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Quantum Information Physics II
TR2021-
Revised March 28, 2022
Davi Geiger and Zvi M. Kedem Courant Institute of Mathematical Sciences New York University, New York, New York 10012
Abstract
We study quantum entropy, a measure of randomness over the degrees of freedom of a quantum state and quantified in quantum phase spaces. We show that it is dimensionless, a relativistic scalar, and it is invariant under coordinate and CPT transformations. We show that the entropy evolution of a coherent state is increasing with time. We augment time reversal with time translation and show that CPT with time translation can transform particles with decreasing entropy evolution for a finite time interval into anti- particles with increasing entropy evolution for the same finite time interval. We revisit transition probabilities of a two state Hamiltonian and show how they relate to entropy oscillations. We also explore the possibility that entropy oscillations trigger the annihilations and the creations of particles.
CONTENTS
- Introduction
- Quantum Entropy in Phase Space
- Entropy Invariant Properties
- Continuous Transformations of the Phase Space
- CPT Transformations
- Lorentz Transformations
- QCurves and Entropy-Partition
- The Coordinate-Entropy of Coherent States Increases With Time
- Time Reflection
- Entropy Oscillations
- An Entropy Law and a Time Arrow
- Conclusions
- Acknowledgement
- References
of specifying a quantum state, Wehrl’s entropy is an attempt to adapt a classical entropy to a quantum state based on the Husimi’s quasiprobability distribution and using coherent states as an overcomplete basis representation of a classical phase space. Von Neumann’s entropy assigns a zero entropy to any quantum (pure) state and thus does not address the randomness of the observables as we propose. Wehrl’s entropy does not satisfy the third Kolmogorov axiom of mutual exclusivity of events, and as a consequence does not satisfy the monotonicity or the complement rules. Also, Wehrl’s entropy will not be invariant under the Lorentz group transformations or under pointwise transformations of the position. Thus because of the above shortcomings, Wehrl’s entropy will not quantify exactly the randomness of the quantum-phase observables.
QUANTUM ENTROPY IN PHASE SPACE
Given a state j𝜓i𝑡 and its density operator 𝜌𝑡 = j𝜓i𝑡 h𝜓j𝑡 , we consider the quantum coordinate phase space to be the space of simultaenous projections of all possible states to the basis j r i j p i, i.e., the state j𝜓i𝑡 is described in quantum phase space by the pair ¹h r j𝜓i𝑡 h p j𝜓i𝑡 º. The coordinate-entropy in quantum phase space was defined in [7] as
S =
𝜌r ¹ r 𝑡º 𝜌𝑘 ¹ k 𝑡º ln ¹𝜌r ¹ r 𝑡º𝜌𝑘 ¹ k 𝑡º º d^3 r d^3 k
where Sr = ∫^ 𝜌r ¹ r 𝑡º ln 𝜌r ¹ r 𝑡º d^3 r , and analogously for Sk, 𝜌r ¹ r 𝑡º = h r j 𝜌𝑡 j r i = j𝜓¹ r 𝑡ºj^2 and 𝜌𝑘 ¹ k 𝑡º = h k j 𝜌𝑡 j k i = j 𝜙˜¹ k 𝑡ºj^2 , with 𝜓¹ r 𝑡º and 𝜙˜¹ k 𝑡º representing in QM the wave function and in QFT the coefficients of the Fock states. The momentum is described by the change of variables p = ℏ k , so that the entropy is dimensionless and invariant under changes of the units of measurements.
A natural extension of this entropy to an 𝑁-particle QM system is
𝑆 =
d^3 r 1 d^3 k 1 d^3 r 𝑁 d^3 k 𝑁 𝜌r ¹ r 1 r 𝑁 𝑡º 𝜌k ¹ k 1 k 𝑁 𝑡º ln ¹𝜌r ¹ r 1 r 𝑁 𝑡º𝜌k ¹ k 1 k 𝑁 𝑡ºº =
d^3 r 1
d^3 r 𝑁 𝜌r ¹ r 1 r 𝑁 𝑡º ln 𝜌r ¹ r 1 r 𝑁 𝑡º
d^3 k 1
d^3 k 𝑁 𝜌k ¹ p 1 k 𝑁 𝑡º ln 𝜌k ¹ k 1 k 𝑁 𝑡º
where 𝜌r ¹ r 1 r 𝑁 𝑡º = j𝜓¹ r 1 r 𝑁 𝑡ºj^2 and 𝜌k ¹ k 1 k 𝑁 𝑡º = j𝜙¹ k 1 k 𝑁 𝑡ºj^2 are defined in QM via the projection of the state j𝜓𝑡 i𝑁^ of 𝑁 particles (the product of 𝑁 Hilbert spaces) onto the position h r 1 j h r 𝑁 j and the momentum h k 1 j h k 𝑁 j coordinate systems.
