Spontaneous breakdown of the time reversal symmetry

The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized $\epsilon$-prescription where the non-uniform convergence of the limit $\epsilon\to0$ leaves behind a spontaneous breakdown of the time reversal symmetry.


I. Introduction
The initial conditions are always carefully separated from the equations of motion because we can freely adjust the former but the latter is given by Nature. However this is an oversimplified point of view when dealing with a large, many-body system and observing or controlling a small fraction of the degrees of freedom only. The laws, discovered in this manner, do not correspond to a clean, elementary theory, owing to the unobserved degrees of freedom, called environment. They rather capture an effective dynamics of the observed degrees of freedom, that will be called system, under the influence of its environment, and the effective dynamics obviously depends on the environment initial conditions.
The environmental initial conditions are part of the effective system dynamics, rather than being an independent input to solve the effective equation of motion. The point is that the equation of motion in effective theories usually contains higher order derivatives and additional initial conditions are needed to find a solution. This new information should obviously come from the environment. But this latter is unobserved and we do not possess this information. A proposal is put forward below to generalize the ǫ-prescription and to represent the initial conditions dynamically.
Once the initial conditions are placed into the action then they may influence the symmetry of the dynamics in a non-trivial manner. We shall follow the fate of time related symmetries. The time translation invariance is recovered in the limit t i → −∞, where t i denotes the initial time, if the initial conditions correspond to a stationary state of motion. The state of the time reversal symmetry is more involved. The loss of the time reversal symmetry can be detected by recording of the motion of the system and checking whether the motion, seen when the recording is played forward or backward in time satisfies the same equation of motion. There is an important difference between the motions, run in opposite time, namely their initial conditions. In fact, the initial conditions of the playback motion are the final conditions of the original one. The environment trajectory depends on the environment initial conditions and this dependence breaks the time reversal invariance of the effective system equation of motion, an effect is expected to survive the limit t i → −∞.
The main result of this work is the isolation of the different levels of the breakdown of time reversal symmetry. (i) The initial conditions of a closed system, presented separately from the equation of motion, leave the dynamics symmetrical and break the symmetry of the solutions only. This effect, not being part of the dynamical equations, is a trivial, external symmetry breaking. (ii) We shall represent the initial conditions of a closed system by a generalized ǫprescription, namely by some infinitesimal terms in the action and the choice of the functional space for the variation. The time reversal invariance is broken spontaneously in this scheme because an infinitesimally strong symmetry breaking term in the action generates finite effects during the removal of the regulator of some IR divergences. (iii) The O(ǫ) terms of the action act as an IR regulator, as well, and whatever far can this cutoff be from the scale of observations, the spontaneous symmetry breaking effects of the environment break the time reversal symmetry of the effective system dynamics. Such a symmetry breaking, left behind by a cutoff, is called dynamical symmetry breaking.
The physics of dynamically broken time reversal symmetry is the irreversibility. An irreversible process is hard or soft when it takes place at finite or vanishing frequencies, respectively. The irreversibility manifests itself perturbatively or non-perturbatively, depending on the amplitude of the corresponding motion. The non-perturbative, hard processes correspond to a large amplitude motion in a finite time, such as the instability at coexisting phases or the loss of information during the collapse of the wave function. We rather follow the building up of the soft, perturbative irreversibility, dissipation, which is related to our limited control of a large many-body system: The observations, performed in a finite time leave uncertainties in the measured frequencies and amplitudes. This is the origin of the thermodynamical time arrow, best understood by coarse graining [1]. One can actually regard the effective dynamics as a result of a coarse graining in space and/or time and this view suggests that the origin of soft irreversibility is the coarse graining of the environment initial conditions.
We shall use the Closed Time Path (CTP) formalism both in the quantum and the classical cases because one the one hand, it is sufficiently powerful to solve the effective equation of motion and on the other hand, it allows the dynamical breakdown of the time reversal symmetry by the treatment of the initial conditions as part of the dynamics and supports dissipative forces which are local in time. This scheme has been introduced long time ago in quantum mechanics, [2]. The same scheme has already been used in different contexts, such as to derive relaxation in many-body systems [3], to develop perturbation expansion for retarded Green-functions [4], to find manifestly time reversal invariant description of quantum mechanics [5], to describe finite temperature effects in quantum field theory [6][7][8], to find the mixed state contributions to the density matrix by path integral [29], to follow non-equilibrium processes [9], to derive equations of motion for the expectation value of local operators [10,11] and to describe scattering processes with non-equilibrium final states [12]. The distinguished feature of the CTP scheme, a reduplication of the degrees of freedom, fits so naturally to quantum mechanics that one wonders if such a modification is not advantageous already in classical mechanics. The result of such an inquiry is the classical CTP scheme [13,14], used below.
The presentation starts in Section II with a brief recapitulation of the problems one should address in considering a sufficiently efficient scheme, such as the variational method, to describe classical effective theories. It is found advantageous to represent the initial conditions by the choice of the space of trajectories where the variations are performed. This is an infinite dimensional space and may contain surprising features, caused by non-uniform convergence as the dimension of the space tends to infinity. The symmetry breaking aspects of non-uniform convergent limits are discussed in Section III. The spontaneous breakdown of the time reversal symmetry is described in Section IV within the framework of the CTP scheme for classical mechanics. Section V deals with the dynamical breakdown of the time reversal symmetry, the energy balance equation and contains a brief presentation of some model calculations. A brief discussion of the emergence of the finite life-time and the decoherence in quantum effective theories is given in Section VI. The conclusions are summarized in Section VII, followed by an Appendix with the derivation of the CTP Green function for a classical harmonic oscillator.

II. CLASSICAL EFFECTIVE THEORIES
A generic problem in classical mechanics is to find the dynamics of a coordinate x, called system, which interacts with its environment, described by the coordinates y = (y 1 , . . . , y N ) which are not followed [15,16]. We assume that the full, isolated system, described by the coordinates x, y, follows a reversible and conservative dynamics, defined by the action, S[x, y] = S s [x] + S e [x, y], whose equations of motion are second order differential equations in time. We need auxiliary conditions to make the solution of the equation of motion unique. They are usually initial conditions, specified for the system and the environment at the initial time, t = t i and the system trajectory is followed until the final time, t = t f . The effective equation of motion, a closed equation to be satisfied by the system trajectory, is obtained in two steps. First one solves the environment equation of motion for a general system trajectory, x(t).
After that the solution, y[t; x], is substituted into the system equation of motion. The and the environment equation of motion, imply The effective action, S[x, y[x]], is not useful in systems with soft irreversibility on several counts. The local forces of the variational scheme are holonomic, therefore non-conservative forces are non-local in time and can not describe simple friction forces. Furthermore, to recover a time independent dissipative equation of motion we must perform the limit t i → −∞ which renders the the state of the environment of a dissipative system rather complex. Therefore we need a generalization of the action principle which can handle initial condition problems, as opposed to boundary conditions, used in the traditional variation method. The trade of the final coordinates to the initial velocities is not a trivial change because the variational equation for the final coordinate, ∂S/∂y f = p f = 0, is unacceptable restrictive. Both the the local nonconservative forces and the initial conditions can be accommodated in the variational scheme by reduplicating of the degrees of freedom.
The impact of the initial conditions in the resulting scheme is similar to the spontaneous or the dynamical symmetry breakings. Therefore before embarking the issue of the generalized variational scheme we survey briefly the ways in which the spontaneous and dynamical symmetry breaking takes place in large systems.

