Quantum Entropy

Home , Fock state

arXiv:2106.15375v1 [quant-ph] 29 Jun 2021 e okUiest,NwYr,NwYr 10012 York New York, New University, York New orn nttt fMteaia Sciences Mathematical of Institute Courant aiGie n v .Kedem M. Zvi and Geiger Davi unu Entropy Quantum 1 Abstract All the laws of particle physics are time-reversible. A time arrow emerges only when ensembles of classical particles are treated probabilistically, outside of physics laws, and the second law of thermodynamics is introduced. In quantum physics, despite its intrinsically probabilistic nature, no mechanism for a time arrow has been proposed. We propose an entropy for quantum physics, which may conduce to the emergence of a time arrow. The proposed entropy is a measure of randomness over the degrees of freedom of a quantum state. It is dimensionless, it is a relativistic scalar, it is invariant under coordinate transformation of position and momentum that maintain conjugate properties, and under CPT transformations; and its minimum is positive due to the uncertainty principle.

I. INTRODUCTION

Today’s classical and quantum physics laws are time-reversible and a time arrow emerges in physics only when a probabilistic behavior of ensembles of particles is considered. In contrast, no mechanism for a time arrow was proposed for quantum physics even though it introduces probability as intrinsic to the description of even a single-particle system. A probability P 0 is assigned to each value of 0 A ( ) ∈ for an observable A. For a ﬁnite A = 08 8=1,...,# , the Shannon entropy, H = { } # P 08 log P 08 , is a measure of information about A. − 8=1 ( ) 2 ( ) PExtending the concept of entropy to continuous variables, continuous distributions, and to quantum mechanics (QM) has been challenging. For example, von Neumann’s entropy [17] requires the existence of classical statistics elements (mixed states)in order not to vanish, and consequentlyit assigns zero entropy to one-particle systems. Therefore, it is not possible to start with von Neumann’s entropy if one wants to assign an entropy that measures the randomness of a one particle state and then

2 extend it to multiple particles and a quantum ﬁeld. There are two frameworks for quantum physics: QM and quantum ﬁeld theory

(QFT). A particle system is described in QM by a quantum state kC , which is a ray | i in Hilbert space, while in QFT such a state is described by an operator R acting on the vacuum state 0 . Our proposed entropy is applicable to both frameworks. | i In classical physics, Boltzmann entropy and Gibbs entropy, and their respective H-theorems [11], are formulated in the phase space, reflecting the degrees of freedomofasystem. InQM,thepositionandthemomentumareconjugate operators and their eigenstates allow to describe a phase space. In QFT the position is demoted to a variable, and the Fourier transform of the fields introduces a spatial frequency variable that together with position compose a quantum field phase space coordinate system. We require the entropy (i) to account for all degrees of freedom of a state, (ii) to be a measure of randomness of such a state, (iii) to be invariant under the applicable transformations, such as Lorentz transformations. The paper is organized as follows. In Section II, we propose an entropy measure of randomness of a quantum state. In Section III we prove its minimum. Then, we prove invariant properties under: continuous coordinate transformations in phase space in Section IV A, discrete CPT transformations in Section IV B, and special relativity in Section IV C. Section V concludes the paper.

II. QUANTUM ENTROPY

The entropy must account for both continuous and discrete degrees of freedom (dof). TheentropyforcontinuousdofsisdevelopedinsectionIIAanditisassociated with the space coordinates. Together with the spatial frequency (the conjugate variable to the spatial variable) they form the continuous phase space. The discrete dof is associated with the spin, which is an internal dof. The entropy for a spin

3 1 phase space is developed in section IIB for spins 0 and 2. The entropy of a state is then the sum of the entropy of the continuous dofs and the entropy of the spin.

A. Entropy for the Continuous Degrees of Freedom

To address the continuous coordinates degrees of freedom and the uncertainties associated with them, we associate with every QM state k the projection onto | iC the QM eigenstates of the conjugate operators rˆ and pˆ , i.e., r and p . Note that | i | i the continuous degrees of freedom are completely deﬁned by one of the choices of continous coordinates, r or k, as one of the projections can be recovered from the other one via a Fourier transform. However, due to the uncertainty principle, the uncertainties or randomness of the continuous coordinates of a particle are captured in the phase space described by k r,C = r kC , q p,C = p kC . By Born’s ( ( ) h | i ( ) h | i) 2 2 rule, dr r,C = k r,C and d? p,C = q˜ p,C are the probability densities of ( ) | ( )| ( ) | ( )| the phase space representation of the state. We deﬁne the continuous entropy of a particle to be

