A

Introduction to Supersymmetric Theory of Stochastics

Abstract

Many natural and engineered dynamical systems, including all living objects, exhibit signatures of what can be called spontaneous dynamical long-range order (DLRO). This order’s omnipresence has long been recognized by the scientific community, as evidenced by a myriad of related concepts, theoretical and phenomenological frameworks, and experimental phenomena such as turbulence, noise, dynamical complexity, chaos and the butterfly effect, the Richter scale for earthquakes and the scale-free statistics of other sudden processes, self-organization and pattern formation, self-organized criticality, etc. Although several successful approaches to various realizations of DLRO have been established, the universal theoretical understanding of this phenomenon remained elusive. The possibility of constructing a unified theory of DLRO has emerged recently within the approximation-free supersymmetric theory of stochastics (STS). There, DLRO is the spontaneous breakdown of the topological or de Rahm supersymmetry that all stochastic differential equations (SDEs) possess. This theory may be interesting to researchers with very different backgrounds because the ubiquitous DLRO is a truly interdisciplinary entity. The STS is also an interdisciplinary construction. This theory is based on dynamical systems theory, cohomological field theories, the theory of pseudo-Hermitian operators, and the conventional theory of SDEs. Reviewing the literature on all these mathematical disciplines can be time consuming. As such, a concise and self-contained introduction to the STS, the goal of this paper, may be useful.

I Introduction

i.1 Dynamical Long-Range Order

It is well established experimentally and numerically that many seemingly unrelated sudden processes in astrophysics Aschwanden (2011), geophysics Gutenberg and Richter (1955), neurodynamics Beggs and Plenz (2004); Chialvo (2010), econodynamics Preis et al. (2011), and other branches of modern science exhibit power-law statistics, the very reason why the Richter scale is logarithmic. This is simply one example of the spontaneous long-range dynamical behavior (LRDB) that emerges in many nonlinear dynamical systems (DSs) with no underlying long-range interactions that could potentially explain such behavior. Two other well-known examples of LRDB are the infinitely long memory of perturbations known as the butterfly effect Lorenz (1963), and the algebraic power-spectra commonly known as noise or the long-term memory effect Kogan (1996) found in many existing DSs, including apparently all living objects Dana et al. (2009); Musha and Mitsuaki (1997).

It was understood that the LRDB must be a signature of some type of spontaneous dynamical long-range order (DLRO). The existence and omnipresence of this DLRO has long been recognized by the scientific community, as evidenced by a myriad of related concepts, including chaos Ruelle (2014); Motter and Campbell (2014); Shepelyansky (2014), turbulence Ruelle (1995); Davidson (2004), dynamical complexity Lewin (1999), self-organization Kauffman (1993), pattern formation Hoyle (2006), and self-organized criticality Bak et al. (1987).

Several successful approaches to various realizations of DLRO have been established. For example, the concept of deterministic chaos is a centerpiece of the well-developed dynamical system (DS) theory. Nevertheless, there existed no universal theoretical understanding of DLRO. In particular, no rigorous stochastic generalization of the concept of deterministic chaos existed previously, whereas all natural DSs are never completely isolated from their environments and are thus always stochastic.

A class of models with the potential to reveal the mathematical essence of the ubiquitous DLRO is the stochastic (partial) differential equations (SDEs). Indeed, SDEs most likely have the widest applicability in modern science. In physics, for example, SDEs are the effective equations of motion (EoM) for all physical systems above the scale of quantum degeneracy/coherence. In quantum models, SDEs are used in a variety of ways. For example, SDEs play a central role in quantum optics (see, e.g., Breuer and Petruccione (2007) and the references therein). In many-body quantum models, SDEs are used in the investigation of non-equilibrium quantum dynamical phenomena in the form of the effective EoM of the collective quantum modes Mandt et al. (2015) and order parameters Tien (2013). They also represent a useful tool for quantum statistics Ringel and Gritsev (2013). In other scientific disciplines, SDEs are even more fundamental, as they appear at the level of the very formulation of dynamics, unlike the EoM in physics, which descend from least action principles.