III. NON-UNIFORM CONVERGENCE AND SYMMETRY BREAKING
The realistic description of experiments usually involves a large number of dynamical variables, namely coordinates and Fourier modes. The quantity which controls the number of the variables is commonly called cutoff and its removal defines an infinite system. The cutoff is usually a dimensional number and is chosen to be far away from the scale of observations to make the predictions approximatively independent of its choice. One the one hand, this procedure makes our description natural in the sense that it eliminates an unreasonably large or small number in our expressions. But on the other hand, the resulting infinite system is defined formally and it may possess surprising, counter intuitive features, owing to a lack of sufficiently smooth convergence during the removal of the cutoff.
Certain mathematical operations, such as the multiple limits may develop dependence on the order of their execution in case of non-uniform convergence. If this happens then the multiple integrals may become dependent on the order of integration and the execution of the limits and the derivatives of parametric integrals may produce different results, depending whether they are made before or after integration. Well known examples of non-uniformly converging integrals are given by distributions: The regulated distribution is a regular function which depends on a parameter ǫ and the limit, ǫ → 0, is carried out after the integration.
Non-uniform convergences appear usually through the amplification of a divergence, controlled by a cutoff. One can distinguish different kinds of non-uniform convergences, the infinitely many variables may arise from the infinitely small or infinitely large distances or times. These four classes define infinite systems with new features which are unexpected for the intuition, based on point-wise convergence because they are not localizable in finite space or time regions.
A. Long distance cutoff: Phase transition A cutoff in space controls the number of degrees of freedom and its typical implementation is the finite size, L, and ts removal, L → ∞, is the thermodynamical limit. Phase transitions take place if the time dependence of observables do not converge uniformly in thermodynamic limit. Observations, carried out in arbitrary large, but finite time in an infinitely large system may give different results than the infinitely long time observations in large, but finite system. The result is the coexistence of different formal measures in the statistical ensembles of infinite systems, a non-trivial phase structure. The choice of a phase at the coexisting point is the new feature, offered by the infinite systems only.
It is easy to follow this phenomenon for a special class of phase transitions, the spontaneous symmetry breaking which is due to the slowing down of the order parameter. Let us consider a macroscopic, solid body, consisting of N particles of mass m which interact with translation and rotation invariant forces. To understand qualitatively the spontaneous breakdown of translation and rotation symmetries in macroscopic bodies we separate the translational and rotational motion by distinguishing laboratory and body-fixed, co-moving coordinate systems, the latter is defined by having vanishing center of mass velocity and a diagonal tensor of inertia The collective coordinates X, θ, φ, α consists of the vector of the translation and the Euler angles of the rotation which bring the laboratory frame into the body-fixed coordinate system. These variables serve as order parameters for translations and rotations and their effective dynamics is described by the Hamiltonian written in terms of the total momentum and angular momentum, P and L, respectively. The representations of a finite Heisenberg algebra are unitary equivalent and the ground state of the Hamiltonian (6) is a singlet with vanishing expectation value for the order parameters. But the excitation spectrum becomes dense for large systems and the classical description of the order parameters become and excellent approximation for macroscopic bodies. The result is the slowing down of the order parameters: The order of magnitude of the velocity of the collective coordinates of a system of mass M , given in grams, and characteristic size L, expressed in centimeters, in a pure state where the energy of each degrees of freedom is k B T /2 isẊ ≈ T /M · 10 −8 cm/s and θ,φ,α ≈ T /M /L · 10 −8 /s, where the temperature is given in Kelvin. The order parameters can safely be considers as stationary on the macroscopic scale. The spontaneous symmetry breaking can be detected in a sufficiently large system without following the time evolution, by observing equilibrium states averages only. One introduces an external, explicite symmetry breaking of strength h and checks the status of the symmetry in the limit h → 0. A qualitative volume dependence of the order parameter of the symmetry h → −h is given by where the volume V controls the closeness of a pole to the real, physical h axis, cf. The cutoff which controls the number of degrees of freedom located arbitrarily close to each other in space is a minimal distance, a. Its removal, the renormalization procedure, may generate non-uniform convergence and introduce some unexpected structure in the space-time continuum.
The easiest way to understand the emergence of the non-uniform convergence in the renormalization of quantum field theories is the BPHZ renormalization scheme [17][18][19] which can briefly be summarized as follows. One starts with the perturbation series of the observables, Green functions in general, in the regulated, bare theory and modifies the Lagrangian in a recursive manner in each order of the expansion to define a convergent theory. Let us write the contribution of a Wick-rotated Feynman graph to a one-particle irreducible vertex function in the form of the loop-integral, where p and q denote the external momentum and the integration variables, respectively andg stands for the set of parameters of the theory, rendered dimensionless by the the cutoff,g = ga [g] . The Wick rotation, the analytic extension of the loop-integral for imaginary energy, is needed to decouple the problem of null-space singularities, considered later, from the large q divergences, treated here. The overall divergence of this integral is the divergence which comes from the integration domain where all components of q diverge with the same speed. The power counting is a simple algorithm to find the degree of the overall divergence, it is given by the mass dimension, [G a ], = c = 1.
The loop integrals with [G a ] < 0 are finite and need no special attention. However one modifies the Lagrangian by adding a counterterms for each graph with overall divergence, [G a ] ≥ 0. The counterterms are defined by the first [G a ] order of the Taylor expansion of G a in p around p = 0 and their impact on the perturbation series is the subtraction of the Taylor expansion terms from the graph in question. As a result, the graph (8) contributes to the modified theory by G Ba (p) = G a (p) − P a (p), where P a (p) is polynomial of p, consisting of the first [G a ] order of G a . The complicated part of the BPHZ scheme is a rather involved, recursive proof that the removal of the overall divergences at the order of the perturbation expansion where they appear leaves no sub-divergences behind and makes the limit a → 0 of the observables convergent and finite. The distinguished feature of the BPHZ scheme is that it can be carried out before the integration, by directly subtracting the divergent terms from the integrand [19]. The subtracted loop-integral has overall degree of divergence −1 therefore it is finite and plays no role anymore in the renormalization. This makes this scheme attractive because it allows us to remove the cutoff before the integration and the renormalized Feynman graphs can be given in terms of finite integrals, without the explicite use of the cutoff. Such a removal of the divergences is realized by the use of appropriately chosen, cutoff-dependent bare parameters,g →g a , and the loop-integrals of the renormalized perturbation expansion are defined by G R (p) = lim a→0 G Ba (p) of A simple way to show that the integral (9) converges uniformly in the renormalization is to find an integrable bound, i.e. a function F (p, q) ≥ |I(q, p,g a , a)|, with dqF (p, q) < ∞.
The uniform convergence of the loop-integral is important because it allows us to interchange the order of removal of the cutoff, a → 0, and the integration and thereby to define the renormalized perturbation expansion in terms of integrands without cutoff, at a = 0. Furthermore it makes the loop-integral independent of the order of the integration over the energy and the momentum. It is easy to check that the Wick-rotated loop-integrals with negative overall degree of divergence converge uniformly. It may happen that there are graphs in the theory with non-negative overall degree of divergence which are accidentally finite [20]. These graphs need no counterterms but may converge in a non-uniform manner. If this happens then the renormalized theory, defined by the limit a → 0, differs from the theory which is obtained by simply setting a = 0 in the subtracted integrand. Such a surprising and interesting phenomenon, that a cutoff leaves a finite trace in the dynamics even after it has been removed, is called in a somehow unfortunate manner anomaly. Physical phenomenas, such as the anomalous breakdown of chiral invariance and the neutral pion decay [21,22], the anomalous breakdown of scale invariance and the proton mass [23] and the Abraham-Lorentz force [24] owe their existence to such a restricted convergence of the renormalization procedure.
The space-time continuum of the renormalized theories possesses features beyond the Bolzano-Weierstrass theorem, they are characterized by the (finite) counterterms to accidentally finite vertex functions. It is a difficult task to identify all properties which are changed by these counterterms however it is much simpler to find the differences by inspecting equations which express some symmetries. This might be the reason that the impact of the non-uniform convergence during the removal of the UV cutoff is traditionally called dynamical symmetry breaking.
The phase transitions and spontaneous symmetry breakings are usually driven by the potential energy, the kinetic energy being less important for the long distance modes with a remarkable exception, the deconfinement transition in finite temperature Yang-Mills theories [25]. The dynamical symmetry breaking and the singularities of the one-particle sector, to be discussed next, are always related to the kinetic energy.