3 3 3 S = dr r,C d? p,C ln dr r,C d? p,C \ d r d p , −¾ ( ) ( ) ( ) ( ) 3 3 = dr r,C d: k,C ln dr r,C d: k,C d r d k , −¾ ( ) ( ) ( ( ) ( ) )

= Sr S: , (1) +

1 1 where k = p is the spatial frequency, d: k,C = d? p,C is the spatial frequency \ ( ) \3 ( ) probability density (so that the inﬁnitesimal probability in a inﬁnitesimal volume 3 is invariant under the change in variables), Sr = dr r,C ln dr r,C d r, and − ( ) ( ) ´ analogously for Sk. The proposed entropy is motivated by the work of Gibbs [11] and Jaynes [13]. The dimensionless phase space volume element is d3r d3k, and the dimensionless

probability density in phase space is d r, k,C = dr r,C d: k,C . Thus the entropy ( ) ( ) ( ) 4 is dimensionless and invariant under changes of the units of measurements. A natural extension to an #-particle system in QM is

3 3 3 3 ( = d r1 d k1 ... d r# d k# dr r1,..., r# ,C dk k1,..., k# ,C −¾ ( ) ( )

ln dr r1,..., r# ,C d k1,..., k# ,C × ( ( ) k( )) 3 3 = d r1 ... d r# dr r1,..., r# ,C ln dr r1,..., r# ,C −¾ ¾ ( ) ( ) 3 3 d k1 ... d k# dk k1,..., k# ,C ln dp k1,..., k# ,C , −¾ ¾ ( ) ( ) 2 2 where dr r1,..., r# ,C = k r1,..., r# ,C and dk k1,..., k# ,C = q k1,..., k# ,C ( ) | ( )| ( ) | ( )| # are defined in QM via the projection of the state kC of # particles onto the position r1 ... r# and the spatial frequency (momentum) k1 ... k# coordinate h | h | h | h | # systems. The state kC is defined in Fock spaces, the product of # Hilbert spaces, and requiring the construction of combinatorics to describe indistinguishable particles. In QFT, fields are described by the operators R r,C , where r,C become pa- ( ) ( ) rameters describing space-time, and Q k,C is the Fourier transform of R r,C . A ( ) ( ) representation used in QFT for a system of particles is based on Fock states with

occupation number of the form =@1 ,=@2 ,,...,=@8 ,... , where =@8 is the number of

particles in a QM Hilbert space state @8 . A special state is the vacuum sate 0 | i | i with no particles. One can create a Fock occupation state, or simply a Fock state, for sets of quantum numbers from the vaccum state and the creation operator as 1 = =@8 =@1 ,=@2 ,,...,=@8 ,...,=@ 0†@8 0 . = = ! ( ) | i Y8 1 @8

=p The number of particles of a Fock state is # 8=1 =@8 . Then, a QFT state is described in a Fock space as a linear superpositionP of Fock states, i.e., state = | i

< U< =@1 ,=@2 ,,...,=@8 ,... , where < is an index for each possible conﬁguration P 2 of a Fock state, U< ℂ,and1 = < U< . A general state in Fock space does not ∈ | | | i have a speciﬁc number of particlesP since it can be in a superposition of two Fock

5 states with diﬀerent number of particles. The QFT operators act on the the states producing a phase space state given by R r,C state ,Q k,C state . We then ( ( )| i ( )| i) deﬁne the probability density function in position and momentum space as

QFT 2 d r,C = R r,C state = state R † r,C R r,C state . r ( ) | ( )| i| h | ( ) ( )| i Analogously, we have dQFT k,C = Q k,C state 2 = state Q k,C Q k,C state . k ( ) | ( )| i| h | †( ) ( )| i One should not interpret these probability density distributionsin QFT as associated with the measurements of finding a particle in a given position or spatial frequency, but rather as distributions of the information about position and space frequency of the state of the field. The information and the uncertainties of the position and momentum of the field of particles in this state are captured by the distributions dQFT r,C and dQFT k,C . The QFT entropy of the position and momentum r ( ) k ( ) information is then given by

= QFT QFT QFT QFT 3 3 S dr r,C dk k,C ln dr r,C dk k,C d r d k −¾ ( ) ( ) ( ) ( ) = Sr S: , + QFT QFT 3 where Sr = d r,C ln d r,C d r, and analogously for Sk. − r ( ) r ( ) In the rest´ of the paper the superscript QFT will generally be dropped as it will be clear whether we are using the representation of QM or of QFT.