The theory of stochastic dynamics has a long history. Many important insights into stochastic dynamics have been provided so far (see, e.g., Øksendal (2010); Kunita (1997); Baxendale and Lototsky (2007); Arnold (2003); Nobuyuki and Watanabe (1989); Crauel and Gundlach (1999); Kapitaniak (1990); Le Jan (1984) and the references therein). Nevertheless, the mathematical essence of DLRO remained elusive.

i.2 Topological Supersymmetry of Continuous Time Dynamics

One way of deducing the potential theoretical origin of DLRO is provided by the following qualitative yet solid argument. From the field-theoretic point of view, LRDB is indicative of the presence of a gapless excitation with an infinite correlation length/time. There are only two possible scenarios for such a situation to occur: the accidental or critical scenario and the Goldstone scenario. In the accidental scenario, the parameters of the model can be fine-tuned to ensure that a certain excitation has zero gap. This is exactly the situation with (structural) phase transitions, where an excitation called the soft mode becomes gapless exactly at the transition temperature (or other parameter). This allows the system to move effortlessly from a previously stable vacuum to a new vacuum. Immediately following the transition, the soft mode “hardens” again, i.e., it acquires a finite gap that signifies the dynamical stability of the new vacuum. In other words, only at exactly the transition point the soft mode is gapless and thus has an infinite correlation length/time.

The accidental scenario for DLRO contradicts the fact that DLRO is robust against moderate variations in the parameters of the model. For example, a slight variation in the magnitude of the electric current flowing through a dirty conductor will not destroy the noise. In other words, in phase diagrams, DLRO occupies full-dimensional phases and not the lower dimensional transitions/boundaries between different full-dimensional phases. This observation unambiguously suggests that the Goldstone scenario is the only possibility for the field-theoretic explanation of DLRO. More specifically, the Goldstone theorem states that, under the conditions of the spontaneous breakdown of a global continuous symmetry, the ground state is degenerate and that, in spatially extended models, this degeneracy tailors the existence of a gapless excitation called the Goldstone–Nambu particle. As such, DLRO may be the result of the spontaneous breakdown of some global continuous symmetry.

It is understood that the symmetry responsible for DLRO cannot be a conventional bosonic symmetry because DSs with no bosonic symmetries can also exhibit DLRO, e.g., be chaotic. In other words, DLRO must be a result of the spontaneous breakdown of some fermionic symmetry or supersymmetry. It has long been known that supersymmetries are indeed present in some classes of SDEs. The work on supersymmetric theories of SDEs began with the Parisi–Sourlas stochastic quantization procedure Parisi and Sourlas (1979, 1982); Cecotti and Girardello (1982, 1982); Drummond and Horgan (2012); Gozzi (1984); Olemskoi, Khomenko, Olemskoi (2006); Kurchan (1992); Dijkgraaf, Orlando, Reffert (2010); Kleinert and Shabanov (1997); Zinn-Justin (1986), which leads from a Langevin SDE, i.e., an SDE with a gradient flow vector field, to a model with supersymmetry. The Parisi–Sourlas quantization procedure was later identified as a realization of the concept of Nicolai maps Nicolai (1980a, b) and “half” of the supersymmetry as a corresponding Becchi–Rouet–Stora–Tyutin (BRST) or topological supersymmetry, which is a definitive feature of Witten-type topological or cohomological field theories Frenkel, Losev, Nekrasov (2007); Birmingham et al. (1991); Labastida (1989); Witten (1988a, b, 1982, 1981). Similar supersymmetries have been studied in classical mechanics Gozzi (1998, 1989, 1990); Deotto and Gozzi (2001); Gozzi (1994); Deotto et.al. (2003); Niemi and Pasanen (1996); Niemi (1996) and its stochastic generalization Tailleur, Tänase-Nicola, and Kurchan (2006).

From the perspective of the theory of ubiquitous DLRO, the consideration of specific models is clearly insufficient. In reality, EoM are never exactly Langevin or classical mechanical, and the generalization of the discussion to all or at least most of SDEs is necessary. In other words, the supersymmetry responsible for DLRO must be an attribute of all SDEs to be able to account for omnipresence of DLRO in nature. The remaining question in the Goldstone scenario of the theory of DLRO now is whether such supersymmetry exists.