C. Short time cutoff: Canonical commutation relation
The description of single degree of freedom needs infinitely many modes in time. The discretization of the time with the time step ∆t maximizes the frequency of observing the system at Λ = 2π/∆t and allows us to approach the continuous time in a mathematically well defined manner. The new feature of the continuous time evolution, introduced by such a limit, is the Planck constant, the possibility of having non-trivial canonical commutation relation.
We shall work with a harmonic oscillator, defined by the Lagrangian, and consider the the trace of the time evolution operator, regulated for long time by a large, but finite propagation time, t f − t i , and for short time by ∆t, the difference between the consecutive times the oscillator can be observed. The path integral expression, where ∆ω = 2π/(t f − t i ) and ω k = k∆ω. First we remove the IR cutoff by performing the limit t f − t i → ∞. One can simplify this expression for sufficiently large t f − t i by splitting the values of k into two sets, for 1 ≤ n < cN and for cN ≤ n < N where c is a small, N -independent number. The latter sum over the latter set is O(1/N ) and can be neglected for large N and the replacement 2 ∆t sin ω k ∆t 2 → ω n can be made in the sum. Another justification of such a replacement is simply noting that the sum has −1 overall degree of divergence therefore it converges uniformly and the summation and the limit ∆t → 0 commute. The result is the expression for the UV regulated propagator. The time variables are continued analytically for real numbers but the physical values remain discrete.
To check the canonical commutation relation we consider the function The insertion of the momentum, p ℓ , into a matrix element is equivalent with the presence of the multiplicative factor m(x ℓ − x ℓ−1 )/∆t in integrand of the path integral, therefore The function (16) can be expressed in terms of the regulated propagator, revealing that the limit ∆t → 0 is not uniform, C(t, ∆t) = O(t) for t ≪ ∆t, and lim t→0 C(t, 0) = i . It is not surprising that the canonical commutation relation which implies an unbounded spectrum of the momentum requires that the UV cutoff be removed before t → 0. Note that the linear UV divergence of the Green function, indicates that the trajectories which dominate the path integral have diverging velocity [26]. Let us consider a coordinate x, following a local, harmonic dynamics, described by the Lagrangian, L = xKx/2+jx, where K is a differential operator in time. The null-space of K, the modes Kx(t) = 0, play special role in the dynamics. They are used to satisfy the auxiliary conditions of the motion and drop out from the variational principle because the action is degenerate within this functional subspace. They are actually at the edge of being unstable, runaway trajectories. In fact, let us use the eigenfunctions, Kϕ λ (t) = κ λ ϕ λ (t), to write the time dependent functions in the form where the sum is replaced by an integral in the continuous part of the spectrum. The equation of motion, λx λ = −j λ , shows that the restoring force of the equilibrium position is proportional to λ, indicating that a finite amplitude of the source induces a diverging response close to the null-space. To have stable motion the continuous spectrum integral must converge around the null-space. If a discrete spectral line coincides with the null-space then the regularization of the product of the two distributions with coinciding singularities gives the secular term which rises linearly in time.
The naive Green function of the harmonic oscillator (11), is divergent due to the null-space modes. The power counting, applied for the integration domain around the massshell, predicts logarithmic divergences which need regularization. One may regulate the Green function, by simply skipping the frequency intervals Ω − 1/T < |ω| < Ω + 1/T in the integration. This regulator preserves time reversal invariance and turns into the principal value prescription after the removal of the cutoff. The null-space modes which handle the initial conditions are left out by this sharp cutoff hence we should rather use another regulator. This should be a smooth cutoff which leaves the null-modes present in the dynamics with a regulated, non-vanishing restoring force. The traditional ǫ-prescription, the infinitesimal, imaginary shifts of the poles in the complex frequency plane, satisfies this need.
To reveal the non-uniform convergence, implied in the ǫ-prescription, we use keep both regulator in the retarded Green function, which can easily be calculated for large T , where Note the similarities of Figs. 1 and 2, arising from the relation G(z) = F (z) − 1 mod 2. The time reversal breaking cutoff must therefore be removed after the long time limit of the observations is reached.

IV. SPONTANEOUS BREAKDOWN OF THE TIME REVERSAL SYMMETRY
A generalization of the action principle is described in this Section which handles the initial conditions as part of the dynamics. We start with the brief introduction of the generalized action principle [13,15], followed by the demonstration of the spontaneously broken time reversal invariance for a harmonic oscillator.

A. Classical chronon-dynamics
We address now two of the issues, raised in Section II, namely the need of the handling of non-holonomic forces and the use of initial conditions in the variation method of classical mechanics. The system coordinate plays a double role in the function U (x,ẋ) of holonomic forces: The derivatives with respect to x orẋ produce the force and the value of x andẋ control the contribution to the energy. These two roles can be separated by the help of an active and a passive system coordinate, x and x p , respectively. We define semi-holonomic forces by the generalization of (4), They cover the effective forces since the effective equation of motion, (1) can be written as A new problem, left behind by the introduction of a passive copy of the system, is the treatment of the passive copy within the variational scheme. It can be solved, together with problem of the initial conditions, by extending the motion for twice as long time. First we let the system follow its time evolution from the given initial condition at the initial time, t i , to the final time, t f . After that we perform a time inversion on the state of motion of the system and follow the motion back in time until original initial conditions are recovered, as shown in Fig. 3. The trajectory, spanned in such a manner,x is a closed loop. We assume here that our system is subject of holonomic forces only, described by the Lagrangian L(x,ẋ), the effective theories will be considered later, in Section V A. Then the action, corresponding to the trajectorỹ where T denotes time reversal, T : . We split the trajectoryx(t) into two segments, and write where both x + (t) and x − (t) satisfy the same initial conditions. The time reversal can be represented by exchanging the trajectories,x T = τx, where The system undergoes a time reversal transformation, the time arrow is flipped at t = t f , and the motion is followed back to the initial state.
The CTP doublet,x, will be called chronon since it allows us to represent both directions of the flow of the time within the action principle. A chronon trajectory with will be called classical because it characterizes a classical motion with well defined position. The transformation of action (30) under the time reversal, will be called chronon conjugation symmetry.
The action (28) is degenerate for the classical trajectories, (32), and in the null-space of the equation of motion of a harmonic system. We need a non-degenerate action functional when the initial conditions are handled by the action hence we add an infinitesimal piece to the CTP Lagrangian, with The splitting term, L ǫ , forms an analytic structure in the frequency space to handle the non-uniform convergence at the null-space. The immediate advantage of removing the degeneracy with an imaginary term is that the chronon conjugation symmetry remains a simple condition, When the variational equations are calculated for an initial condition problem then the boundary contribution at t f are not vanishing. To suppress these contributions we impose the condition The functional space of the trajectories for the variational calculation will be defined in such a manner that this condition, together with the choice (35) lead to Note that t f is not a relevant parameter, the classical trajectory, x(t) = x ± (t), found by solving the equations of motion is independent of the choice of t f for t < t f .