B. Entropy for the Spin Degrees of Freedom

The dofs associated with the spin are captured by the vector or bispinor representation of the states in both frameworks, QM and QFT. The spin magnitudes of a particle species are well deﬁned and for re-normalizable quantum theories they 1 must be either 0 (Higgs boson), 2 (e.g., electrons), or 1 (e.g., photons). In string theory, higher order spin values are possible, including the graviton with spin 2. It is not possible to simultaneouslyknow the spin of a particle on all three dimensional

6 directions and this uncertainty, or randomness, is exploited by the Stern–Gerlach experiment [10] to demonstrate the quantum nature of the spin. The spin matrix associated with a particle can be described (e.g., [6, 14]) as

2 2 2 2 ( = (GGˆ (H Hˆ (IIˆ and ( = ( ( ( , ì + + G + H + I (0,(1 = i\(2 , where 0,1,2 is a cyclic permutation of G,H,I, [ ] 2 ( ,(0 = 0 , for 0 = G,H,I. (2) [ ] It is usual to represent a spin state of a particle species with spin value B through 2 a set of orthonormal eigenstates bB,< of the operators ( ,(I , with associated ( ) eigenvalues B B 1 ,< , where the eigenvalues of (I are in the range < = B, B ( + ) − − + 1,...,B. We will refer to the entropy associated with the spin degrees of freedom of a particle as the spin-entropy. 1 We evaluate the entropy of spin 0 and 2 particles as a measure of the spin uncertainty, which must be invariant to the choice of the 3D coordinate system.

= 1 Theorem 1 (Spin-Entropy). For particles with spin B 0, 2 the spin-entropy is 2B ln2π.

Proof. The uncertainty relation for the spatial orientation of a particle spin can be characterized by the polar and the azimuthal angles \ and q. The conjugate pair of variables q,B\ cos \ forms a spin phase space with volume 4πB\. We follow ( ) the geometric quantization approach to spin, see for example [14], and treat these conjugate variables as operators satisfying the canonical commutation relation

q,B\ cos \ = i\ . (3) [ ] However, at the northern pole (cos \ = 1) and the southern pole (cos \ = 1) the − angle q is not deﬁned. Thus, in the basis q that diagonalizes q we have two oper- | i ators, B\ cos \ = B\ i\ m for the northern hemisphere and B\ cos \ = B\ i\ m − mq − − mq for the southern hemisphere. The spin commutation relations (2) can be derived

7 from the canonical commutation relation (3) by writing (G = B\ sin \ cos q,(H =

B\ sin \ sin q,(I = B\ cos \, which is a Cartesian representation derived from a polar one for spin magnitude B\. The eigentstates of (I are then kB,< q = q bB,< = ( ) 1 i B < q 1 i B < q e ( + ) for the northern hemisphere and kB,< q = q bB,< = e (− + ) for √2π ( ) √2π

the southern hemisphere, where < = B,...,B. The two solutions differ by a phase − i2Bq (gauge) transformation of e− . This representation explicitly shows that a well de- 2 fined spin state along I (fixed <) has the probability density dB,< q = kB,< q = ( ) | ( )| 1 = B 2π , uniform along q. The general state is then kB q <= B UB,

1. Spin 0. There is no intrinsic randomness and the entropy must be 0.

2. Spin 1 . One can reach any spin state from any other spin state via a 2 2 2 × unitary rotation transformation in the SO 3 group. Thus, a rotation-invariant ( ) spin-entropy must be a constant independent of what the spin state direction is. Aligning the I coordinate with any given state produces the intrinsic − entropy of a uniform density along q in the range 0, 2π , yielding entropy [ ) 2c 1 1 dq ln = ln2π (which is the minimum entropy of dB q , over − 0 2c 2c ( ) ´ h i the possible set of coeﬃcients UB,<.)

Combining the two cases in one expression completes the proof.