Traces of this supersymmetry can be found in the literature. In Gawedzki and Kupiainen (1986), for example, the authors considered a non-potential generalization of the Langevin stochastic dynamics and noted that half of the supersymmetry survives a non-potential perturbation. Nevertheless, to the best of the knowledge of the present author, this supersymmetry in the context of all SDEs has not been addressed previously. One possible reason for this is the pseudo-Hermitianity of the stochastic evolution operator of a general SDE. Specifically, the theory of pseudo-Hermitian evolution operators appeared only relatively recently Mostafazadeh (2002a, b, c, d, 2013) as a generalization of the theory of -symmetric evolution operators Bender et al. (1998); Bender and Boettcher (1998); Fernandez et al. (1998); Bender et al. (1999); Mezincescu (2000). It was only after the theory of pseudo-Hermitian operators became available that studies on topological supersymmetry in the context of the general SDEs could be resumed. The idea that the spontaneous breakdown of this supersymmetry pertinent to all SDEs may be the mathematical essence of DLRO, or rather of one of its realizations known previously as self-organized criticality, was reported in Ovchinnikov (2011). Further work in this direction Ovchinnikov (2012, 2013, 2013, 2016) resulted in the formulation of what can be called the supersymmetric theory of stochastics (STS). The goal of this paper is to present the current state of the STS in a self-consistent manner. This paper can be viewed as a compilation of a few previous works and as a compilation that corrects several mistakes made during the early stages of the development of the STS and that clarifies a couple of points that were previously swept under the carpet. This paper also presents a few new results, including a discussion of the pseudo-time reversal symmetry.

Given the multidisciplinary character of STS, it would take an enormous amount of work to review all the relevant results from DS theory, cohomological field theory, the classical theory of SDEs, and physics. This goal is not pursued in this paper, and references are provided on only the most relevant results that are known to the author and that the material presented here is directly based on. The author would like to apologize in advance if some important related works have escaped his attention.

i.3 Relation to Existing Theories

The topological supersymmetry breaking picture of DLRO aligns well with the previous understanding of the concept of dynamical chaos. For example, the nontrivial connection between chaos and topology is at the heart of the topological theory of chaos Gilmore (1998). Furthermore, it was also known that, in some cases, the transition into chaos must be a phase transition of some sort, as evident from certain universal features of the onset of chaotic behavior Hilborn (2000). The only unexpected insight from the supersymmetry breaking picture of DLRO is the fact that its mathematical essence is in a sense opposite to the semantics of the word chaos. Indeed, chaos literally means “absence of order”, whereas the phase with the spontaneously broken supersymmetry is the low-symmetry or “ordered” phase. This is why DLRO may be a more accurate identifier for this phenomenon than, say, stochastic chaos. In this paper, both terms will be used interchangeably.

STS in a nutshell is the following. An SDE defines the noise-configuration-dependent trajectories in the phase space. The collection of all these trajectories can be viewed as a family of noise-configuration-dependent phase space diffeomorphisms. Instead of studying the trajectories, one can equivalently study the actions, called the pullbacks, that these diffeomorphisms induce on the exterior algebra of the phase space. The original trajectories can be reconstructed from these pullbacks so that the later contain all the information on the SDE-defined dynamics.

The pullbacks have one very important advantage over the trajectories. Unlike trajectories in the general case of a nonlinear phase space, the pullbacks are linear objects and can thus be averaged over the noise configurations. Such a stochastically averaged pullback is the finite-time stochastic evolution operator (SEO). Thus, it becomes clear where the supersymmetry originates from: all the diffeomorphism-induced pullbacks and consequently the finite-time SEO are commutative with the exterior derivative, which is thus a (super-)symmetry of any SDE. In other words, the existence of this supersymmetry in all SDEs is merely the algebraic version of the most fundamental and indisputable statement that continuous dynamics preserves the continuity of the phase space.