B. Harmonic systems
The simplest dynamical system is defined by a quadratic action, where the conjugate chronon,x =x trĝ , withĝ is used andĵ = (j + , j − ). The "metric tensor",ĝ, can be used to raise or lower the chronon indices, g σσ ′ g σ ′ σ ′′ = δ σ ′′ σ , by preserving the chronon conjugation symmetry, (36). The chronon Green function is defined asD =K −1 , and the trajectory is given byx = −Dĵ. If the source is physically realizable, j + = j − = j, then both copies perform the same motion and one finds the chronon exchange symmetry, for the Green function. This relation, together with a similar one forK yield the block structure for either ∆ = D or ∆ = K. The retarded and advanced components are defined by ∆ r a = ∆ n ± ∆ f and the relations follow from the straightforward inversion for commutative blocks. The antisymmetric part of the quadratic form of the action yields a boundary term in the variation and can be canceled. The symmetry, , gained in such a manner assures the these real functions have simple transformation properties under the exchange of the indices, ∆ n (t, , and we have the positive and negative time parity terms of the harmonic action in the diagonal and the off-diagonal blocks, respectively. The time translation symmetry is recovered in the limit t f − t i → ∞ where the Fourier transformation, produces real ∆ n (ω), ∆ i (ω) and imaginary ∆ f (ω). Note that the imaginary part of the action, K i , is infinitesimal and the finite infinitesimal part of the Green function, D i , drops out in classical mechanics, playing role in the quantum case only. The blocks D n and D f are the near and far field Green functions in the case of the electromagnetic field. In non-field theoretical models the splitting of the retarded Green function into the sum of an odd (D f ) and an even (D n ) part corresponds to the separation of the modes with vanishing and non-vanishing restoring force, respectively. The physics of this separation will be clarified in Section V B where it is shown that the energy, exchanges in the limit t f − t i → ∞ between the a harmonic system and the external source, corresponds to the modes with vanishing restoring force only.
It is instructive to check how dissipative forces can be accommodated in a harmonic system. This is not possible within the framework of the traditional action principle because the time inversion parity of the antisymmetric time derivative, ∂ t , is −1 which makes the kernel of any time inversion-odd term in the action antisymmetric. But the chronon conjugation makes the time derivative,∂ t = τ ∂ t , symmetric. In particular, the action, yields the equation of motion [28] mẍ ± = −mΩ 2 x ± − kẋ ∓ , up to the infinitesimal terms of S ǫ .

C. Generalized ǫ-prescription
We discuss now the generalization of the standard ǫ-prescription to handle the general auxiliary conditions first in the case of a harmonic oscillator (11), in the presence of an external source, L → L + jx, followed by few remarks about the weakly interacting models.
The solution of a linear equation of motion is written as a sum, x = x p + x h , of a particular solution, x p , of the inhomogeneous equation with the source, satisfying the trivial initial conditions and a solution of the homogeneous equation, x h , to satisfy the desired, non-trivial initial conditions. It is assumed in the usual ǫ-prescription that the source has a continuous spectrum and j(ω) is an analytic function except some isolated poles with finite imaginary part. Hence the frequency integrals can be calculated by the help of the residuum theorem and the trivial initial conditions are recovered for x p . Such an analytic structure is generalized by allowing the source to acquire a discrete spectrum component with support within the null-space, j = j c + j d , chosen in such a manner that it reproduces the non-trivial initial conditions, x h = −D r j d and x p = −D r j c . The particular solution, x p , has a continuous spectrum owing to the Heaviside function, Θ(±t), appearing in the frequency integrals and the discrete spectrum component of the trajectory is x h .
Let us now consider the harmonic oscillator, defined by the chronon action, whose initial conditions are represented by The details of the calculation of the Green function forx p =x −x h are given in Appendix A in the presence of an UV and an IR regulator, a finite step size, ∆t, in time and the length of observation, t f − t i , respectively. The initial coordinate is set to zero by the choice of the functional space and the initial velocity is found to be vanishing for finite values of the cutoffs. The common coordinate, , is eliminated first at finite UV cutoff and this step is followed by performing the continuum limit, ∆t → 0. We find at this point a Fourier series which represents the Green function and is non-diagonal in the frequency space, owing to the broken translation symmetry in time.
The Fourier sum can be approximated by Fourier integrals for large enough t f − t i and the Green function becomes diagonal in the frequency space, or equivalently, as long as the inequalities are assured. The first inequality protects the final condition against eroding, caused by the ǫ-prescription, and limits the time in between the observations and t f . The second bound assures that the discrete sum of the Fourier series can be approximated by the Fourier integral and places us deep in the non-uniform limit, displayed in Fig 2. Finally, the inequality between the first and the last expression, t i ≪ t, is needed to decouple the initial conditions. The Green function describes the effects of external perturbations which propagate on-shell in time. But the separation of the real, symmetric part of the harmonic oscillator action by the regularization (22) introduces the principal value prescription for D n , c.f. eqs. (47), and makes it off-shell. The remaining blocks of the Green function, D f , and D i , remain on-shell, they correspond to the propagation of the perturbation through t f , c.f. Fig 3, for excitations in the final time.
Once the Green function is found then the action is given by the expression, withK =D −1 and x σ h = x h . An equivalent expression is The inversion givesK(ω) = mĤ(ω, Ω), where or The Fourier transformation back to time brings the action, (49), into the form (34) with t i = −∞, t f = ∞ and where P denotes the principal value prescription. Few remarks are in order at this point: The aforementioned analysis can be extended for weakly interactive systems. Let us consider the model described by the coordinate x = (x 1 , . . . , x N ) and the action The CTP equation of motion, withD −1 n (ω) = m nĤn (ω, Ω n ), can be found by iteration, the tree-graphs of the first three orders are depicted in Fig.  4. The homogeneous solutions can be taken into account as a shift of the external source,ĵ n →ĵ n + (D n ) −1x hn . One can prove by the repeated application of the equation, σ ′ D σ σ ′ = D r , that the Green functions,D n , can be replaced by D r n in the equation of motion which becomes for physical external source, j σ n = j n . To avoid the secular terms we apply the adiabatic switching approximation, the interactions are turned on gradually in time, g → g(t), where the coupling strength is negligible at the initial conditions, g(−∞) = 0. The time dependence in the coupling constant dissolves the discrete spectrum contributions, generated by the iteration at integer multiples of Ω, into the continuous spectrum.

D. Induced time arrow
The initial or final conditions break the time reversal symmetry independently of the equation of motion, in a manner which is external to the dynamics. They are represented in the chronon action by L ǫ with ǫ = 0 + or ǫ = 0 − , respectively, by terms which break the time reversal invariance of the action. The regulated retarded Green function, (22), converges in a non-uniform manner during the removal of the null-space regulator, T → ∞, c.f. Fig. 2 and amplifies the infinitesimal symmetry breaking, (52), to finite effects, (47). This is superficially similar to the nonuniform convergence of the order parameter in the thermodynamical limit in case of spontaneous symmetry breaking. We further corroborate this similarity by constructing an order parameter for the time reversal symmetry. A localized, time reversal invariant external perturbation of the harmonic oscillator, j(t) = j 0 δ(t − t 0 ), generates a response, δx(t), which is not time reversal invariant due to the initial or final conditions. We define a local order parameter of time reversal symmetry, indicating the direction of the flow of time, set by the boundary conditions. The dynamics transforms it to a homogeneous order parameter, The poles if the Green function, D r (ω), come in pairs, {ω p , −ω * p }, thus in particular ρ = 1 for H r , given by eq. (46). As of the regulator dependence of the order parameter, generated by the Green function (22) is concerned, one finds ρ = F (ǫT ) where the function F (z) is defined by eq. (7). The order parameter diplays the hallmarks of spontaneous symmetry breaking, namely the non-commutativity of the removal of the IR cutoff and the external symmetry breaking and the left over a finite symmetry breaking effect in the infinite system when the external symmetry breaking is removed.
The order parameter of the time reversal invariance provides us a time arrow, induced by the initial conditions, the direction of time pointing away from the time of the initial conditions.

V. DYNAMICAL BREAKING OF THE TIME REVERSAL SYMMETRY
Let us suppose that a variation in time of the system trajectory, δx(t), generates a perturbation, δy(t), of the environment trajectory due to the system-environment interactions which is assumed to be time reversal invariant. The non-vanishing order parameter, (57), of the environment indicates that δy(t) is not time reversal invariant and its reaction back to the system, δx ′ (t), has a component which breaks the time reversal symmetry. The key property from our point of view is that this component is independent of the system auxiliary conditions and its sign is determined by environment time arrow only. Such a transmutation of the environment time arrow to the system, demonstrated in details in this Section, originates from the ǫ-prescription of the environment, the regulator of the nullspace divergences. Thus it can be considered as a dynamical breakdown of the time reversal symmetry, a particular realization of the symmetry breaking effects of a regulator, discussed in Section III B.

