Symmetric and antisymmetric forms of the Pauli master equation

When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter — this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case and the opposite in the time-antisymmetric case). We demonstrate that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric form of PME favours full conversion of antimatter into matter. At this stage, it is impossible to make an experimentally justified choice in favour of the symmetric or antisymmetric versions of thermodynamics since we have no experience of thermodynamic properties of macroscopic objects made of antimatter, but experiments of this kind may become possible in the future.


A generic quantum system and perturbation analysis of its unitary evolution
System Hamiltonian. While considering evolution of a generic quantum system, it is common to distinguish two components in its Hamiltonian: j 0 and the disturbance   = t ( ) 1 1 that characterises interactions of these eigenstates |j〉 and may be time-dependent 2,8,9,12 . Different states (i.e. eigenstates) may belong to the same energy levels ε ε = j j 1 2 , but these states are still marked by different indices j 1 ≠ j 2 . The characteristic time associated with the leading Hamiltonian is much smaller than that of the interactions: . Here,  denotes a norm estimate for the Hamiltonian . The energy eigenstates form a complete orthogonal basis 〈 j|k〉 = δ kj where ε j represents energy of the j th eigenstate. The number of eigenstates, 2n, is presumed large (eigenstates can be continuous). The Hamiltonian disturbance  1 is responsible for interaction of the eigenstates due to non-diagonal elements. If  1 depends on time, the characteristic time of this dependence is assumed to be of order of τ 1 . The diagonal elements of  1 are not of interest and we put 1 Mathematically, this assumption simplifies the analysis but does not restrict generality since the diagonal elements of  1 can always be merged with the diagonal elements of  0 .
Invariant properties of the system. Two types of states are distinguished: the matter states and the antimatter states. The matter states are indexed by j , = …  k n 1, , , while the antimatter states are indexed by j, = − … − k n 1, , . The indices j and k run over all 2n states. We also use the indicator function C j = ± 1 defined by , the Hamiltonian (1) allows for conversion between the corresponding matter states and antimatter states. The states |+ j〉 and |− j〉 are presumed to represent the CP transformations of each other j although the physical coordinates and their parity (P) transformation, which reverses the directions of these coordinates, are not explicitly considered here. The CP transformation also involves the charge conjugation (C), which changes particles into antiparticles and wise versa. The time reversal operator (T) reverses the direction of time.
The complex phase |α j | = 1 is subject to a number of physical constraints and, in most cases, can be eliminated by incorporating the phase angle into the states. Since our consideration is generic, we keep the phases α j , which, however, can be omitted as they do not affect our consideration and results. Under conditions considered here, the CP-and CPT-invariant Hamiltonians satisfy: Scientific RepoRts | 6:29942 | DOI: 10.1038/srep29942 These symmetric properties of quantum systems are discussed in standard monographs 13,14 .
It must be noted that not all matter and antimatter states can evolve into each other (or form a quantum superposition), since evolution of quantum systems must preserve a number of conservative properties (such as the electric charge). Hence only mutually convertible states, which preserve the conservative properties, are considered here. Mutual conversions of matter and antimatter inevitably alter the baryon numbers (for example, conversion of a neutron into an antineutron changes this number from + 1 to − 1). The possibility of the baryon number violations is known as the first Sakharov 15 condition of baryogenesis and is conventionally presumed in theories dealing with the balance of matter and antimatter.
Asymptotic expressions for unitary evolution. The evolution of the quantum system is governed by the Schrodinger equation where ψ(t) is the wave function, the Hamiltonian  t ( ) is Hermitian, the evolution operator  t t ( , ) 0 is unitary so that its time inverse corresponds to its conjugate transpose  In quantum perturbation theory 8,9,12,16 , the time evolution operator is conventionally represented by the series that after substitution into equation (8) yields The terms in the expansion can be easily evaluated when Δ t = t − t 0 is sufficiently small; specifically, assuming that τ τ ∆   t 0 1 is sufficient for the derivation of PME. In this case we implying that there also exists a weak dependence of  ∆t ( ) on t. Evaluation of the non-diagonal elements results in The diagonal elements take the form Since Q jk (1) and Q jk (2) are linked by the equation  By default, the sums over indices k and j are evaluated over all 2n eigenstates. It is easy to see that, at the leading orders (up to O(λ 2 )), the magnitudes  ∆ j t k ( ) 2 form a symmetric matrix since U V ∼ 1 is Hermitian. This property, however, is not valid at the higher orders: generally, the matrix ∆  j t k ( ) 2 is not symmetric. While the asymptotic representations of  ∆t ( ) given above are universal, these representations do not define uniquely the form of the master equation, which depends on additional assumptions. Large systems are subject to the process of decoherence, whose properties determine probabilistic behaviour of the system.