ENTROPY INVARIANT PROPERTIES
Continuous Transformations of the Phase Space
In the QM setting, we investigate a point transformation of coordinates and a translation in phase space of a quantum reference frame [1]. Consider a point transformation of position coordinates 𝐹 : r ↦! r^0. It induces the new conjugate momentum operator [3]
p ˆ^0 = iℏ
rr^0 ¸ 12 𝐽 ^1 ¹ r^0 ºrr^0 𝐽 ¹ r^0 º
where 𝐽 ¹ r^0 º = 𝐽 ¹𝐹 ^1 º¹ r^0 º =^ 𝜕 r 𝜕¹ rr 00 ºis the Jacobian of 𝐹 ^1 at r^0.
Theorem 1. The entropy is invariant under a point transformation of coordinates.
Proof. Let S be the entropy in phase-space relative to a conjugate Cartesian pair of coordinates ¹ r p º. Let p^0 be the momentum conjugate to r^0. As the probabilities in
and 𝑃ˆ is the momentum operator conjugate to 𝑋ˆ. When the reference frame is translated by 𝑝 0 along 𝑝, the state j𝜓𝑡 i in the momentum representation becomes 𝜙^ ˜¹ 𝑝 𝑝 0 𝑡º = h𝑝 𝑝 0 j𝜓𝑡 i = h𝑝j 𝑇ˆ𝑋 ¹ 𝑝 0 º j𝜓𝑡 i, where 𝑇ˆ𝑋 ¹ 𝑝 0 º = ei𝑝^0 𝑋ˆ^ , and 𝑋ˆ is the position operator conjugate to 𝑃ˆ.
Theorem 2 (Frames of reference). The entropy of a state is invariant under a change of a quantum reference frame by translations along 𝑥 and along 𝑝.
Proof. Let j𝜓𝑡 i be a state and S its entropy. We start by showing that S𝑥 =
1 d𝑥^ j𝜓¹𝑥 𝑡ºj^2 ln^ j𝜓¹𝑥 𝑡ºj^2 is invariant under two types of translations:
(i) translations along 𝑥 by any 𝑥 0
S𝑥¸𝑥 0 =
d𝑥 j𝜓¹𝑥 ¸ 𝑥 0 𝑡ºj^2 ln j𝜓¹𝑥 ¸ 𝑥 0 𝑡ºj^2 = S𝑥
which is verified by changing variables under the infinite integration interval.
(ii) translations along 𝑝 by any 𝑝 0
𝜓𝑝 0 ¹𝑥 𝑡º = h𝑥j 𝑇ˆ𝑋 ¹ 𝑝 0 º j𝜓𝑡 i =
h𝑥j 𝑇ˆ𝑋 ¹ 𝑝 0 º j 𝑝i h𝑝j𝜓𝑡 i d 𝑝
h𝑥j 𝑝 ¸ 𝑝 0 i 𝜙˜¹ 𝑝 𝑡º d 𝑝 =
p^1 2 π
ei^ 𝑥^ ¹^ 𝑝¸𝑝^0 º^ 𝜙˜¹ 𝑝 𝑡º d 𝑝 = 𝜓¹𝑥 𝑡º ei^ 𝑥 𝑝^0
implying j𝜓𝑝 0 ¹𝑥 𝑡ºj^2 = j𝜓¹𝑥 𝑡ºj^2.
Similarly, by applying both translations to Sp = ∫^ 1^1 d 𝑝 j 𝜙˜¹ 𝑝 𝑡ºj^2 ln j 𝜙˜¹ 𝑝 𝑡ºj^2 we conclude that Sp is invariant under them too. Therefore S = S𝑥 ¸ Sp 3 ln ℏ is invariant under translations in both 𝑥 and 𝑝.