A. Effective chronon theory
The power of the CTP formalism becomes obvious in effective theories [15]. The chronon action, of the full, closed system yields the effective action, where the trajectory y[x] is the solution of the environment equation of motion, and satisfies the environment initial conditions. Note that owing to the independent variations of y + (t) and y − (t), the solution, y[x + , x − ], contains more information than y[x], arising from solving eq. (2). According to Fig. 3 the couplings between the members of a chronon, δ 2 y[x]/δx + δx − , arise from the coupling of the trajectories at the final time, t f . Those couplings which survive the limit t f → ∞ correspond to phenomenas which decouple from the system and take place in the asymptotic long time state of the environment, decoupled from the system by the adiabatic switching.
One can separate the original system dynamics in the effective action by writing denotes the influence functional [29]. Another way to write the effective action is where δ 2 S 2 [x]/δx + δx − = 0. The interpretation of S 1 and S 2 can be inferred from the equations of motion. It is specially advantageous to write these equations by using the parametrization x ± = x ± x d /2 because it is sufficient to calculate the effective action in O(x d ) for classical trajectories, satisfying eq. (32). The variational equation for x d at x d = 0, indicates that that the one-point action, S 1 , includes the conservative part of the effective dynamics and the two-point action, S 2 , covers local, non-conservative, e.g. semiholonomic forces. We can find the same effective action by Legendre transformation, as well, by borrowing ideas from quantum mechanics. First we define the generator functional, where the system and the environment coordinates are eliminated by their equation of motion, The n-point Green functions are defined by the functional Taylor expansion, n! t f ti dt 1 · · · dt n D σ1,...,σn (t 1 , . . . , t n )j σ1 (t 1 ) · · · j σn (t n ), and can be found by iteration. The system trajectory is given by the one-point function, and the higher order Green functions describe the dependence of the trajectory on the source, the higher order graphs in the iterative solution of the equation of motion, c.f. Fig. 4. The effective action, (62), is recovered as the Legendre transform of W [ĵ], whereĵ is eliminated on the right hand side by inverting eq. (70).

B. Energy balance equation
We have seen in Section IV C that the symmetry with respect to translation in time is recovered in the effective theory in the limit t f − t i → ∞. This is a characteristic feature of soft irreversibility where the energy loss to the environment in equilibrium is due to the uncontrollable small amplitude, slow fluctuations rather than external time dependence in the dynamics. To fit dissipation into a time translation invariant dynamics we derive the energy balance equation for semiholonomic forces.
Let us start again with the harmonic oscillator, (11), coupled to an external source, L → L + jx. The work, done by the oscillator on the external source by following the trajectory x = x p + x h , can be expressed as a frequency integral, The coefficient of D r (ω) in the integrand is an odd function of the frequency and suppresses the the contribution of D n , allowing the replacement D r → D f , and D f is given by eq. (47) and one finds the energy loss The lesson is that the energy exchange takes place in asymptotically long time in the null-space of the oscillator only. This is a natural result since only the null-space modes may be non-vanishing in the absence of the external source and may accumulate the energy excess or loss. Note that sign(W ) = τ i for x h = 0, meaning that the direction of time in which the energy, accumulated from the source by the oscillator which started with the trivial initial conditions, is lost could have been used to define the time arrow (60). The conserved quantities, defined formally by the help of the Noether theorem, are vanishing due to the chronon inversion symmetry (36). Therefore one should perform the symmetry transformations on one copy only, leaving the other, representing the environment, unchanged [30]. If the decoupled interaction, represented by S 2 , violates certain symmetry then the corresponding conservation law receives a non-vanishing source term and becomes a balance equation. One has to be careful and to identify a quantity which would be conserved in the absence of the system-environment interaction by keeping the contributions of S 1 only to the Noether construction and leaving the contributions of S 2 for the source term.
We assume that the real part of the effective action corresponds to an effective Lagrangian of the form, L(x) = L 1 (x + ) − L 1 (x − ) + L 2 (x), containing time derivatives up to order n d . The equation of motion for x + is where the notation x (j) = d j x/dt n is used. To find the energy balance equation we perform the variation δx + = −ξẋ, δx − = 0 and write the linearized action of ξ(t), after some partial integrations in the form where the partial derivative, ∂ − t , acts on x − (t) only. Its equation of motion for a classical trajectory, x = x + = x − , is given byḢ where for n d = 1 and for n d ≥ 2. The source term, is due to the coupling δ 2 y[x]/δx + δx − of the doublers at t f and represents that part of the interaction energy which is lost to the environment at the final time. The rate of change is not definite for large amplitude and fast motion but we shall see in the next Section that κ ≤ 0 for soft irreversibility.

C. Normal mode dynamics
Let us consider a time reversal invariant, conservative harmonic system, described by the coordinate z = (x, y 1 , . . . , z N ), and the action The effective action forx =ẑ 1 is calculated by the help of the generator functional (67). First we diagonalize the action by the orthogonal transformation, connecting z with the normal modes, w. Next, the generator functional, is calculated by by solving the equations of motion for the normal modes and replacing them into W [ĵ], Finally, the effective action, is found withK The solution of the effective equation of motion,x = −Dj, is obtained in terms of the Green function,D =K −1 . We use a physically realizable source, j σ = j, giving rise to the response, x ± = x = −D r j, where D r = (K r ) −1 , according to the inversion rules, (41). The equation of motion, satisfied by this trajectory is K r x = −j, with where M j and Ω j denote the j-th normal mass and frequency, respectively and j A 2 1j = 1. The structure of the effective equation of motion can better be understood by noting that the external source forx is coupled to several normal modes in (84) hence the energy, injected into the system byĵ is spread over the normal modes and generates a rather complicated response, reminiscent of interacting systems [49]. The form of the retarded Green function shows that ℜK r (ω) is non-vanishing and time reversal invariant except in the normal mode spectrum. At the same time ℑK r (ω) is localized in the frequency space within the ǫ neighborhood of the spectrum and it breaks the time reversal invariance. The sign of ℑK r (ω) changes when the environment initial conditions are replaced by final conditions, ǫ → −ǫ. The discrete peaks of ℑK r (ω) represents the breakdown of the time reversal invariance in the effective dynamics, induced by the environment initial conditions. Note that the auxiliary condition time arrow is homogeneous, it is the same for the normal modes and the effective theory. The coupling of the system coordinate to its environment can be characterized by the spectral function, where the possible dependence on the order of performing the summation and the integration is revealed by the non-uniform convergence in the second line as ǫ → 0. As a simple example take a mixed spectrum, where the environment is represented by a Drude-type spectral weight, The roots of the equation of motion kernel (92), ω (±) = ±ω s − iǫ and ω (0) = −iΩ D , define the spectrum of an irreversible effective theory.
To understand better the dependence of the first equation in (92) on the order of the summation and the integration we make a coarse graining in time and restrict the system dynamics for a finite time by considering the trajectories where c(t) ≈ 1 for |x| ≪ T , c(t) ≈ 0 for |t| ≫ T , T being a smooth IR cutoff function with time reversal invariance, c(−t) = c(t). This amounts to the spread, of the discrete frequency lines in the frequency space. Let us introduce the minimal separation of the discrete normal frequencies, ∆ω = inf |ω j −ω j ′ | and distinguish the following two cases [31]: • ∆ω > 0: The observations, carried out in time T ≫ 1/∆ω, can resolve all normal modes and the effective theory for x T is conservative. We can reproduce such observations with an effective theory where the integration over the spectral variable is performed first, followed by the summation. The linear equation of motion operator is given by (90), the motion is reversible at frequencies which do not belong to the normal mode spectrum. The system coordinate, x1, acts as an external source on the environment coordinate xn, n > 1, c.f. Fig. 3.
• ∆ω = 0: The normal frequency spectrum has an accumulation point and the arbitrary long but finite time measurements leave infinitely many normal modes unresolved. The infinitely many unresolved normal modes act as an uncontrollable sink of the system energy and the effective dynamics for x T with finite T contains dissipative forces. The summation must be carried out first in the first equation in (92), leaving the integration for the second step.
The soft irreversibility can be found in the iterative solution of weakly interactive models, too. Let us take for instance the model, described by the action (54), whose iterative solution can be written as a series of tree-graphs, as shown on Fig. 4. Consider the contribution of the Green function, D r n = D n n + D f n , n > 1, corresponding to the line AB in the third graph. The system coordinate acts as an external source on the environment coordinate x n and the contribution D + nσ ′ (t 1 , t 2 )z σ ′ 1 (t 2 ), to z σ n (t 1 ), shown by dashed lines in Fig. 5, represents the oscillations of x n at t f = ∞, the sink for the system energy when σ ′ = −.
The leading order solution with trivial environment initial conditions is for n > 1 and its insertion into the chronon action yields the influence functional for x = x 1 , The introduction of the spectral function, allows us to write where the Fourier transform ofĜ(t − t ′ , Ω) is given by eq. (46). The corresponding effective equation of motion for Although there are nonlinear terms the discussion of the breakdown of the time reversal invariance and the emergence of dissipative forces, presented for the harmonic model, applies.

