3.1. Brief Review of Polymer Quantization and Quantum Mechanics on
In ordinary quantum mechanics, any representation of the fundamental commutation relation of position and momentum leads to identical physical predictions by virtue of the Stone–von Neumann uniqueness result [
47]. This property relies on the (weak) continuity of the so-called Weyl elements, that is the one-parameter unitary groups associated to the position and momentum operators, which in the Schrödinger case are given by
and
. The Weyl elements satisfy the following properties
The task of quantization is to find a representation of the ensuing Weyl algebra on some suitable Hilbert space. The conventional choice is simply
with the standard inner product. In this representation, both Weyl elements are weakly continuous and thus the derivatives of the Weyl elements with respect to their parameters and hence the associated infinitesimal generators exist as operators on this Hilbert space. In recent years, the so-called polymer representation has gained attention due to its inevitable presence in Loop Quantum Gravity (LQG), see for instance [
26,
27,
48,
49], and hence possible connection with Planck-scale physics. Especially in the context of Loop Quantum Cosmology (LQC), as exemplarily treated in [
28,
29,
30,
50,
51,
52,
53], the polymer representation offers a variety of interesting aspects that are not included in the Wheeler–de Witt approach. The crucial difference lies in the discontinuity of at least one of the Weyl elements. The polymer representation in one dimension can be also formulated in terms of an
space but no longer over
but instead involves the so-called
Bohr compactification. In order to see this one considers the additive group
and uses that the corresponding characters
of
labelled by
form an isomorphic group that we denote by
. Note that a locally compact Abelian group
G is compact if and only if
is discrete, so we equip
with the discrete topology, where this brief introduction closely follows the notation in [
54]. Consequently, the dual of
, more precisely its set of characters, is the afore-mentioned Bohr compactification
of the original group
G, which carries a natural probability measure in terms of the Haar measure
[
38,
54,
55]. The polymer Hilbert space
and its most convenient orthonormal basis are then given by
where the basis is given in
q-polarization. In this representation, that we call the A-type representation following the notation from [
38], the fundamental operators act upon the basis elements according to
The inner product is defined via the Haar measure on
:
Due to the uncountably infinite number of basis elements,
is non-separable. Furthermore, in constrast to the Schrödinger representation the Weyl element
is not weakly continuous at
since any two states
are orthogonal for all
with
. Hence, the position operator as the generator of
does not exist on
. The momentum operator
, however, exists based on the weak continuity of
. By means of an Application of the Fourier–Bohr transform of the basis elements denoted by
we obtain the dual basis of the Hilbert space
with
denoting the real numbers with discrete topology and
the associated counting measure
The dual basis
is orthonormal with respect to the counting measure, that is
Likewise, the basis
is uncountable and
non-separable as a consequence. It can be shown that he polymer Hilbert space
decomposes into a direct sum of separable Hilbert spaces [
55]. We have
where
are so called superselection Hilbert spaces and their basis elements
are obtained by realizing that the basis label
can be decomposed into an integer
n and a superselection parameter
in the following manner. Different aspects on how this parameter arises from representation theory on
and the physical implications of different values of
in the context of various physical phenomena are briefly reviewed in
Appendix A.
The term
superselection refers to the fact that the fundamental operators
and
that generate the algebra of observables do not map between individual superselection sectors if
is constrained to integer values. These superselection Hilbert spaces carry an inner product akin to the ordinary choice for the square-integrable functions on a unit circle,
, where
is associated to the quasi-periodicity of basis functions. This freedom in the choice of representation is also briefly discussed in
Appendix A. At this point we adopt the conventional notation also used in [
35,
36] in terms of
as the angle variable and
as its associated (angular) momentum. A connection to the notation often used in the literature for polymerized quantum systems can be established by rescaling
and a corresponding rescaling of the limits in the inner product. In this case
can be associated to some fundamental length scale
originating from a more profound theory, providing the lattice spacing of the discrete momentum variable. If we were to polymerize the translation generator
, this fundamental discreteness would be found in the eigenvalues of the position operator, analogous to, but much simpler than, the occurrence of an area gap in LQG [
26,
27] and the upper bound on matter density in LQC [
29,
33] respectively. However, the rescaling in terms of integer multiples of
does not add any additional insights and can be shown to lead to the same
qualitative results in terms of the structural properties of the master equation. Note however, that the polymer scale explicitly enters the master equation in terms of the eigenvalues of the momentum operator, hence we do
not observe scale invariance in the model. The elementary holonomy and momentum operators in the model considered here act on the basis elements given in (
29):
For the holonomy operators we directly see that
would map out of a given superselection sector. All further calculations will be performed in the
basis states shown in (
29), which are the momentum eigenstates. In the next subsection we will use the individual superselection sectors in order to derive a quantum master equation for an open
scattering model for which we will specify the corresponding interaction Hamiltonian.