The idea to study pullbacks induced by random maps averaged over noise configurations appeared first, to the best of this author’s knowledge, in DS theory, where the analogue of the finite-time SEO is known as the generalized transfer operator Ruelle (2002). From this perspective, the STS can be viewed as a continuation of DS theory. On the side of the Parisi–Sourlas quantization procedure, the path integral representation of the Witten index of the STS is a member of the cohomological field theories. Furthermore, the SEO of the general SDE is pseudo-Hermitian; thus, the STS is within the domain of applicability of the theory of pseudo-Hermitian operators. In other words, the STS is a multidisciplinary mathematical construction. It combines a few major mathematical disciplines that are naturally synergetic within the STS. This synergy can ensure fruitful cross-fertilization during future work on the STS. To date, the STS has already provided a few novel findings, therein making it interesting from several points of view, as discussed below.

For DS theory, an interesting result from the STS is the established equivalence between the so-called sharp-trace of the generalized transfer operator, the stochastic Lefschetz index of the corresponding SDE-defined diffeomorphisms, and the Witten index of the STS. From the perspective of the conventional theory of SDEs, a valuable result from the STS is the demonstrated equivalence between the Stratonovich interpretation of SDEs and the (bi-graded) Weyl symmetrization procedure. For a field theorist, there are two potentially interesting results from the STS. First, the cohomological field theories, or rather the methodology developed within them (e.g., the localization principle and topological invariants as expectation values on instantons), together with the theory of pseudo-Hermitian evolution operators, may find multiple applications in almost all branches of modern science. Second, there are very few known analytical mechanisms that can result in the spontaneous breakdown of supersymmetry Intriligator and Seiberg (2007), which is basically one of the main reasons behind the introduction of the concept of explicit (or soft) supersymmetry breaking Chung et al. (2005). The STS provides yet another such mechanism: the topological supersymmetry in (deterministic) chaotic DSs is spontaneously broken by the non-integrability of the flow vector field.

From a wider perspective, SDEs find applications in almost all modern scientific disciplines, ranging from social sciences and econodynamics to astrophysics and high-energy physics. Therefore, the STS in general and this paper in particular may be interesting to specialists working in any of these areas of science.

i.4 Models of Interest and the Structure of This Paper

The following class of SDEs that covers most of the models in the literature will be of primary interest:

(1)

Here and in the following, summation over repeated indexes is assumed; is a trajectory of the DS in a -dimensional topological manifold called the phase space, ; is the flow vector field from the tangent space of at the point ; are noise variables; and is a set of vector fields. The position-dependent/independent are often called multiplicative/additive noise. The notation is introduced to separate the flow perturbed by the noise from the deterministic flow, . As will be discussed in Section III.2, the SDE in Equation (1) is the Stratonovich SDE along the lines of stochastic calculus on manifolds (see, e.g., Polettini (2013) and the references therein). It will also be argued that the STS appears to point to the possibility that the Stratonovich approach is the only correct choice for continuous time models.

The parameter represents the temperature or rather the intensity of the noise. As will be made clear below in Section III, the vector fields define the noise-induced metric on : . Therefore, in situations wherein the number of vector fields equals the dimensionality of the phase space, these vector fields can be identified as veilbeins (see, e.g., Chapter 7 of Nakahara (1990)). In the general case, however, the number of s must not necessarily be equal to the dimensionality of the phase space.

Most of the discussion will be directed toward models with Gaussian white noise. The probability distribution of its configurations is

(2)

with being a normalization constant such that

(3)

Here, the functional or infinitely dimensional integration is over all the configurations of the noise. The stochastic expectation value of some functional is defined as

(4)

The fundamental correlator of the Gaussian white noise is

(5)
Figure 1: Piece-wise constant approximation for Gaussian white noise. Each is a random Gaussian variable. The time in the figure flows from right to left. This is conventional in both quantum theory and the theory under consideration, as discussed at the end of Section II.3.1.