D. Classical toy model
We can better isolate the role of the environment in generating irreversibility in the harmonic model of the Lagrangian [32], We write the action in the form [34] S and the elimination of the environment leads to the effective action, where the inverse Green function,K ef f =D −1 0 −Σ =D −1 , contains the self energyΣ = gĜg. The advantage of this model, compared to that of the previous Section is that the kinetic energy does not mix the system and the environment coordinates and allows us to have independent ǫ-prescription, given by ǫ s and ǫ e , for the system and the environment, respectively. This in turn makes possible to have different time arrows for the system and the environment for g = 0 since τ i = sign(ǫ). When the system and the environment modes couple, g = 0, then the finite, imaginary Σ f (ω) dominates the O(ǫ s ) terms inK(ω) and we have sign(iK f ef f (ω)) = −sign(ǫ e ω), c.f. the second equation in (47). Therefore the inversion, (41), produces the equation sign(iD f (ω)) = sign(ǫ e ω) or τ i = sign(ǫ e ), the transfer of the environment time arrow to the system.
The solution of the effective equation of motion can easily be given in terms of the Green function, When applied to the source, j σ = j, the first factor ofD 0 produces due to the block structure (40). Each further, successive application of an operator produces the retarded component with the result the perturbative proof of the first equation in the inversion rules, (41). The influence functional, can conveniently be parametrized by means of the spectral function, The self energy assumes the formΣ whereĜ(ω, Ω) is given by eq. (46), in particular and the expansion of the exponent in the right hand side of eq. (108) in t ′ − t andΣ(ω) in ω yields the influence Lagrangian, The equation of motion, generated by the variation of x d , at x d = 0, is with Σ r = Σ n + Σ f and contains up to O( The time reversal of the environment induces ǫ → −ǫ,Σ →Σ * , Σ r → Σ a and breaks the time reversal invariance at frequencies, belonging to the environment spectrum. This is similar to spontaneous symmetry breaking, namely the replacement ǫ → h, and T → V brings the order parameter (58) into M , mentioned in Section III A. We can separate the following cases: 1. Discrete environment spectrum without condensation point: The dynamical breakdown of the time reversal symmetry is not universal. The experiments, performed in a sufficiently long time can explore the time reversal invariant system dynamics, realized by the frequency modes which do not belong to the environment spectrum.
2. Discrete environment spectrum with condensation point: The continuous spectral function becomes a good approximation for finite time observations at the condensation point and indicate the irresistible loss of energy for those modes.
3. The continuous environment spectrum: We find soft irreversibility at any frequency. The analysis of the discrete spectrum reveals that the loss of energy is due to the environment, whose modes are degenerate with the system modes.
The soft irreversibility is the result of the mixing of the environment modes into the system dynamics, described by Σ f . This mixing can clearly be seen in the geometric series (107), where Σ r = Σ n + Σ f and each Σ f stands for environment excitations. These excitations decouple from the system after the system-environment interactions is adiabatically switched off because they correspond to the environment null-space.
It is instructive to calculate the energy balance, by using the results of Section V B for continuous spectrum. Let us assume the validity of the expansion of the effective action in the time derivative, and write the resulting effective Lagrangian in the form, . Partial integration brings this Lagrangian into the form The corresponding energy balance equation, (78), contains the system energy and the source term The total energy loss, the integral of κ during the motion, allows us to define the energy loss, for each frequency. It is useful to consider the Green function in the context of the normal mode decomposition, c.f.(90), where the coupling between the doublers make D f (ω) imaginary and the positivity of the norm in the coordinate space, z, guarantees the inequality for ω > 0. As a result we have the bound ℑK f (ω) = −ℑK f (−ω) > 0 for ω > 0. This leads to the inequality K f j (−1) j ≤ 0 and makes the system energy non-increasing,Ḣ ≤ 0, within each frequency mode. This result holds only for trivial environment initial conditions and it is easy to see that the source term, κ, is non-definite for non-trivial initial conditions.

E. Runaway trajectories and acausality
The effective equation of motion is an integro-differential equation whose solution is difficult to find. The harmonic models represent a yet soluble but already non-trivial example. Consider for instance the model, defined by the action (82), where the effective effective equation of motion for z 1 , can be written in the form where the kernel K r = K n + K f is defined by eq. (90). For a finite model, N < ∞, the finite, ǫ-independent part is a 2(N + 1)-th higher order integro-differential equation which can be solved in the frequency space. The general solution requires the knowledge of 2(N + 1) auxiliary conditions, coming from the system and the environment. But we know two of them only, those of the system, and we are not in the position to find the appropriate solution of the effective equation in the absence of the remaining 2N data. This problem is solved in the CTP formalism by representing the initial conditions by the generalized ǫ-prescription. We shall comment on two surprising features of the way, this scheme generates the missing integration constants, namely the absence of the runaway trajectories and the resulting acausality of the solutions. The higher order derivatives in K r generate several poles for D r (ω) which come in pairs, {ω p , −ω * p }. The oscillation is always time reversal invariant, the broken time reversal symmetry is revealed by its envelope, the damping. If there are poles with ℑω p > 0 then the corresponding residuum diverges as t → ∞. But the Green function, calculated by means of the residuum theorem excludes the runaway trajectories.
The suppression of the runaway modes can be viewed as final condition on the trajectory and it can naturally generate acausality. The causality of a motion is characterized by a causal time arrow, to be constructed by comparing two identical systems which evolve in different environment. The system coordinate, x, evolves with vanishing external source, j(t) = 0 for system 1 and the external source, j(t) = j 0 δ(t − t 0 ), acts on system 2. The causal time arrow, τ c = ±1, points in the direction in which the response of the system to the external source, ∆x(t) = x 2 (t) − x 1 (t), is non-vanishing, ∆x(t 0 − τ c t ′ ) = 0, for t ′ > 0. If ∆z(t) = 0 in both directions then the motion has no well defined time arrow and is called acausal. The causal time arrow can therefore be given by with sign(0) = 0, c.f. (60). The time arrow can always be set in finite systems by imposing initial or final conditions and it points away from the time where the auxiliary conditions are set. In fact, the integration of the equations of motion of a finite, closed system which are local in time always produces causal dynamics. But localized peaks of the spectral weight, used together with the residuum theorem to evaluate the retarded Green function, may produce acausality in infinite system with continuous spectrum.