Different forms of the Pauli master equation
This section derives two alternative forms of Pauli master equation (PME): conventional symmetric and non-conventional antisymmetric. In the next section these forms will be shown to correspond to the symmetric and antisymmetric extensions of thermodynamics. PME was originally suggested by Pauli 8 , and has been repeatedly re-derived using varying techniques and assumptions 2,9,12,17 . The original approach developed by Pauli is most suitable for the derivations in this section for a number of reasons. First, Pauli's approach explicitly discriminates the directions of time by repeated application of decoherence, which corresponds to setting the state of random phases at the beginning of many sequential time intervals. Since human intuition is deeply linked to inequality of directions of time it is common to introduce discrimination of directions of time implicitly by implying "good" initial conditions. As remarked by Price 4 , this implicit treatment is not desirable in applications, where the direction of time needs to be analysed and not postulated a priori. Second, Pauli approach is based on wave functions, which seem to be more convenient for the present analysis, which involves multi-time correlations, than density matrices. Symmetric PME. According to Pauli's approach to the master equation, the decoherence events occur at the times t 0 , t 1 , … , t β , … , t e , which, as illustrated in Fig. 1, are spaced by the characteristic decoherence time for reasons discussed below. In the symmetric case decoherence occurs forward in time for all (matter and antimatter) states. The energy eigenstates form the preferred basis for decoherence: the phase of a decohered eigenstate becomes independent of the rest of the distribution. The effect of the decoherence on the density matrix ρ(t) is removing all non-diagonal elements (as specified by the Zwanzig projection operator 18 ): Irrespective of the previous state of the system, this corresponds to transformation of the wave function into a mixture where 2n random phases Θ k indicate that ψ at t = t β + 0 is not a superposition but a mixture of 2n wave functions ψ (k) . Each function ψ (k) in (18) corresponds at t = t β + 0 to the k th diagonal term of the density matrix ρ k,k = 〈 ψ (k) |ψ (k) 〉 and satisfies Here, we use random phases Θ k as notation that indicates mixed states of quantum system 19 . In this case, Θ k can be interpreted as special quantum states. This is not exactly the same but very close in its measured effect to Pauli's work where phases were assumed to be physically randomised. At the moment t = t β decoherence converts the overall wave function ψ of the system (which can be in any state, mixed or pure, at t < t β ) into a mixture of 2n pure states corresponding to the eigenstates of  0 . The decoherence events change phases but not the amplitudes of the wave functions.
Due to linearity of the quantum evolutions governed by (8) where D jk is evaluated in (14) and Hermitian properties of  1 and  are taken into account to establish that where τ Δ ~ 1/Δ ε is proportional to the characteristic density of quantum levels in the energy space (the characteristic energy distance Δ ε between quantum levels is very small in large systems). The probability change for every p (k) over the interval Δ t = t β+1 − t β is determined by equations (19)(20)(21) and takes the form With introduction of the overall probability, and taking into account that Δ t is small, equation (22) is summed over all k to give the Pauli master equation 8 The the right-hand side form of the equation explicitly involves formula for w k k in (21).
Antisymmetric PME. In the case of antisymmertic decoherence, the matter states decohere at the moments t 0 < t 1 < … < t β < … < t e but the antimatter states recohere at the same moments (recoherence is decoherence backwards in time t -this is illustrated in Fig. 2. Within every unitary evolution interval t β ≤ t ≤ t β+1 , the joint effect of matter decoherence at t = t β and antimatter recoherence at t = t β+1 is representation of the wave function in the following form where 2n random phases Θ k of the decohered values are statistically independent from each other, while the corresponding 2n wave functions ψ (k) (t) are subject to unitary evolution within the interval interval t β ≤ t ≤ t β+1 and satisfy the boundary conditions Note that the property specified by these equations cannot be expressed in terms of the conventional single-time density matrix ρ(t), since correlations at different time moments are needed in (25)-(28). In general, this problem requires consideration of a two-time density matrix (e.g. ρ(t 1 , t 2 ) = |ψ(t 1 )〉 〈 ψ(t 2 )|) but using wave functions seems to be more convenient and is perfectly sufficient for our goals. Note that, unlike in the case of symmetric decoherence, the antimatter states have coherent components at t = t β + 0 just as the matter states have coherent components at t = t β+1 − 0.
While the formulae (8)- (14) for the unitary evolution operator  are the same as in the symmetric case, the wave function ψ(t), which is evaluated below, is different from (20) due to differences in the boundary conditions. While some of the terms remain similar to (20) at the leading order to ensure compliance with the respective boundary conditions in (28). The additional exponential multipliers exp(± iε j Δ t) appear due to the phase change between the states | j〉 taken at t = t β + 0 and t = t β+1 − 0. As previously, the real parts of the diagonal terms must be evaluated up to the second order The diagonal contributions from the antimatter states are evaluated in a similar manner Taking squares of the the wave functions results in and the coefficients w j k are still specified by (21). Evaluation of the sum , while taking into account that p(t β + 0) = p(t β − 0) is continuous (for any t β ) and that  . The off-diagonal elements differ from those in SPME only by their signs = ± W w j k j k for j ≠ k but the diagonal elements are generally different ≠ ± W w k k k k . In absence of matter the evolution of the antimatter states according to APME represents, as expected, a time reversal of the evolution of these states according to SPME. Note that, in addition to this expected result, the derived equation also evaluates another, highly non-trivial statistical property -how the matter and antimatter states interact with each other.