The infinite-dimensional integrations in Equations (4) and (5) can be given a more concrete meaning by splitting the time domain into a large number of intervals with infinitesimal duration and then taking the continuous time limit, namely, . Before taking this limit, each noise configuration can be viewed as a piece-wise constant function (see Figure 1) on each interval, i.e., the value of the noise variable for . The discrete-time version of the probability distribution of the Gaussian white noise in Equation (2) is

(6)

and that of the correlator in Equation (5) is

(7)

whereas all the other (even) order correlators are

(8)

The theory of stochastic dynamics defined by Equation (1) can be constructed in two steps. The first step is to understand the deterministic temporal evolution defined by the ordinary differential equation (ODE) obtained from the SDE in Equation (1) by fixing the noise configuration. This problem will be addressed in Section II, where a few concepts closely related to the continuous-time dynamics will also be introduced. The second step is the stochastic generalization of this deterministic evolution, which will be addressed in Section III.

The realistic noises are more complicated than Gaussian white noise, which is, of course, a mathematical idealization. In Section IV, the path integral representation of the theory will enable the generalization to noise of any form. Further generalization to the spatially extended models with infinite-dimensional phase spaces will also be discussed briefly in Section IV.3. Having established general technical aspects of the STS, the discussion will concentrate on the analysis of the structure of the ground states in Section V. The classification of ergodic stochastic models on the most general level related to topological supersymmetry breaking will be proposed. This in particular will help reveal the theoretical picture of the stochastic dynamics on the border of “ordinary chaos”, known previously under such names as intermittency, complexity, and self-organized criticality. Finally, in Section VI, the paper will be concluded with a brief discussion of a few potentially fruitful directions for future work.

Ii Continuous-Time Dynamics and Related Concepts

ii.1 Dynamics as Maps

For a fixed noise configuration, Equation (1) is an ODE with a time-dependent flow vector field in its Right-Hand Side (R.H.S.). This ODE defines a two-parameter family of maps of the phase space onto itself, namely, :

(9)

These maps have straightforward interpretations: is the solution of the ODE with the condition . Clearly,

(10)

The only difference here with the stationary flows described, e.g., in Chapter 5 of Nakahara (1990) is that the maps depend on both the initial and final moments of evolution, i.e., and , and not only on the duration of the evolution, i.e., . This is the result of the dependence of the noise configuration on time, which breaks the time-translation symmetry. Following stochastic averaging over the Gaussian white noise, which does possess time-translation symmetry, this symmetry of the model will be restored (see Section III).

Only physical models in which the maps (for finite time evolution) are invertible and differentiable will be considered. On the mathematical level, this means that and ’s are sufficiently smooth in such that the Picard-Lindelöf theorem (see, e.g., Coddington and Levinson (1955)) on the existence and uniqueness of the solution of an ODE for any initial condition is applicable. In other words, all maps are diffeomorphisms.

To avoid the necessity of addressing various subtle mathematical aspects not directly related to the subject of interest, the fixed noise configuration will be assumed as a continuous function of time. However, this continuity is not necessary. The noise configuration only needs to be integrable in the sense that there must exist a such that . For Gaussian white noise, is called the Wiener process.

A physicist’s proof of the invertibility of maps defined by Equation (10) is as follows. A physical ODE provides only one outcome at for each initial condition at . The same must be true for the time-reversed physical ODE, which provides only one at for any at . In other words, the map is a one-to-one map, i.e., it is invertible.

If at time the DS is described by a total probability function , the expectation value of some function is

(11)

According to Equation (9), this expectation value at a later time moment is

(12)

This view on dynamics can be clarified through the following example. Consider and an ODE of the simple form , where is a constant vector field. The corresponding diffeomorphisms are . For being one of the coordinates, i.e., , Equation (12) states that , just as it should.

One can now make the transformation of the variable of integration in Equation (12), i.e., ,

(13)

Here, is the operation of the variable transformation applied to the coordinate-free object consisting of and the collection of all the differentials ,

(14)

where is the Jacobian of the tangent map, ,

(15)

with

(16)

being the coordinate representation of the tangent map.

Equation (13) suggests that the forward temporal evolution of the variables of the DS is equivalent to the backward temporal evolution of the coordinate-free object representing the total probability distribution (TPD)

(17)

In algebraic topology, this object is known as a top differential form (D-form), the infinite-dimensional linear space of all D-forms is denoted as , and the operation in Equation (14) is called the action or the pullback induced by on .