3.2. Derivation of the Master Equation of an Open Scattering Model
A minimal decoherence model unarguably consists of free particles in system and environment with a given interaction Hamiltonian
bilinear in the combined variables. In our case, the analogue of the position operator is unitary and hence bounded, contrary to standard quantization procedures, a fact which at some points during the derivation of the master equation will turn out to be beneficial regarding the appearance of singularities in a straightforward approach, as was already briefly discussed in [
15]. In order to feature a common thread in the several steps of the derivation of a master equation, we would like to give the reader a concise overview in the form of the following flowchart in
Figure 1:
In terms of the A-representation in which
is discontinuous and the position (or angle) operator does hence not exist, we associate a coupling constant
to each individual particle in the environment. This is a technicality related to the physical dimension of the coupling constant. In this model, the coupling constant is not dimensionless, since the interaction Hamiltonian has a fixed dimension by definition. In the course of the perturbative expansion, the strategy is completely analogous to the dimensionless case. At the level of the master equation, this choice of interaction enables us in principle to introduce a spectral density in order to emulate a continuum of (effective) particle masses in the environment, which in some cases is a crucial ingredient in the derivation of a Lindblad-type master equation. An interaction Hamiltonian associated to a dynamic position monitoring [
3,
20] or
localization includes a coupling of the position variables of system and environment, respectively, hence our
choice of total Hamiltonian is given by:
where the index
corresponds to the individual particles in the (thermal) environment,
is the respective coupling and
is the particle number. The assumption that
N is large is based on a technicality that relates to the thermodynamic limit, more precisely to the replacement of the sum over couplings by an integral over a spectral density. This will become clear in the process of deriving and discussing the final master equation. Furthermore, the notation
was adopted to keep distinguishability between the Schrödinger- and the interaction picture operators. The prefactors of
for every such combination of holonomies ensure that the Hamiltonian has an appropriate limit for vanishing
. The connection to similar models in a Schrödinger-like representation [
3,
7,
20] is given by the replacement:
In this sense,
describes a lattice spacing where
is the non-rescaled case described in
Appendix A. However intermediate values are not meaningful in the sense that they would map out of the chosen superselection sector. This procedure of emulating operators via differences of holonomies is well-known in the literature, see for example [
55], but has never been applied to the canonical open quantum mechanical decoherence models [
3,
7,
8] to the knowledge of the authors. On top of this, for the model considered here, the continuum limit might not be well-defined in the midst of our calculations. This is not surprising since there are divergences associated to the fact that momentum eigenstates are non-normalizable in Schrödinger quantum mechanics, see for instance the discussion in [
15]. This can however be in principle dealt with in the framework of rigged Hilbert spaces [
56]. In the model considered here these kind of divergencies are absent because we work in the polymer but not in the Schrödinger representation and as a consequence the position variable
q needs to be expressed in terms of holonomies, often referred to as the polymerization of the position variable. The total Hilbert space has the following structure:
Note that the superselection parameter
agrees for all environmental degrees of freedom, since adding a particle label to the parameter does add notational complexity while leaving the results largely unchanged. Considering the continuum limit in terms of a spectral density would be a more complicated affair, however. Given a total Hamiltonian, the next step towards a master equation of the form (
23) is to transform the interaction Hamiltonian into the interaction picture. The most straightforward route is to solve the Heisenberg equations of motion for the operators involved in the interaction:
where an explicit time argument denotes the interaction picture quantity, with
being the Schrödinger picture operator. Due to the same structure of
and
respectively, it is sufficient to investigate the interaction picture of either part of the interaction Hamiltonian, the other one follows immediately. The fact that
and
together with the time-independence of
and
lets us conclude:
Hence we can directly give a closed-form expression in terms of an exponential:
The fact that the above forms of (
35) and (
36) are the correct solutions can be explicitly checked by direct computation of the derivative or by virtue of ’traditionally’ evaluating the iterated commutator bracket, see
Appendix B. The next step consists of the computation of the environmental two-point correlation functions, where we have chosen a thermal state for the environment, which is the most convenient choice and has the benefit of time-homogeneous correlation functions, that is
. Thus we need to evaluate
, where
denotes the environmental expectation value with an
N-particle thermal environment
. For this purpose we use the framework introduced in
Section 3.1 where we can construct a thermal state that can act as an