Comparison of the two forms of PME
First we note that the forms of PME, symmetric (24) and antisymmetric (35), can both be written as where the modified indicator-function ′ C j is defined differently for symmetric PME (SPME) and antisymmetric PME (APME) by This form is useful for analysis of the common features of these equations.

Invariant properties.
These properties can be summarised by the following proposition: First we note that both constraints imposed on the Hamiltonian, (6) and (7) are the same for diagonal elements j = k resulting in ε j = ε −j . For off-diagonal elements, the CP invariance yields Here we use the definition of w k j by (21) and take into account that |α j | = 1 in (6) and (7). Finally, the symmetry of the kinetic coefficients = w w k j j k determines that both of the invariant properties, CP in (6) and CPT in (7), result in the same set of constrains While in general the constraints imposed on the interaction Hamiltonians by CP invariance and by CPT invariance produce different evolutions of quantum systems, this difference is not revealed in the PME when the decoherence time is sufficiently short, as stipulated by (16). The statement of the proposition is then easily proved by substituting − j for j in (24) and − j for j and − t for t in (35). This proposition indicates that SPME can be referred to as the CP-invariant PME and APME can be referred to as the CPT-invariant PME, although using these terms should not lead to confusion of invariant properties of PME with those of the Hamiltonian (and with symmetries of unitary evolutions corresponding to the Hamiltonian). The same statement applies to the two possible extensions of thermodynamics.