F. Classical point charge
One finds the features, mentioned before in a well known, weakly coupled interactive model, namely in the effective theory of a point charge in classical electrodynamics. The interactions between the charges is well known, we shall therefore restrict our attention to the self-interaction of a single point charge. One would have thought that there is no self interactions for strictly point charges because the world line of a massive particle can not pierce its own light cone. But the electromagnetic interactions are ill-defined for a point charge, owing to the short distance singularity of the Coulomb field and this problem requires a short distance regularization, followed by a renormalization. We shall find that the regulator generates self interactions and there is a local, cutoff-independent interaction, the Abraham-Lorentz force, left behind after the removal of the UV cutoff. It is the result of the non-uniform convergence of a loop-integral and stands for the relativistic friction force of radiation.
The action of the electromagnetic field (EMF) and a single point charge is defined by the action S = S ch +S EMF +S i , where Q(p 2 ) being the regulator for the EMF [24]. The influence functional, can easily be found by using the regulated Maxwell equations. The action (126) is similar to the usual action-at-adistance action [35][36][37][38] except that it contains the radiation field, too. The EMF Green function, is regulated by means of a smeared Dirac-delta, δ ℓ (z). The singularity in the far field needs special attention: First, because it arises from the coincidence of the singular points of the two distributions in the off-diagonal blocks. Second, any smearing of sign(x 0 ) breaks Lorentz invariance. We keep sign(x 0 ) unchanged and require δ ℓ (0) = 0. A simple regularization which satisfies all requirements is δ ℓ (z) = δ(z − ℓ 2 ). The expansion of (126) in (s − s ′ )/ℓ is straightforward and produces the non-local influence Lagrangian up to terms O(x 2 ). The convergence of the integral is non-uniform as ℓ → 0 and the Taylor expansion in u must be performed in before the integration. To find the integral in terms of the derivatives of the world line we have to separate the problematic terms and deal with them one-by-one. For this end we add u 2ẍ and u 3 ...
x with coefficients which make the integral uniform convergent and subtract the integral of these new terms separately, The first term of the influence functional renormalizes the mass, m = m B + βe 2 /2c 3 ℓ, and the renormalized equation of motion is with The integral in this expression is uniform convergent and O(ℓ), hence can be ignored as the cutoff is removed. One can find another "anomaly" beyond the Abraham-Lorentz force, it is a light cone anomaly. Namely, the modification of the retarded Green function, amounts to the shift, m → m+e 2 /2c 2 s 0 , of the mass. In other words, the EMF field which is placed slightly off shell by the regulator remains sensitive to the off-shell modifications of its dynamics even after the removal of the cutoff. The origin of the loss of the symmetry with respect the modification (133) of the EMF dynamics comes in a manner which is characteristic to non-uniform convergent integrals, namely an O(ℓ) modification of the Green function multiplies an O(ℓ −1 ), divergent piece in the regulated loop-integral, leaving behind an O(ℓ 0 ), cutoff independent result. The influence functional of a point charge contains a single scale parameter, it is the the classical electron radius, r 0 = e 2 /mc 2 . The usual electromagnetic interactions are recovered by the force (132) for ℓ ≫ r 0 but a singularity, reminiscent of the Landau pole of QED is found at ℓ ∼ r 0 . The UV side of the crossover, ℓ ≪ r 0 , supports unusual, acausal interactions. This can be understood by noting that the cutoff-independent, O(ℓ 0 ) part of the self-force, the first term in the bracket of eq. (132), generates runaway solution which is stabilized by the second, non-local term in the right hand side of the equation. This integral is uniform convergent and becomes negligible for small ℓ, leaving behind a third order equation of motion with runaway solution whose suppression generates acausality [39].

VI. FINITE LIFE-TIME AND DECOHERENCE
The dynamical breakdown of time reversal invariance induces genuine quantum effects, too. The soft irreversibility stands for the possibility of any mode of the system to loose energy to the environment. In the quantum case this leads to the leakage of the norm of the system state into the environment, the dynamical origin of the finiteness of the life-time of the system excitations. Furthermore, the slightest system-environment interactions generate a system-environment mixing according to the degenerate perturbation expansion, the system state becomes mixed by decoherence. To address these features we need the original, quantum CTP formalism [2].

A. Quantum chronon-dynamics
The CTP formalism was introduced by J. Schwinger for the perturbation expansion for the expectation values, in quantum mechanics [2]. Here |ψ(t i ) denotes the state at the time t i and U (t, t ′ ) is the time evolution operator. The reduplication of the degree of freedom originates from the simultaneous presence of the bra and ket in the expectation value, two states, developing in opposite direction in time but representing the same physical system. Let us follow the naive quantization of the chronon dynamics of a closed system, introduced in Section IV A. It is based on the Hamiltonian, of a particle, assumed to be moving in a potential, and the replacement of c-numbers with operators, [x σ , p σ ′ ] = σδ σσ ′ i , where the sign factor σ is included to take into account the opposite orientation of time for x + and x − . Next we introduce the chronon "wave function", ψ(x), and calculate the path integral expression of the "transition amplitude" between two coordinate eigenstates, where the integration is over trajectories with end pointsx(t i ) =x i ,x(t f ) =x f and the action in the exponent is given by eqs. (34)- (35). The standard procedure to derive the equation of motion for A, which usually gives the Schrödinger equation, now leads to the Neumann equation, indicating that ψ(t,x) = A(t,x, t i ,x i ) is actually the density matrix. This is not surprising after noting the formal similarity between the doubling x → (x + , x − ) and ψ → (|ψ , ψ|) in classical and quantum systems. Such an interpretation reveals a fundamental difference between ψ(x) and the wave functions in quantum mechanics: The expectation value, A ψ , is additive in the quantum state, A ψ1+ψ2 = A ψ1 + A ψ2 , without interference. The state ψ(x) contains all quantum effects, i.e. all interference has already been placed into ψ(x). The only expression for A ψ which is additive in ψ and respects the condition (37) of virtual variation is A ψ = Tr [ψA]. From now on we can follow the traditional scheme and introduce the quantum version of the generator functional (67), where the trace stands for the the condition (37) and ρ i denotes the initial density matrix. Its path integral representation is The Green function of a harmonic oscillator is identical to the classical one, c.f. Section A 4.
where the bare (Wilsonian) effective action can be found in a manner, similar to the classical case, , where the influence functional is given by and the decomposition (65) can be used again. It is instructive to consider a generalization of the CTP formalism, based on the density matrix, rather than its trace, as in eq. (137). It is the Open Time Path scheme, and the path integral expression for the reduced density matrix is withx(t f ) =x f and y + (t f ) = y − (t f ) and the effective action of the CTP formalism gives The decoherence in the coordinate diagonal representation, the suppression of the contributions of well separated chronon trajectories, is driven by ℑS 2 . The couplings between the doubler trajectories in S 2 which generate semiholonomic forces in classical mechanics now stand for the contributions of several final environment states in the trace of (137), make the system state mixed and represent the system-environment entanglement. In particular, the O(x 2 ) terms in (53) represent an infinitesimal decoherence and the corresponding initial state is mainly the ground state but contains an infinitesimal amount of mixing. The two trajectories of a chronon are identical in classical physics, c.f. eq. (32). It is easy to see that the unitarity of the time evolution preserves eq. (32) on the level of the averages. In fact, the average of the coordinate at time t o < t f can be calculated in two equivalent, t f -independent manners, owing to the identities c.f. the remark about the t f independence of the classical trajectory, made at the end of Section IV A. This allows us to identify x d = x + − x − with the quantum fluctuations in the coordinate basis, to be suppressed in the classical limit.

C. Propagator
The generator functional, (137), motivates the introduction of the generalized time ordered product [6], and the two point function of any observable, defined by i D σσ ′ (t, t ′ ) = Tr[A σ (t)A σ ′ (t ′ )ρ i ], displays the block structure (40). The spectral function is defined by the Wightmann function, where |n are eigenstates of the Hamiltonian and the last inequality follows from the positivity of the norm. The spectral function, D −+ , is vanishing for ω < 0 if the motion starts with the ground state, ρ i = |0 0|, since only m = 0 contributes to the spectral function. Then the equation iD i (ω) = sign(ℜω)D f (ω) follows or equivalently, the Feynman propagator, D F = D ++ , is related to the retarded or advanced Green functions, Since D r (ω) is a real function of iǫ an equivalent form of this equation is D r a (ω) = D F (±|ℜω| + iℑω).
Similar a relations hold for the inverse Green function, iK i (ω) = sign(ω)K f (ω), furthermore the inequality (147) assures the bound (122). The effective action can be approximated by the quadratic expression, whereK −1 =D for small fluctuations. The time scale of the leakage of the system into the environment, 1/|ℑω 0 |, where D −1 F (ω 0 ) = 0 is identical to the damping time scales of the classical trajectory due to the relation (149), written forK. The inequality (122) implies D i (ω) < 0 which, together with the second equation in eqs. (41), assures the bound, K i (ω) > 0. As a result, the O(x d2 ) term of the effective action suppresses the chronon trajectories with well separated doublers and generates decoherence. The Gaussian suppression factor of x d (ω) has the width 2π /K i (ω)t during the time t. Thus ℓ d = 2π |ℜω 0 |/K i (ω 0 ) can be considered as the characteristic length of decoherence, the length of displacement which generates a suppression of x d by the factor 1/ √ e within an oscillation of the phase of the quasi-particle. The final result is that soft irreversibility, finite-life-time of the quasi-particles and decoherence, all share the same dynamical origin, namely K f .