For massless spin 1, there are only two possible helicities and as in the case of 1 spin 2 , any state can be reached from one state by rotating the coordinate system. There is still the uncertainty on the polarization, yielding the same entropy as for 1 = spin 2 particles. For the massive case, we observe that a I-state with

be reached from state

8 with

C. Entanglement of two fermions

= 1 Let us consider a system with two identical particles, A and B, each with B 2 in an entangled state

iU b = e cos \U b b sin \U b b | i + − − − + where indicates the I-component. The entanglement is reﬂected by the mixture ± of the product of single particle states b b and b b , with probabilities + − − + 2 2 P \U = cos \U and P \U = sin \U . This state has an uncertainty associ- + − ( ) − + ( ) ated with the uniform probability along the angle q for each particle’s I-state and our proposed entropy is 2c PZ \U PZ \U ( \ = dq dq ( ) ln ( ) spin U 2 2 ( )Z − , ¾0 2c 2c ∈{ +X− − +} ( ) ( ) 2 2 2 2 = 2ln2c cos \U lncos \U sin \U lnsin \U . (4) − + Works exist exploring the information content of entangled physical systems, e.g., [3, 5, 16]. It considered the von Neumann entropy for b , which is zero, but after | i tracing out a particle state a non-zero entropy emerges, equal to the second term of (4). Our state’s entropy, after a measurement of one particle, is theentropy of theother particle’s state, i.e., ln 2c; after all there is more randomness in the entangled state b then on the single-particle state. The rest of the entropy of b is absorbed by the | i | i measurement apparatus that would need to be modeled to be properly accounted. The balance of information when considering two fermions colliding and trans- forming into possibly other new particles must include the individual spin-entropy term, not present in the von Neumann’s entropy.

9 Beyond the dof of spins, such as in the study of the coupled harmonic oscillators [16], our proposed entropy will diﬀer from the von Neumann one after the tracing, by including non-constant terms associated with the individual oscillators. They also must be accounted for in the balance of information on physical processes.

III. THE MINIMUM ENTROPY VALUE

The third law of thermodynamics establishes the value of 0 as the minimum classical entropy. However, the minimum of the quantum entropy must be positive due to the uncertainty principle, where the position and the velocity cannot be both fully speciﬁed.

Theorem 2. The minimum of the entropy of a particle with spin B = 0, 1 is 3 1 2 ( + ln π 2B ln2π. ) +

Proof. The spin-entropy value 2B ln2π is a constant per particle and must be added to form the total entropy. The continuous entropy (1) is the sum of two entropy terms, the spatial component and the spatial frequency component, which satisfy the entropic uncertainty principle Sr S? 3lneπ\, where S? = S: 3ln \, as + ≥ + shown in [2, 4, 12], i.e., Sr S: 3lneπ. + ≥

Higgs bosons (B = 0) in coherent states have the lowest possible entropy 3 1 ln π . We think that the entropy is a better measure of the uncertainty ( + ) than the variance since (i) this lower bound of the entropic uncertainty principle for unbounded operators is tighter than the bound of the standard uncertainty principle [15], as shown in [2, 4]; (ii) the minimum value from the variance based uncertainty principle for the operators q and cos \ is zero as the operator q is bounded. The dimensionless element of volume of integration to deﬁne the entropy will not contain a particle unless d3r d3k 1, due to the uncertainty principle, and this ≥ 10 may be interpreted as a necessity of discretizing the phase space. We note that the minimum entropy of the discretization of (1) is also 3 1 ln π , as shown in [7]. ( + )

IV. ENTROPY INVARIANT PROPERTIES

In this section we establish entropy invariant properties supporting its applica- bility to analyze quantum systems. In each section we use either the QM setting or the QFT setting but the results are valid for both settings. We show that the entropy is invariant under (i) continuous coordinate transformations in Section IVA, (ii) discrete CPT transformations, in Section IVB, and (iii) Lorentz transformations in Section IV C. Since the spin-entropy remains a constant per particle, it is invariant under all such transformations.

A. Phase-Space Continuous Transformations

We investigate two types of continuous transformations of the phase space: a point transformation of coordinates and a translation in phase space of a quantum reference frame [1]. This section is described in QM setting. Consider a transformation of position coordinates : r r . In QM, such a 7→ ′ coordinate transformation must be a point transformation, which induces the new conjugate momentum operator [8]

1 1 pˆ′ = i\ r − r′ r r′ , (5) − ∇ ′ + 2 ( )∇ ′ · ( )

mr r′ 1 where r′ = ( ) is the Jacobian of − . ( ) mr′ Theorem 3. The entropy is invariant under a point transformation of coordinates.