Consistency with thermodynamics. The definition of entropy
coincides with the conventional definition of entropy for SPME but changes the signs of contributions of the antimatter states for APME. This definition of entropy does not involve the degeneracy factors, as the summation is performed over all quantum states (and not over different energy levels, which can be degenerate). Note that the placement of ′ C k in (36) and (39) does not allow for interpretation of ′ C k as effective degeneracy factors. The entropy S defined by (39) satisfies the following H-theorem:

Proposition 2 The entropy S monotonically increases in evolution of probabilities predicted by the Pauli master equation (both symmetric and antisymmetric), unless the system is in equilibrium where the entropy remains constant.
The proof of the proposition is achieved by evaluating dS/dt using (36) Here we use the normalisation of p j in (23), equivalence of the summation indices j and k as well as the symmetry of the coefficients = w w k j j k defined in (21). We also note that ′ = C ( ) 1 j 2 and ′ = ′ C C 1/ j j . Each term (j, k) in the last sum is no less than zero -this can be easily seen by considering two alternatives ′ = ′ C C j k and ′ = − ′ C C j k . The symmetric version of this H-theorem (i.e. all ′ = C 1 j ), which is conventional 8 , also requires that the kinetic coefficients are symmetric = w w j k k j . The proof given here is suitable for both SPME and APME. The essential property of PME is consistency with thermodynamics as stipulated in the following proposition.

Proposition 3
Pauli master equations (PME) are consistent with thermodynamics: symmetric PME corresponds to symmetric extension of thermodynamics from matter to antimatter and antisymmetric PME corresponds to the antisymmetric extension of thermodynamics from matter to antimatter. Specifically, this consistency implies: The first property directly follows from PME (36). The second property is valid since (21), which is valid when decoherence does not interfere with the main eigenstates τ τ  d 0 . The states with the same energy as the initial state are conventionally called "on-shelf states". If τ d is too small, equation (14) can allow for transitions between states with different energies ε k ≠ ε j since D jk in (21) deviates from 2πδ(ε j − ε k ) or, alternatively, can freeze any evolution of the system since the change in probabilities is proportional to Δ t 2 In many cases, Ω(∞ , − ∞ ) is quite small for a single interaction event. As expected, when Ω becomes sufficiently large, the SPME solution converges to the state with maximal entropy p +1 = p −1 = 1/2. That is, if applied on a large scale, this model predicts equilibration of matter and antimatter that, at equilibrium, should be present in equal proportions. The APME prediction is radically different -this equation transfers probability from the antimatter states to the matter states until the antimatter states can no longer be present (have zero probability). On a large scale, this model predicts that there could not be equilibrium balance between matter and antimatter unless antimatter is fully converted into matter. This can be summarised in form of the proposition: The evolution specified in the proposition follows from the H-theorem and the definition of entropy by equation (39). The maximal value of entropy (subject to physical constraints) is achieved 1) for SPME when p j > 0 are the same for all interacting on-shelf states and 2) for APME when p j > 0 are the same for all interacting on-shelf matter states and p j = 0 for all antimatter and remaining matter states. The statement of the proposition does not contradict the declared similarity of matter and antimatter. In fact, both SPME and APME are based on similarity of matter and antimatter but interpret this similarity differently. In case of APME (but not SPME) this interpretation involves a time reversal: antimatter is converted into matter forward in time but, in reversed time, the same process converts matter into antimatter.