D. Quantized toy model
The quantum version of the toy model of Section V D has already been throughly examined [33,40]. The calculation of the CTP effective theory proceeds as in the classical case, the influence Lagrangian (112) is recovered and the classical equation of motion remain valid for the expectation value of the coordinate. The non-increasing nature of the energy (119) is assured by the positivity of the norm and the Schwinger-Dyson resummation, (105), can be interpreted as the indication of the mixing of the degenerate system and environment excitations.
The influence functional, defined by eq. (140), has a richer structure then in the classical case owing to quantum fluctuations: There are loop-contributions which represent the quantum fluctuations in the environment, the higher order terms in x d play role in shaping the quantum fluctuations and driving decoherence. Let us separate the free and the interactive part of the environment dynamics by writing the environment action in the form S e [x, y] = where the free generator functional is defined by It is easy to see that the iteration of the classical equation of motion, (55), generates the O( ), tree-graphs in the perturbation series for δW [ĵ]/δĵ.

E. Test particle in an ideal quantum gas
Let us now move to a more realistic problem of a quantum test particle, interacting with an ideal gas. This problem has already been considered by using the traditional effective action approach in imaginary time [41] and by means of the CTP formalism [42]. The quantum master equation [43] has been used extensively, and the dephasing has been described in the pioneering work [44], followed by the inclusion of relaxation [45][46][47]. The quantum Boltzmann equation has recently been derived for one-dimensional gas [48], as well.
We summarize here the calculation of the effective Lagrangian of a test particle in an ideal fermi gas in the leading order of the Landau-Ginzburg double expansion [34]. The action, S = S p + S g + S i , is given by where M denotes the mass of the test particle which propagates under the influence of an external potential V . The gas is described by the field ψ and its density is coupled to the test particle by the potential U . The influence functional (140), contains the propagator of the gas particles, where n q denotes the occupation number andψ †Γ [x]ψ = σ σS i [x σ , ψ σ † , ψ σ ]. We assume that the potential localizes the particle strong enough to justify the expansion in the coordinate and seek the leading, O(x 2 ) contributions to the influence functional, The tadpole contribution is canceled by introducing a homogeneous, neutralizing classical background charge and G σ1σ2 (x 1 , x 2 ) = iξ n sF σ1σ2 (x 1 , x 2 )F σ2σ1 (x 2 , x 1 ) denotes the density two-point function and j σ (t, y) = U (y − x σ (t)). The two-point function can be expressed by well known one-loop integrals, G n ωq = G + ωq + G + −ωq , G f ωq = G − ωq − G − −ωq and iG i ωq = G − ωq + G − −ωq , with n s = 2s + 1 denoting the spin degeneracy factor. The integrals are analytic functions of the dimensionless variables x = ω/|q|v F , y = |q|/k F where v F = k F /m, k F = 3 6π 2 n/n s , n denoting the density of the gas. The second expansion of the Landau-Ginzburg scheme, the expansion in the time derivative, is possible for |x| ≪ 1. The characteristic frequency of the particle-gas interaction is |qẋ| hence this condition reads |ẋ| ≪ v F . The influence functional, gives rise the influence Lagrangian, where ∆x = ∞ n=1 x (n) n! ∂ n iω , x = x(t) and x (n) = d n x/dt n . The the block structure (40) allows us to write it in the form which gives after the expansion of the time derivative Due to x d = 0 the expectation value x satisfies of the real part of the Euler-Lagrange equation for x d , which includes a mass renormalization, M R = M + δM , and a friction force. The imaginary part of L inf l generates decoherence, a suppression factor, exp −ℑS inf l , in the path integral. The terms involving x d2 andẋ d2 represent the decoherence strength in the coordinate and in the momentum basis, respectively. Finally, few notes in passing: The calculation of the Abraham-Lorentz force, presented in Section V F, can be carried over non-relativistic charges in QED without any change owing to the charge neutrality of photons. The friction force is of order ∂ 3 t in this case because the Newtonian, ∂ t , friction force is suppressed by boost invariance of the photon state. It is interesting that the boost-invariance breaking Newton friction force is the only linear dissipative force respecting causality. Furthermore, the dissipation and decoherence can be found in the effective dynamics of the density and the current of the ideal gas itself, [49].

VII. CONCLUSIONS
The role of the environment initial conditions in generating soft irreversibility was studied in this work. Our particular scheme, the CTP formalism, is selected by consistency, namely once the environment initial conditions play role in the effective dynamics, it is better to include all initial conditions into the dynamics. This way of encoding the intial conditions leads to null-space divergences, to be regulated by the ǫ-prescription. The frequency integrals converges in a non-uniform manner during the removal of this cutoff, ǫ → 0 + , and this feature is essential in the breakdown of the time reversal symmetry.
We have distinguished three levels of the breakdown of the time reversal symmetry. The first level is the breakdown of the symmetry in the solution of a reversible dynamics by the initial conditions. This is a trivial, external symmetry breaking. The second level is a spontaneous symmetry breaking within closed systems, driven by the ǫ-prescription. The third type, the dynamical symmetry breaking, was found in effective theories where it was due to the finite left-over of the environment ǫ-dependence. It is not easy to pin down the effects of non-uniform convergence because they are not related to finite scale and we believe that this difficulty justifies our approach to soft irreversibility.
The local, dissipative effective forces have been derived in the classical CTP formalism for harmonic and weakly coupled systems and the energy balance equation has been obtained. The energy is found to be non-increasing for each frequency mode in harmonic system if the environment starts in a stable, stationary state.
Our approach, based on the Green functions, allows us to define two time arrows dynamically. One is the initial condition time arrow, the direction of the time away from the initial conditions. The other is the causal time arrow, the direction of the cause-effect relation. The dynamical breakdown of the time reversal symmetry consists of the domination of the system time initial condition arrow by the environment and makes the time of the system to flow in a direction which is determined by the environment.
If the effective equation of motion has a runaway solution then it must be suppressed in the future to represent the solution by the retarded Green function. When this happens then we find pre-acceleration and the opposite orientations for the two time arrows, in other words acausality. This can be imagined as a coherent effect of a large environment which pumps energy into a system mode in a way which seems as a time reversed image of dissipation, to cancel the runaway solution.
The present approach leaves several questions open. The different levels of the breakdown of the time reversal invariance has been established by means of the ǫ-prescription and depends on the details of this scheme. Is the resulting scenario universal, valid in any formalism or we see some non-physical features of a particular analytic structure on the complex frequency plane? In the case of an anomalous quantum field theory the anomalous symmetry breaking has been accepted after several, different regulators have reproduced the same effect. A similar, alternative derivation of the dissipative forces in effective theory would be needed to lend more thrust to the picture, outlined here. Another question concerns the relation between soft and hard irreversibility. Which features, found for soft irreversibility, apply for irreversible processes, taking place at fixed scale, such as in Wilson's cloud chamber?
Finally it is worth reminding the reader that we had nothing to say about the time arrow problem, the origin of the direction of the time in the Universe [50]. The dynamics with spontaneous symmetry breaking has a strong amplification mechanics which can convert an infinitesimal symmetry breaking to a finite effect, in agreement with the proposition, put forward in ref. [51], though counter examples exist, as well [52] . Therefore all we can say is that it is rather natural that the time arrow is homogeneous in the Universe but its particular direction remains to be better understood.