Proof. Let S be an entropy in phase space relative to a conjugate Cartesian pair of coordinates r, p . Weignoretheconstantterm 3ln \. Let p denotethemomentum ( ) − ′ 11 conjugate to r′. As the probabilities in inﬁnitesimal volumes are invariant under point transformations,

2 3 2 3 2 3 2 3 k′ r′ r d r′ = k r d r and q˜′ p′ p d p′ = q˜ p d p . (6) | ( ( ))| | ( )| | ( ( ))| | ( )| Thus, Born’s rule that the probability density functions are k r 2 and q˜ p 2 | ′( ′)| | ′( ′)| holds. The Jacobian satisﬁes

3 3 det r′ d r′ = d r . (7) ( ) Combining (6) and (7), 1 r′ k = k′ r′ = k r r′ , (8) h | i det r ( ) ( ( )) ( ′) p so that the iniﬁnitesimal probability r k 2 det r d3r = k r 2 d3r is in- | h ′| i| ( ′) ′ | ′( ′)| ′ variant. = 1 ir p Considering the Fourier basis p r 3 e− · combined with (8) leads to h | i 2c 2 ( ) 3 q˜ p = p k = det r′ d r′ p r′ r′ k ( ) h | i ¾ ( ) h | i h | i 1 = ir′ p 3 3 det r′ k′ r′ e− · d r′ . 2π 2 ¾ ( ) ( ) ( ) p It was noted in [8] that in the momentum space there is a transformation : p p , speciﬁed by (5) up to an arbitrary function 6 p = det 1 p . This 7→ ′ ( ′) ( − )( ′) freedom to specify 6 p is the result of the volume elements scaling according to ( ′) det r , that is 6 p d3p = d3p. Similarly to (8) let ( ′) ( ′) ′ 1 p′ k = q˜′ p′ = q˜ p p′ , h | i 6 p ( ) ( ( )) ( ′) implying p

2 3 2 3 p′ k 6 p′ d p′ = q˜′ p′ d p′ , | h | i| ( ) | ( )| which is an inﬁnitesimal probability invariant in momentum space. Thus, we can scale by scaling det 1 p by any function, say 5 p , while also scaling ( − )( ′) ( ′) 12 1 p′ k according to . Then, the scaled will satisfy the conjugate properties √ 5 p h | i ( ′) and be a solution. Thus

3 3 Sr Sp = d r d p dr r,C d? p,C ln dr r,C d? p,C + −¾ ( ) ( ) ( ) ( ) 3 3 1 = d r′ d p′ d′ r′,C d′ p′,C ln d′ r′,C d′ p′,C −¾ A ′ ( ) ?′ ( ) 6 p det r A ′ ( ) ?′ ( ) ( ′) ( ′)

1 3 3 det − r′ = Sr Sp d r′ d p′ d′ r′,C d′ p′,C ln ( ) ′ + ′ −¾ A ′ ( ) ?′ ( ) 6 p ( ′) = 1 = Sr′ Sp′ lndet − r′ d′ ln 6 p′ d′ Sr′ Sp′ , + − h ( )i A ′ + h ( )i ?′ + where

1 = 1 lndet − r′ d′ dA′ r′,C lndet − r′ , h ( )i A ′ ¾ ′ ( ) ( ) = 3 ln 6 p′ d′ d′? p′,C ln 6 p′ d p′ . h ( )i ?′ ¾ ′ ( ) ( ) And 6 p must be chosen as to satisfy ( ′) = 1 = 1 ln 6 p′ d′ ln det − r′ d′ ln . h ( )i ?′ h ( )i A ′ det r ′ d′ ( ) A ′ We next investigate translation transformations. When a quantum reference frame

[1] is translated by G0 along G, the state kC in position representation becomes | i iG %ˆ k G G0,C = G G0 kC = G )ˆ% G0 kC , where )ˆ% G0 = e 0 , and %ˆ is ( − ) h − | i h | (− )| i (− ) the momentum operator conjugate to -ˆ . When the reference frame is translated by ?0 along ?, the state kC in momentum representation becomes q˜ ? ?0,C = | i ( − ) i? -ˆ ? ?0 kC = ? )ˆ- ?0 kC , where )ˆ- ?0 = e 0 , and -ˆ is the position h − | i h | (− )| i (− ) operator conjugate to %ˆ.

Theorem 4. Consider a state kC . Its entropy S is invariant under a change of a | i quantum reference frame by translations along G and along ?.