Discussion
While conventional flow of time is deeply imbedded into our intuition, it was Boltzmann 7 who connected the perceived direction of the "flow of time" with the second law of thermodynamics. He suggested that if there was a section of Universe where entropy decreases in time, the local population would perceive the past as the future and the future as the past in that part of the Universe. Even now this statement would seem very strange to many people. APME appeals to similar ideas and, for many people, may contradict the intuitive perception of time, which makes the use of this model more difficult. This is a disadvantage, but there are advantages associated with APME and the antisymmetric (or CPT-invariant) approach to the thermodynamics of antimatter. First, it connects two fundamental asymmetries of our world-the absence of antimatter and the preferred direction of thermodynamic time-by predicting that conversion of antimatter into matter forward in our time is strongly favoured by thermodynamics. Demonstrating that APME tends to convert antimatter into matter is one of the major results of the present work (assuming that matter/antimatter conversions are allowed by the superselection rules -see the previous section).
The microscopic world is mostly CP-invariant. This world inevitably interacts with thermodynamic surroundings and these interactions should also be CP-symmetric. We see macroscopic effects of these interactions in form of thermodynamic irreversibility but do not detect microscopic effects of these interactions, since they do not conflict with the quantum CP invariance. However environmental interactions that generate thermodynamic time can become visible at a microscopic level in CP-violating systems. These interactions are seen as apparent CPT violations even if the quantum system is strictly CPT-preserving 19 . Under these conditions, the ubiquitous nature of thermodynamic interactions may lead to questioning CPT invariance. The second advantage is that the antisymmetric approach offers a very convenient interpretation that strictly upholds the CPT invariance: the apparent CPT violation appears only because exact application of the CPT transformation requires to change environmental matter into antimatter, which is practically impossible.
The system considered in the present work is subject to much stronger thermodynamic interference than that considered in the CP-violating Kaon decays 19 . This strong and persistent influence of decoherence in the derivation of PME ensures thermodynamic compliance for the evolution of the system but suppresses the difference between CP-and CPT-invariant Hamiltonians. It seems, however, that invariant properties of decoherence, which remain largely unknown, should have some physical links with invariant properties of the microscopic world. The effect of decoherence on simulations of realistic interactions of particles and antiparticles should involve radiation and may need to incorporate relativistic quantum mechanics, where differences between microscopic symmetries can be more persistent.
While there are some significant advantages in considering APME and CPT-invariant thermodynamics, these advantages do not prove that it is the antisymmetric (and not symmetric) approach that corresponds to the real world. Absence of antimatter in the universe may have different explanations. Would it be possible to establish, at least in principle, which version of thermodynamics corresponds to the real world? It seems that the answer is generally positive but the following important points need to be taken into account.
The thermodynamic properties are not revealed in simple microscopic systems; this requires a system of sufficient size and complexity. A simple system placed into a thermodynamic environment does not create thermodynamic behaviour on its own, but is subject to the thermodynamic properties of the environment. Hence having relatively few isolated antistates or placing these antistates into a conventional thermodynamic environment does not create an antimatter-controlled thermodynamic system and does not solve the problem. The challenge is to create a thermodynamic object (i.e. object that is sufficiently complex and, conventionally, cannot be in a coherent state) that is made not from matter but from antimatter. This object should be sufficiently insulated from the environment so that the inter-object interactions are overwhelmingly stronger than any environmental interference. The difficulty of this task should not be underestimated but some encouraging news is arriving from high-energy colliders 20,21 . It is becoming possible to create antinuclei 22 and even antiatoms 21 , but still only in very small quantities. The other possible object of interest is quark-gluon plasma, which can be created in high-energy collisions -despite its small size, this object seems to have some thermodynamic properties 20,22 . In any case, the progress in high-energy experiments moves forward quickly and a time when the thermodynamic properties of antimatter can be assessed experimentally may not be too far ahead.

Conclusions
This work introduces the antisymmetric version of the Pauli master equation (APME). The symmetric version of the equation (SPME) is conventional. The proved H-theorem demonstrates consistency of the symmetric and antisymmetric versions of the PME with the symmetric and antisymmetric extensions of thermodynamics from matter into antimatter. The symmetric versions of these approaches are CP-invariant while the antisymmetric versions are CPT-invariant (under conditions specified in Proposition 1). These properties do not necessarily correspond to, and must not be confused with, microscopic invariant properties of the quantum Hamiltonians. The analysis of the present work demonstrates that SPME predicts evolution towards the same probabilities of matter and antimatter states, while APME points to full conversion of antimatter into matter. In the absence of experimental knowledge about thermodynamic properties of antimatter, we cannot make an experimentally justified choice in favour of the symmetric or antisymmetric versions, however continuing progress of high-energy physics will, hopefully, be able to resolve this dilemma in the future.