N-particle environment for our single system degree of freedom. As usual, we start out with a thermally weighted exponential of the respective free particle Hamiltonian
with
where
is the Boltzmann constant and
T is the temperature parameter. Consider a single particle in a thermal state, which we can then easily extend to
N identical particles for the environment and hence skip the particle label
for now. First expand the Hamiltonian in the basis of (angular) momentum eigenstates:
where
(or often
if considered in the context of auxiliary theta functions) is the Jacobi theta function,
and
for brevity. Note that
is non-vanishing and real for all
regardless of the temperature parameter, hence the density operator for a single free particle on a circle in a thermal state with temperature
T is given by the expression:
In the limit of
or
, respectively, we can easily see that
and the density operator reduces to the vacuum state, that is
as in standard quantum mechanics. Since any given
(note that we have omitted the subspace index in the previous calculations for a better overview) only acts on the
-subspace of the environmental Hilbert space and maps to a state orthogonal to the previous one, we can directly conclude that
. This of course is nothing but another justification of one of the steps we took in the course of the derivation of the master equation. It amounts to the statement that
, in particular for the case
, since operators with different indices act on different environmental subspaces. Thus it is sufficient to find
according to the following, where contributions including two instances of
or
can be dropped due to orthogonality of the
basis states:
where in the first step, we decomposed the
N-particle Hilbert space scalar product into its individual contributions. In the second step we isolated the only non-trivial contribution to this product, note that
only acts on the respective subspace, all other elements can be pulled through, where we then use orthonormality of the single-particle basis states. This gives the definitive proof that only correlation functions with
give a non-vanishing contribution,
. Each individual non-
instance of the contributions including the Jacobi theta function sum up to unity, simply due to the form of the partition function, see Equation (
37). Consequently, we abbreviated the normalization factor as
and renamed the summation index from
to
n but kept this label on the coefficients, as for example
explicitly contains the mass
. This parameter, which we did not yet a priori fix among the environmental degrees of freedom, exhibits an important aspect and will be brought up again later in this work. In the last step we just applied the very definitions of
and
, respectively and abbreviated the result in terms of the so-called
noise- and
dissipation kernel functions
and
respectively. The origin of this nomenclature will be discussed in the next section, where we will shed light onto the fact that this straightforward interpretation breaks down amidst the polymerization of the configuration variable. Note that
can be checked to hold explicitly for our particular correlation functions. The fact that there is only a single system’s degree of freedom and
eases our way towards the final result.
At this stage we can directly insert the correlation functions
into the Born–Redfield master Equation (
23). We intend to keep the explicit time-dependence from the upper limit of integration as long as possible, taking the limit
is generally non-trivial and only possible for certain models and spectral densities, the latter will be discussed in more detail in
Section 3.3. Given the concrete form of the environmental correlation functions and depending on the temperature of the environment, it might be reasonable to introduce another aspect of the Markov approximation and extend the integral limits accordingly, but for now we shall remain in Redfield form. For the benefit of the reader we recall the individual contributions in the Redfield master equation:
In the light of the definitions in (
24) and (
25) together with the realization that the sum over
only affects the environmental contributions, for the system consists only of a single degree of freedom, we can further simplify. Utilizing that the interaction picture operators of the system’s part of the interaction Hamiltonian are given in full analogy to the environment, we get:
As can be checked explicitly, it holds that
. This can be easily proven by adjoining the previous result and with the help of the Weyl algebra relations. Straightforward insertion into the definitions of (
24) and (
25) then yields the environmental monitoring operators for the open
scattering model:
and similarly for the second set of operators:
Furthermore we can set
and
, respectively. As mentioned, this is specifically possible in the model here since the summation indices between the system’s interaction picture operators
and the environmental correlation functions
are not intertwined due to the fact that the
are diagonal and the system only has a single degree of freedom. For a brief moment we will resert to the more compact notation
and
instead of the above form. This lets us see the structure of the resulting master equation more clearly, we will reconsider the explicit, written-out form when we examine the physical properties of the individual contributions in
Section 3.3. Let us briefly recall the abbreviations we made along the way:
Gathering all individual terms we can finally insert these into the Born-Redfield Equation (
23) and obtain:
This is one of the standard forms of the equation, however when it comes to attributing physical meaning to the equation, it is more feasible to cast it into a different form, expressed as a combination of double-commutators and commutators of anti-commutators, respectively. For this purpose, we use the following identities:
where
denotes the anticommutator. Note that the above two expressions only differ by the sign in front of the commutator-anticommutator combination, which precisely fits the difference in the sign of the argument of
in the definitions of the operators
and
in (
24) and (
25), respectively. This lets us immediately conclude that in the final master equation, the double-commutator is smeared with the cosine contribution whereas the commutator-anticommutator term is smeared with the sine-contribution from the environmental correlation functions, the mixed contributions cancel essentially by virtue of the way the correlation functions are defined. If we now gather all terms, we get the following expression for the final master equation of the open
scattering model:
In the limit of a vanishing ’temperature’ parameter
T we reduce the initial environmental density operator to a pure state and correspondingly are left with the vacuum contribution
instead of the sum over all integers:
3.3. Physical Properties of the Master Equation of the Open Scattering Model
In the context of the four canonical models of open quantum systems [
3], there is a specific interpretation to the individual terms in the master equation, see for instance [
7]. The nomenclature of the real and imaginary part of the environmental correlation functions as noise- and dissipation kernel already are suggestive. It is indeed the noise kernel that usually determines the decoherence dynamics. Suppose we are interested in the temporal derivative of the expectation value of the system’s momentum. Then we can replace the expectation value with a trace and apply the derivative to the density operator and consequently replace this derivative by the right hand side of the master equation. The resulting equation can be greatly simplified by realizing that the trace is cyclic. This is true in our case and generally needs to be checked explicitly if unbounded operators are involved. Under the assumptions of the validity of the Born–Markov master equation one can deduce a system of generally coupled differential equations for the various moments of the position and momentum variables, respectively. Let us quickly recall the case of collisional decoherence as discussed in the introduction and the form of its master equation derived in [
15,
16]:
where
is the momentum transfer function with
and some collision rate
. It is immediately clear that this equation has Lindblad form, which is evidently not the case for the equation derived here (
38), since there is a residual time-dependence in the effective system operators. In case of
and
, where the expectation values with respect to the system’s density operator has been denoted with a lowercase
for more clarity, the resulting differential equations in the collisional decoherence model amount to:
which vanishes in the case of a symmetric function
, suggesting
. A similar computation [
15,
16] for the position operator yields
, just as one would expect based on intuition from classical physics. We now shall repeat this analysis for the master equation in the previous section and derive the equations of motion for the respective first moments. For now it is sufficient to express the expectation value in terms of proportionalities, at the end of the derivation we will again collect all proper prefactors. We obtain
Hence, we are interested in the explicit expressions for
and
respectively, where we set
for brevity. Note that the unitary evolution generated by the free particle Hamiltonian
does not contribute in this computation since
commutes with itself and the trace is invariant under cyclic permutations of its arguments. This yields
where we have first permuted every instance of
to the rightmost position and consecutively cancelled the first and fourth term in the trace by inserting the commutators. As a last step we used
. On a similar note we can simplify the second contribution:
Note that we left the interaction picture’s operators in
untouched so far, the remaining trace can now be evaluated with the Weyl relations between the one-parameter unitary groups associated to the canonical variables known from ordinary quantum mechanics. More precisely, we need to simplify commutators of four different kinds:
where we repeatedly used the Weyl relations to rewrite the commutators in the second step of each line. The exponential prefactor is universal to
both operators
and
, which is evident if we recall the operator ordering that we chose beforehand and the fact that these operators contain an exponential of
with which neither
nor
commute. On a similar note the anticommutators can be obtained by carefully rearranging terms, using the Weyl relations and the fact that
and
commute among themselves
If we collect all contributions for the commutator we end up with
and in the case of the contribution involving the anticommutator we get
On top of this, the expectation value of the commutator and anticommutator brackets that do not include the holonomies can be further simplified to represent an analytic function of
when we consider
complexifier coherent states [
35,
57]. In the course of our analysis, we do not observe the higher-order corrections from a more general class of states due to the truncation at linear order in
performed in consecutive steps, hence we opt for the more compact notation at this point. Furthermore, let us abbreviate the second expectation value with