Note that the diffeomorphism in Equation (13) is for the inverse temporal evolution as compared to the time flow in the SDE. This seeming confusion of the time direction can be clarified as follows. The pullbacks act in the opposite direction compared to the diffeomorphisms inducing them. This is the reason for the term pullback. The graphical representation of this situation is given in Figure 2. There, one introduces an infinite number of copies of the phase space for each time moment, , and dynamics is defined as a two-parameter family of diffeomorphisms between these copies: .

Figure 2: Continuous-time deterministic dynamics with a fixed noise configuration can be viewed as a two-parameter family of diffeomorphisms of the phase space onto itself or between the copies of the phase space: . The temporal evolution of a differential form is a pullback induced by the inverse diffeomorphism .

In this path-integral-like picture of dynamics, the pullback in Equation (14) can be given as

(18)

or

(19)

as opposed to

(20)

The relation between the direction of the flow of time for maps and the corresponding pullbacks can be expressed via the following diagram:

(21)

This diagram particularly suggests that the composition law for pullbacks is

(22)

ii.2 Differential Forms as Wavefunctions

The description of a stochastic model in terms of only TPDs as in the previous subsection is insufficient in the general case. This can be observed from the following qualitative example. Consider the simplest Langevin SDE with , , . Consider also the case of the stable Langevin potential , as shown in Figure 3a. It is clear that, after a sufficiently long temporal evolution, this DS will forget its initial condition, and its (only) variable will be distributed according to some steady-state TPD, which is the ground state of this DS (see Section V.2 for details). In contrast, when the Langevin potential is unstable, as in Figure 3b, the DS will never forget its initial condition because a small difference in the initial conditions will grow exponentially. No meaningful steady-state TPD can be prescribed to its unstable variable. This example signifies that the steady-state probability distributions make sense only for stable variables.

Figure 3: (a) A one-variable Langevin stochastic differential equations (SDE) with stable potential, i.e., (blue parabola oriented up), exhibits dynamics (broken dashed arrow) that can be characterized as the gradual settling to a steady-state probability distribution (bell-shaped curve). These dynamics exhibit a loss of the dynamical memory of the initial condition; (b) In the case of an unstable Langevin potential (blue parabola oriented down), the dynamics escape to infinity (dashed arrow pointing left). The dynamics is sensitive to the initial condition. No meaningful steady-state probability distribution can be associated with the ground state in this case.

The previous example may not look physical because, for any initial condition, the DS escapes to infinity and never returns. Perhaps a better example for the same purpose is a (deterministic) chaotic DS, in which the unstable variables exist even after the infinitely long temporal evolution, i.e., even in the ground state of the DS. In DS theory, the existence of these unstable variables is revealed by positive (global) Lyapunov exponents. Such a chaotic ground state must not be a probability distribution in its unstable variables. That this is indeed so will be observed in Section V.3.2 below. The DS theory predecessors of such ground states are the Sinai–Ruelle–Bowen conditional probability functions on the global unstable manifolds Eckmann and Ruelle (1985).

Section III.6 will demonstrate on a more rigorous level that it is a mathematical necessity that the Hilbert space of a stochastic DS be not only the space of the TPDs but rather the entire exterior algebra of :

(23)

with the elements being the differential forms of all degrees (see, e.g., Chapter 5 of Nakahara (1990))

(24)

Here, , is an antisymmetric tensor, is the wedge or antisymmetrized product of differentials, e.g., , and is the space of all differential forms of degree (-forms).

This by no means contradicts the intuitive understanding that it must be possible to associate a TPD with any wavefunction. As will be clear later, the TPD associated with a wavefunction is not the wavefunction itself but rather, as in quantum theory, is the bra-ket combination, which is a D-form and/or a TPD (see, e.g., Section III.3 and the discussion following Equation (96)).

One possible interpretation of the differential forms is the generalized (total, conditional, marginal) probability distributions in the coordinate-free setting. The following example demonstrates how the conditional probability distribution can be represented as a differential form (the dimensionality of is ):

where , , and . Similarly, the TPD introduced previously is

where , with being the Levi-Civita antisymmetric tensor and with being the parity of the permutation of indexes.