CPT Transformations
We will be focusing on fermions, and thus on the Dirac spinors equation, though most of the ideas apply to bosons as well. The QFT Dirac Hamiltonian is
H D^ =
d^3 r 𝛹 y^ ¹ r 𝑡º
iℏ𝛾^0 𝛾® r ¸ 𝑚𝑐𝛾^0
𝛹 ¹ r 𝑡º
A QFT solution 𝛹 ¹ r 𝑡º satisfies »H D𝛹 ¹ r 𝑡º¼ = iℏ^ 𝜕𝛹^ 𝜕¹ r 𝑡 𝑡º and the 𝐶, 𝑃, and 𝑇 symmetries provide new solutions from 𝛹 ¹ r 𝑡º. As usual, 𝛹 C^ ¹ r 𝑡º = 𝐶𝛹 T^ ¹ r 𝑡º, 𝛹 P^ ¹ r 𝑡º = 𝑃𝛹 ¹ r 𝑡º, 𝛹 T^ ¹ r 𝑡º = 𝑇𝛹 ^ ¹ r 𝑡º, and 𝜓CPT^ ¹ r 𝑡º = 𝐶𝑃𝑇𝜓T^ ¹ r 𝑡º, For completeness, we briefly review the three operations, Charge Conjugation, Parity Change, and Time Reversal. Charge Conjugation transforms particles 𝛹 ¹ r 𝑡º into antiparticles 𝛹 T^ ¹ r 𝑡º = ¹𝛹 y𝛾^0 ºT^ ¹ r 𝑡º. As 𝐶𝛾𝜇𝐶 ^1 = 𝛾𝜇T, 𝛹 C^ ¹ r 𝑡º is also a solution for the same Hamil- tonian. In the standard representation, 𝐶 = i𝛾^2 𝛾^0 up to a phase. Parity Change 𝑃 = 𝛾^0 , up to a sign, effects the transformation r ↦! r. Time Reversal effects 𝑡 ↦! 𝑡 and is carried by the operator T = 𝑇 𝐾ˆ, where 𝐾ˆ applies conjugation. In the standard representation 𝑇 = i𝛾^1 𝛾^3 , up to a phase.
Theorem 3 (Invariance of the entropy under CPT-transformations). Given a quan- tum field 𝛹 ¹ r 𝑡º , its Fourier transform 𝛷¹ k 𝑡º , and its entropy S𝑡 , the entropies of 𝛹 ^ ¹ r 𝑡º , 𝛹 P^ ¹ r 𝑡º , 𝛹 C^ ¹ r 𝑡º , 𝛹 T^ ¹ r 𝑡º , of and 𝛹 CPT^ ¹ r 𝑡º , and their correspond- ing Fourier transforms are all equal to S𝑡.
Proof. The probability densities of 𝛹 ^ ¹ r 𝑡º, 𝛹 T^ ¹ r 𝑡º, 𝛹 P^ ¹ r 𝑡º, 𝛹 C^ ¹ r 𝑡º, and
creation and the annihilation operators 𝛼y^ ¹ k º = p𝜔k a y^ ¹ k º and 𝛼¹ k º = p𝜔 a ¹ k º. In this way, the density operator 𝛷y^ ¹ k 𝑡º𝛷¹ k 𝑡º scales with 𝜔k and becomes a relativistic scalar. Also, with such a scaling, the infinitesimal probability of finding a particle with momentum p = ℏ k in the original reference frame is invariant under the Lorentz transformation, though it would be found with momentum p^0 = ℏ k^0.
QCURVES AND ENTROPY-PARTITION
In [7], we introduced the concept of a QCurve to specify a curve (or path) in a Hilbert space parametrized by time. In QM, a QCurve is represented by a triple (^) j𝜓 0 i^ 𝑈^ ¹𝑡º^ δ𝑡
, where j𝜓 0 i^ is the initial state,^ 𝑈^ ¹𝑡º^ =^ e i𝐻𝑡^ is the evolution operator, and » 0 δ𝑡¼ is the time interval of the evolution. Alternatively, we can represent the initial state by ¹h r j𝜓 0 i h k j𝜓 0 iº and in QFT as ¹𝛹 ¹ r 0 º jstatei 𝛷¹ k 0 º jstateiº.
Definition 1 (Partition of E from [7]). Let E to be the set of all QCurves. We define a partition of E based on the entropy evolution into four blocks:
C : Set of the QCurves for which the entropy is a constant. I : Set of the QCurves for which the entropy is increasing, but it is not a constant. D : Set of the QCurves for which the entropy is decreasing, but it is not a constant. O : Set of oscillating QCurves, with the entropy strictly increasing in some subin- terval of » 0 δ𝑡¼ and strictly decreasing in another subinterval of » 0 δ𝑡¼.
Consider stationary states j𝜓𝑡 i = j𝜓𝐸 i e i𝜔𝑡^ with 𝜔 = 𝐸ℏ, where 𝐸 is an energy eigenvalue of the Hamiltonian, and j𝜓𝐸 i is the time-independent eigenstate of the Hamiltonian associated with 𝐸.
Theorem 5. All stationary states are in C_._
Proof. Follows from the time invariance of the probabilities.
