_{L}\otimes\left|\mathcal{E}\right>_{R}=\frac{1}{Z}\sum_{\mathcal{E}}e^{-\frac{\pi
R^{2}}{2}\frac{\mathcal{E}^{2}}{2}}\left|\mathcal{E}+\mathcal{E}_{0}\right>_{L}\otimes\left|\mathcal{E}+\mathcal{E}_{0}\right>_{R}\\\
=\frac{1}{Z}\sum_{\mathcal{E}}e^{-\frac{\pi
R^{2}}{4}(\mathcal{E}-\mathcal{E}_{0})^{2}}\left|\mathcal{E}\right>_{L}\otimes\left|\mathcal{E}\right>_{R}.$
(89)

On the limit $R\rightarrow\infty$, we get a delta function $e^{-\frac{\pi
R^{2}}{4}(\mathcal{E}-\mathcal{E}_{0})^{2}}\rightarrow\delta(\mathcal{E}-\mathcal{E}_{0})$
around $\mathcal{E}_{0}$ on the exponential and thus obtain

$\left|\mathcal{E}_{0}\right>=\left|\mathcal{E}_{0}\right>_{L}\otimes\left|\mathcal{E}_{0}\right>_{R}.$
(90)

Remembering that such constant electric fields are corresponding to $\theta$
originally, we get

$\left|\theta\right>_{M}=\left|\theta\right>_{L}\otimes\left|\theta\right>_{R}.$
(91)

The separation of spacetime into right and left wedges correspond to a
separation of the state into two sectors, both corresponding to the same
parameter $\theta$. That result can also be obtained directly from equation
(4.31) on blomm by applying the Wilson line operator
$\mathcal{W}_{\mathcal{E}}$

$\left|\mathcal{E}\right>_{M}=\mathcal{W}_{\mathcal{E}}\left|0\right>_{M}=\mathcal{W}_{\mathcal{E}}\left|0\right>_{L}\otimes\left|0\right>_{R}\equiv\left|\mathcal{E}\right>_{L}\otimes\left|\mathcal{E}\right>_{R}.$
(92)

These results were calculated using a vacuum prepared as an Eigenstate of the
field theory in Minkowski space. Hence, in this particular example (of
$(1+1)$-dimensional pure photodynamics) the $\theta$’s seem to just factorize
into right and left sectors, without having to be constrained to the
$\theta=0$ case.

As a consequence, take a localized observable $\mathcal{O}$ with support on
the right Rindler wedge. Then $\mathcal{O}$ is non-zero only on
$x\in\mathcal{M}_{R}$, and therefore completely accessible to both an inertial
and a right-accelerated observer that has $\mathcal{M}_{R}$ as its causal
region. Then

$\left<\mathcal{O}\right>_{\Psi}=\int_{\mathcal{B}(\Psi)}[\mathcal{D}A]\,\mathcal{O}\,e^{-S_{E}}\rightarrow\operatorname{Tr}[\mathcal{O}\,e^{-\beta
H_{R}}],$ (93)

because the functional integral reduces to a sum over field configurations
supported on $\mathcal{M}_{R}$. With $H_{R}$ the Rindler Hamiltonian
associated with the $\theta$-dependent action above.

The discussion above is specific to the 1+1 theory because of the relation
adam between the topological current $\epsilon_{\mu\nu}A^{\mu}$ and the charge
at the horizon $\int dxA_{1}$. The charge at the horizon introduces a local
Gauss-law constraint which can be defined and changed locally. As the
Reissner-Nordstrom black holes (where the charge is also ”local” at the
singularity) show, this constraint is valid across horizons. In 3+1 dimensions
there is no Gauss constraint on $\theta$ and $K_{\mu}$ is controlled by Eq. 5,
which has no equivalent local gauge constraint.

We can make the 1+1 theory a bit more ”3D like” if, in analogy to the previous
section A.1 we add the following source terms to the Lagrangian in Eq. 82

$\mathcal{L}\rightarrow\mathcal{L}+J_{\theta}\times\left(\theta+J_{L}(\theta)\right)$
(94)

corresponding to a Rindler-type detector that measures $\theta$ and a device
positioned asymptotically at the lest wedge that tunes $\theta$ according to
some function (this of course breaks the Gauge constraint). In this case, we
reproduce the same situation as in Eq. 81. Equation 93 still holds, but the
entropy density calculated via Eq. 33 from

$\
s(\beta^{\prime})\equiv\operatorname{Tr}_{R}\left[\left[\hat{\rho}\ln\hat{\rho}\right]_{\beta^{\prime}}-\left[\hat{\rho}\ln\hat{\rho}\right]_{\beta^{\prime}\rightarrow\infty}\right]\propto\left(\beta^{\prime}\right)^{-2}\phantom{AA},\phantom{AA}\hat{\rho}=\hat{\rho}_{L}\times\hat{\rho}_{R}$

will not scale with the same $\beta^{\prime-1}=a/(2\pi)$ as expected from the
Unruh effect but will instead obey an equation such as 81. As a consequence,
the ratio of $\beta^{\prime}$ and $\beta$ is not necessarily set by the
microscopic scale,which in this case is the acceleration

Of course, axiomatic quantum field theory axiom is not violated here because
the source terms $J_{\theta},J_{L}$ introduce a preferred coordinate system
which breaks local Lorentz invariance. However, the effect of topological
dynamics in 3D can be argued to be equivalent to a spontaneous generation of
$J_{\theta},J_{L}$. While the scale of $J_{\theta}$ is local (instanton peak)
and of $J_{L}$ is global (instanton tail), as shown in Eq. 81 scale separation
in such a case is not automatically perturbative. This illustrates that, as
argued in section II topological terms can provide a mixing between “local”
detector-scale physics and infrared scales.

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