The geometrical meaning of a -form is a differential of a -dimensional oriented volume. Therefore, a -form can be integrated over a -dimensional submanifold or a -chain, ,

(25)

This quantity can be interpreted as follows. If one introduces local coordinates such that the -chain belongs to the -dimensional manifold cut out by , then Equation (25) is the probability of finding variables within this -chain given that all the other variables are known with certainty to be equal .

It is worth stressing that the interpretation of the differential forms as the generalized probability distributions is valid only locally in the general case. Only in a neighborhood of a given point and with a properly chosen local coordinates can a differential form be thought of as a conditional probability distribution. Globally, however, this may not be possible because there may not exist global coordinates such that a given differential form is positive everywhere on and normalizable. Moreover, if it were possible to interpret all differential forms as conditional probability distributions in the global sense, then there would be no reason to consider the extended Hilbert space in the first place. Indeed, a conditional probability distribution can be constructed from the TPD so that it does not contain any additional information, and it would suffice to describe the DS in terms of the TPD only.

The exact physical meaning of the wavefunctions in the STS is an open question. At this moment, as a working interpretation of the wavefunction, one can adopt the point of view on the wavefunction from quantum theory. Namely, the ket of the wavefunction at a given moment of time is an abstract object that contains information about the system’s past, whereas the bra-ket combination of a wavefunction has the meaning of the TPD.

To finalize the above justification for the use of the extended Hilbert space, it must be stressed that the idea of using the entire exterior algebra as a Hilbert space of a DS is by no means new. This is a well-known method in the supersymmetric theory of Hamilton models in references Gozzi (1989, 1994); Deotto et.al. (2003), where it was even demonstrated to a certain degree that the information of chaoticity of a Hamilton model is better represented by differential forms. Moreover, the mathematical object known as the generalized transfer operator that will play a central role in Section III was designed in the DS theory to probe chaos, and this object was defined on the entire exterior algebra Ruelle (2002).

To establish the law of the temporal evolution of -forms, one assumes that the DS is described by at time moment . By analogy with Equation (13), the quantity in Equation (25) at a later time moment is

(26)

Here, is the generalization of the pullback in Equation (14) to pullbacks acting on . Explicitly,

(27)

where the -form is from Equation (24) and is from Equation (15).

ii.3 Operator Algebra

Lie Derivative

The infinitesimal pullback is known as the physical or Lie derivative

(28)

The infinitesimal map defined by Equation (1) can be given as

(29)

with being the R.H.S. of Equation (1). Accordingly, the infinitesimal tangent map defined in Equation (16) is

(30)

with

(31)

Using Equations (28)–(30) and the definition of the pullback in Equation (27), one arrives at the following expression for the Lie derivative:

(32)

with being from Equation (24).

The finite-time pullback satisfies the following equation:

(33)

where , which follows from Equation (22), has been used together with the definition of the Lie derivative in Equation (28). The integration of this equation with the initial condition results in

(34)

where denotes the operator of chronological ordering. This operator is necessary because at different s do not commute. Equation (34) can be represented in the form of a Taylor series as

(35)

As in quantum theory, in the Taylor series expansion of the finite-time evolution operator, the new infinitesimal evolution operators (the Lie derivatives in this case) accumulate from the left. In other words, operators at later moments of time are always on the left of the operators at earlier moments of time, just like letters in Arabic script. In other words, the time flows from right to left in the operator representation of stochastic evolution. This is exactly the reason why the arrow of time points left in Figures 1 and 2, which may appear unconventional.

Exterior Derivative

One of the fundamental operators of the exterior algebra is the exterior multiplication . This operator can be defined via its action on a -form from Equation (24):

(36)

Viewing the differentials in the definition of a -form in Equation (24) as the operators of exterior multiplication, one can also define the operation of the exterior product of differential forms:

(37)

The other fundamental operator of the exterior algebra is the interior multiplication , which is defined as

(38)

where denotes a missing element. As can be readily verified, the (anti)commutation relations for these operators are

(39)

Here and in the following, the square brackets denote the bi-graded commutator:

(40)

with being the degree of operator , i.e., the difference between the numbers of exterior and interior multiplication operators in . For example, and so that the bi-graded commutators in Equation (39) are actually anticommutators.