Proof. We focus on the spatial and momentum variables and start by showing that 2 2 SG = ∞ dG k G,C ln k G,C is invariant under two types of translations − | ( )| | ( )| ´−∞ 13 2 2 (i) translations along G by any G0 because k G,C becomes k G G0,C , and | ( )| | ( + )|

= ∞ 2 2 SG G0 dG k G G0,C ln k G G0,C + −¾ | ( + )| | ( + )| −∞ ∞ 2 2 = d G G0 k G G0,C ln k G G0,C −¾ ( + )| ( + )| | ( + )| −∞ ∞ 2 2 = dG′ k G′,C ln k G′,C = SG . −¾ | ( )| | ( )| −∞

(ii) translations along ? by any ?0, because applying )- ?0 to k G,C , we get ( ) ( )

∞ k? G,C = G )ˆ- ?0 kC = G )ˆ- ?0 ? ? kC d? 0 ( ) h | ( )| i ¾ h | ( )| i h | i −∞ ∞ = G ? ?0 q˜ ?,C d? ¾ h | + i ( ) −∞ ∞ 1 i G ? ?0 i G?0 = e ( + ) q˜ ?,C d? = k G,C e , ¾ √2π ( ) ( ) −∞ implying 2 2 k? G,C = k G,C . | 0 ( )| | ( )|

Thus, SG is invariant under translations along ?. 2 2 Similarly, by applying both translations to Sp = ∞ d? q˜ ?,C ln q˜ ?,C − | ( )| | ( )| ´−∞ we conclude that Sp is invariant under them too.

Therefore S = SG Sp 3ln \ is invariant under translations in both G and ?. + −

B. CPT Transformations

We now show that the entropy is invariant under the three discrete symmetries associated with Charge Conjugation, Parity Change, and Time Reversal. Those symmetries are studied in QED, QCD, Weak Interactions, the Standard Model, and Wightman axiomatic formulation of QFT [19]. We will be focusing on fermions, and thus on the Dirac spinors equation, though most of the ideas apply to

14 bosons as well. The QFT Dirac Hamiltonian is given by

D 3 0 0 = d rR † r,C i\W W <2W R r,C . H ¾ ( ) − ì · ∇ + ( )

D mR r,C A QFT solutionR r,C satisfies the equation ,R r,C = i\ ( ) and the , ( ) [H ( )] − mC %, and ) symmetries provide new solutions fromR r,C . We denote, as usual, quan- T ( ) C P T tum fields R r,C = R r,C , R r,C = %R r,C , R r, C = )R ∗ r, C , ( ) T( ) (− ) (− ) ( − ) ( − ) and kCPT r, C = %)k r, C , For completeness, we briefly review the three (− − ) (− − ) operations, Charge Conjugation, Parity Change, and Time Reversal.

T 1. Charge Conjugation transforms particles R r,C into antiparticles R r,C = ( ) ( ) R W0 T r,C . The main property that must satisfy is W` 1 = W`T, ( † ) ( ) − − so that R C r,C is also a solution for the same Hamiltonian. In the standard ( ) representation = iW2W0, up to a phase.

2. Parity Change % = W0, up to a sign, eﬀects the transformation r r. 7→ −

3. Time Reversal eﬀects C C and is carried by the operator T = ) ˆ , where ˆ 7→ − applies conjugation. In the standard representation ) = iW1W3, up to a phase.

Theorem 5 (Invariance of the entropy under CPT-transformations). Given a quan-

tum ﬁeld R r,C , its Fourier transform Q k,C , and its entropy SC, the entropies of ( ) ( ) R r,C , R P r,C ,R C r,C ,R T r, C , and R CPT r, C and their corresponding ∗ ( ) (− ) ( ) ( − ) (− − ) Fourier transforms, are all equal to SC.