, that is:
Hence, the effective equation of motion for the expectation value of
is given by:
In order to get a better understanding of the properties of this equation we would like to expand the prefactors of the expectation values in terms of
. This will establish a correspondence between the momentum transfer function
in [
16] and the choice of our environmental interaction Hamiltonian. We treat each line individually, starting with the first one:
There is no contribution of lower (or even inverse) order since
and both summands in the bracket contain such a contribution with
in the argument. The superselection parameter
does start to contribute in
since
and the bracket together with the expectation value is at least of order
. We would like to stress that this part of the master equation was particularly simple to evaluate since there was no involvement of the holonomies whatsoever. An expansion of
and
is not meaningful in the quantum framework since the associated generator does not exist. The strategy will now be to only expand the prefactor of the expectation values including holonomies and investigate the relation between the noise- and dissipation kernel order by order in the Taylor series. We get
Note that these expansions only differ in the mass label
and hence generally do not cancel. It is not possible to take the limit
in the chosen representation for any
as the expansion of
will contain a zeroth order contribution proportional (but not limited) to
, which are at best meaningful in a semiclassical regime because of the discontinuity of the holonomy operators in the superselection sectors due to which the angle operator cannot be implemented on our Hilbert space. This is the point where our result can be related to the symmetry properties of the momentum transfer function found in [
16], the case of a symmetric momentum transfer function in (
40) and hence
is
not included in the polymerized scattering model for a non-vanishing coupling constant. Even the lowest-order contribution of
shows a different behaviour than Schrödinger-type models. Henceforth, we assume that the environment consists of particles with masses identical to the system mass
, which lets us directly simplify the effective differential equation for
since all problematic terms cancel. The conventional approach suggests that the sum over couplings and masses is replaced by an integral over a so-called a-priori
spectral density [
3]. This in a sense corresponds to the thermodynamical limit, the environmental modes (or masses, respectively) are assumed to lie dense enough as to be replaced by a continuous integral instead of a sum. The
choice of spectral density then crucially influences the properties of the model [
8] but can be used to remedy the occurrence of spontaneous recoherence or
Poincaré recurrences, respectively cancel up to arbitrary order in
:
This can immediately be integrated to yield the lowest-order solution for
:
Hence, we get a strong suppression of the momentum expectation value even in the lowest-order expansion in
. The case of
is significantly easier. We can immediately see that the contribution from the non-unitary term in the master equation vanishes by virtue of the cyclicity of the trace, which is evident from the boundedness of every operator involved:
A similar argument holds for the anticommutator:
The only non-vanishing term comes from the unitary evolution:
As can be seen the resulting equation can not be solely written in terms of expectation values of
and
due to the non-closing of the algebra between these two variables, since the relevant algebra is formed by
and not combinations of the latter two. We can however attempt to solve Equation (
44) for
based on the lowest-order solution of
in the context of an expectation value with respect to
complexifier coherent states [
35,
57]. This observation is based on the fact that, as mentioned beforehand, there is no order-by-order expansion of the holonomies, which invalidates the strategy used for the solution of
. Hereby we can use that at lowest order in the semiclassicality parameter, the operators reduce to their classical counterparts and quantum corrections occur in higher orders of the semiclassicality parameter only that shall be neglected here, enabling a straightforward solution of Equation (
44). Direct integration then yields
where
has been chosen. Moreover,
with
in the general case,
is the exponential integral function, closely related to the incomplete Gamma function
(with
two arguments, notably) and
is the ordinary Gamma function, respectively. A visualization for an exemplary choice of parameters can be seen in
Figure 2 and
Figure 3 below, respectively.
Corresponding to the solution for the momentum expectation value in (
43) and visualized in
Figure 2, the lowest-order behaviour of
reflects the strong momentum damping in the sense that the motion of the polymer particle comes to a halt after a characteristic timescale defined by the relation of coupling constants
and masses
. We would like to stress, however, that the above relations between coupling parameter and particle mass are purely exemplary and should be considered a proof of concept. In order to obtain realistic predictions, we would need to make sure to avoid Poincaré recurrences by virtue of an introduction of a proper continuum limit [
3] regarding the environmental degrees of freedom. The choice of a spectral density with appropriate physical properties to effectively describe the environment in the context of a polymerized model is still an open issue to the best knowledge of the authors.