The centerpiece of the theory under consideration is the exterior derivative or de Rahm operator:

(41)

The exterior derivative is a bi-graded differentiation, i.e., for any operators and ,

(42)

In the new notations, the Lie derivative can be given via the Cartan formula

(43)

where is the interior multiplication by , and and have been used.

Hodge Dual

Yet another operation that will be used later is the Hodge star

(44)

defined as

(45)

where is the Levi-Civita antisymmetric tensor and is the determinant of the metric on . As previously mentioned, the natural choice of metric on is the noise-induced metric . In Section III.3.3, it will be noted that, for certain purposes, other metrics on can be used. The Hodge star has the following property:

(46)

where is the operator of the degree of the differential form

(47)

so that

(48)

This can easily be verified using

(49)

where the indexes are lowered by the Euclidean metric and

(50)

In other words, the square of the Hodge star is a unity operator up to a sign. Up to the same sign, the Hodge star is its own inverse:

(51)

and

(52)

The Hodge star naturally defines an internal product on

(53)

for . The internal product is Hermitian positive definite, i.e.,

(54)

Thus, it may serve as a Hermitian metric on . As will be discussed in Section III.3.3, the eigensystem of the pseudo-Hermitian provides its own non-trivial metric on the Hilbert space. It is this metric that must be viewed as the fundamental metric of the Hilbert space of the model and for which the standard notation must be reserved, whereas the round brackets can be used for Equation (54).

Equation (54) can also be used for the definition of the concept of the Hermitian conjugate of an operator

(55)

for any and any operator . Using this definition, it is straightforward to derive

(56)

and the explicit expression for the so-called codifferential, which is the Hermitian conjugate of the exterior derivative,

(57)

Finally, in the forthcoming discussion, the concept of the Hodge Laplacian will also be recalled:

(58)

ii.4 Fermionic Variables

The exterior algebra has an alternative field-theoretic representation in terms of the fermionic variables that will be used in the path integral representation of the theory in Section IV.1 as well as at the end of the next section.

Following reference Witten (1982), one notes that the (anti-)commutation relations in Equation (39) are equivalent to those of Grassmann or anticommuting variables, i.e., , and derivatives over them, i.e.:

(59)

Therefore, one can make the formal substitution

(60)

and a wavefunction in the new notations becomes

(61)

whereas the expression for the exterior derivative,

(62)

reveals why is a “super” operator: it destroys a bosonic or commuting variable and creates a fermionic or anticommuting variable .

Equation (61) can be viewed as a -th term of the Taylor expansion of a wavefunction

(63)

which is now a function of a pair of variables that are the supersymmetric partners with respect to the Operator (62).

Some properties of fermionic variables are similar to those of bosonic variables. For example, one can introduce the fermionic -function

(64)

Here, is an arbitrary function of a fermionic variable, is yet another fermionic variable, the differential is , and

(65)

Note that the definition of the fermionic -function depends on the relative position of the differentials because , where has been used.

The above property of fermionic variables and their -function can be established usingBerezin rules of integration over Grassmann numbers. The latter include identities such as.

Another property of fermionic variables that has a straightforward bosonic analogue is the exponential representation of a fermionic delta function

(66)

where is yet another additional fermionic variable.

Other properties of fermionic variables may be in a sense opposite to their bosonic counterparts, e.g.,

(67)

whereas for bosonic variables, one would have for . There are many other interesting properties and relations associated with fermionic variables (see, e.g., Combescure and Robert (2012)). In the forthcoming discussion, however, only those introduced so far will be used.

Iii Operator Representation

iii.1 Stochastic Generalization of Dynamics

In the previous section, the noise configuration was assumed to be fixed, and the dynamics was essentially deterministic. The next step is to account for all possible realizations of the noise. This goal can be achieved as follows.

The stochastic generalization of Equation (13) is

(68)

and that of Equation (26) is

(69)

where the notation for the stochastic average is from Equation (4) and the new operator is defined as