Proof. The probability densities of R r,C , R T r, C , R P r,C , R C r,C , and ∗ ( ) ( − ) (− ) ( ) 15 R CPT r, C are respectively, (− − ) T d∗ r,C = R r,C R ∗ r,C = R † r,C R r,C = d r,C , r ( ) ( ) ( ) ( ) ( ) ( ) T T T C = † = ∗ = dr r,C R r,C †R r,C R r,C R r,C dr r,C , ( ) ( ) ( ) ( ) ( ) ( ) P 0 0 d r,C = R † r,C W †W R r,C = R † r,C R r,C = dr r,C , r (− ) ( )( ) ( ) ( ) ( ) ( ) T T T d r, C = R r,C ) †)R ∗ r,C = R r,C R ∗ r,C = dr r,C , r ( − ) ( ) ( ) ( ) ( ) ( ) T T CPT † d r, C = R r,C %) † %) R r,C r (− − ) ( )( ) ( ) ( ) T = R ∗ r,C R r,C = dr r,C , (9) ( ) ( ) ( )

where we used † = ) †) = I. Note that because the entropy requires an integration over the whole spatial volume, the densities dP r,C and dCPT r, C will be (− ) (− − ) evaluated over the same volume as d r,C . As the densities are equal, so are the ( ) associated entropies. The properties in (9) are also valid for the corresponding Fourier transform ﬁelds and densities. Note that when Parity Change is applied to a QFT operator, both the spatial and momentum variables behave as odd variables, and when Time Reversal is applied, both spatial and momentum operators are time reversed. So all the derivations above are similarly valid for the Fourier quantum ﬁelds. Thus, both

entropies terms in SC = SA S: are invariant under all CPT transformations. + Note that a proof that the entropy is invariant under CPT and each of the transformations can be readily achieved in QM, without the QFT framework.

C. Lorentz Transformations

We now investigate the behavior of the entropy when space and time are trans- formed according to the Lorentz transformations, using the natural QFT setting.

Theorem 6. The entropy is a scalar in special relativity.

16 3 3 Proof. The probability elements dP r,C = dr r,C d r and dP k,C = dk k,C d k ( ) ( ) ( ) ( ) are invariant under Lorentz transformations since probabilities of events do not depend on the frame of reference. Consider a slice of the phase space with frequency 2 = 2 2 <22 1 3 3 lk k 2 \ . The volume elements l d k and lk d r, are invariant under r + k the Lorentz group [6, 18], that is, 1 d3k = 1 d3k and l d3r = l d3r , implying lk l ′ k k′ ′ k′ = 3 3 = 3 3 = d+ d k d r d k′ d r′ d+′, where r′, k′, and lk′ are the results of a Lorentz 1 transformation applied to r, k, and lk; and dr r,C and lkdk k,C are also lk ( ) ( ) invariant under the group. In fact, such an extra frequency term can be absorbed in the deﬁnition of the quantum ﬁelds that yield the densities (e.g., [18] and see the

following paragraph). Thus, the phase space density dr r,C dk k,C is an invariant. ( ) ( ) Therefore the entropy is a relativistic scalar.

Note that in QFT, one scales the operator Q k,C by √2lk, that is, one scales the ( ) creation and the annihilation operators U k = √lk a k and U k = √l a k . †( ) †( ) ( ) ( ) In this way, the density operator Q k,C Q k,C scales with l and becomes a †( ) ( ) k relativistic scalar. Also, with such a scaling, the inﬁnitesimal probability of ﬁnding a particle with momentum p =\k in the original reference frame is invariant under the Lorentz transformation, though it would be found with momentum p′ =\k′.

V. CONCLUSIONS

We proposed an entropy for a one-particle system in QM and extended it to multiple particles as well as to QFT. The entropy captures the randomness of the probabilistic description of a quantum system over the spatial and the spin degrees 1 of freedom; we studied spin 0 and 2 . We showed that this deﬁnition of the entropy possesses desirable properties an entropy must have, including invariance under coordinate transformations, invariance in special relativity, and invariance under CPT transformations.

17 1 Particles with spin larger than 2 is a topic of future research. In classical physics, the entropy (weakly) increases. We have conjectured in [9] an entropy law, requiring that the entropy increases in quantum systems, analogously to the second law of thermodynamics. QM is thought to reduce to classical mechanics via one or more of the following transformations: the limit \ 0, Ehrenfest theorem, the → WKB approximation, decoherence, though no complete proof exists. In a system of multiple particles, such approximations may lead to statistical mechanics. In that case, the proposed entropy may lead to the Gibbs entropy, and quantum eﬀects may account for the blur needed to prove the H-theorem of Gibbs. Then, the second law of thermodynamics may follow from the entropy law proposed in [9].

VI. ACKNOWLEDGEMENT

This paper is partially based upon work supported by both the National Science Foundation under Grant No. DMS-1439786 and the Simons Foundation Institute Grant Award ID 507536 while the ﬁrst author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the spring 2019 semester “Computer Vision” program.

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