The Coordinate-Entropy of Coherent States Increases With Time
Dirac’s free-particle Hamiltonian in QM [5] is
𝐻 = iℏ𝛾^0 𝛾® r ¸ 𝑚𝑐𝛾^0 (4)
It can be diagonalized in the spatial Fourier domain j k i basis to obtain
𝜔¹ k º = 𝑐
𝑘^2 ¸ 𝑚
2 ℏ^2 𝑐
where 𝜔¹ k º is the frequency component of the Hamiltonian. We focus on the positive energy solutions and so the group velocity becomes
vg ¹ k º = rk𝜔¹ k º = 𝑚ℏ√︃^ k 1 ¸ ¹ ℏ𝑚𝑐𝑘 º^2
In (9) we will use the Taylor expansion of (5) up to the second order, thus requiring the Hessian H ¹ k º, with the entries
H 𝑖 𝑗 ¹ k º =^ 𝜕
(^2) 𝜔¹ k º 𝜕𝑘𝑖 𝜕𝑘^ 𝑗^ =
2 ! ^32 "
δ𝑖 𝑗 1 ¸
𝑖 𝑚𝑐
for the positive energy solution. The three (positive) eigenvalues of H ¹ k º are
2 𝑚 (^2) ¸ 𝜇 (^2) ¹𝑘º 32
𝜆 2 3 = 𝑚ℏ 1 ¸
2 ! ^12
¹𝑚^2 ¸ 𝜇^2 ¹𝑘ºº 12
and N is a normal distribution. Consequently, 𝜓k 0 ¹ r r 𝑡 k 0 𝑡º is the spatial Fourier transform of 𝛷 r 𝑡 k 0 ¹ k k 0 𝑡º.
The probability densities associated with the probability amplitudes in (10) are
𝜌r ¹ r r 𝑡 k 0 𝑡º = (^) 𝑍^12 r
j𝜓k 0 ¹ r r 𝑡 k 0 º N
r j r 𝑡 k 0 i 𝑡 H ¹ k 0 º
j^2
𝜌k ¹ k k 0 𝑡º = (^) 𝑍^12 𝑘
j𝛷 r 𝑡 k 0 ¹ k k 0 ºj^2 (11)
Lemma 1 (Dispersion Transform and Reference Frames). The entropy associated with (11) is equal to the entropy associated with the simplified probability densities
𝜌Sr ¹ r 𝑡º = (^) 𝑍^12 j𝜓 0 ¹ r º N ¹ r j 0 i 𝑡 H ¹ k 0 ºº j^2 𝜌kS ¹ k 𝑡º = (^) 𝑍^12 𝑘
j𝛷 0 ¹ k ºj^2 = 𝜌kS ¹ k 𝑡 = 0 º (12)
Proof. Consider (11). If the frame of reference is translating the position by r 𝑡 k 0 = r 0 ¸ vg ¹ k 0 º𝑡 and the momentum by ℏ k 0 , we get the simplified density functions (12).
Theorem 2 shows that the entropy in position and momentum is invariant under translations of the position r and the spatial frequency k , and that completes the proof.
The time invariance of the density 𝜌Sk ¹ k 𝑡º, and therefore of Sk, reflects the conservation law of momentum for free particles.
We now focus on the case of coherent states, represented by j𝛼i, eigenstates of the annihilator operator. In 1D position space they are represented as 𝜓𝛼 ¹𝑥º = h𝑥j𝛼i = e ^ 𝑝
(^20) 2 π 14 e
12 𝑥 p 2 𝛼^ ^2 , where 𝛼 = p 1 2 ¹𝑥^0 ¸^ i𝑝^0 º. Squeeze states extend to all eigenstate solutions of the annihilator operator by allowing different variances for the Gaussian
solution, and together their representation in 3D position and momentum space are
𝜓k 0 ¹ r r 0 º = h r j𝛼i = 1 23 π^32 ¹det 𝚺º 12
N ¹ r j r 0 𝚺º ei k^0 r^
𝛷r 0 ¹ k k 0 º = h k j𝛼i = 1 23 π^32 ¹det 𝚺 ^1 º 12
N
k j k 0 𝚺 ^1
ei¹ k k^0 º r^0 (13)
where 𝚺 is the spatial covariance matrix.
Theorem 6. A QCurve with an initial coherent state (13) and evolving according to (4) is in I_._
Proof. To describe the evolution of the initial states (13), we apply (10). Then, after applying Lemma 1,
𝜌 rS ¹ r 𝑡º = (^) 𝑍^12 2
N ¹ r j 0 𝚺 ¸ i𝑡 H ¹ k 0 ºº N ¹ r j 0 𝚺 i𝑡 H ¹ k 0 ºº = N
r j 0 12 𝚺¹𝑡º
𝜌S k ¹ k 𝑡º = N
k j 0 𝚺 ^1
where 𝚺¹𝑡º = 𝚺 ¸ 𝑡^2 H ¹ k 0 º𝚺 ^1 H ¹ k 0 º. Then
S = Sr ¸ Sk =
N
r j 0 12 𝚺¹𝑡º
ln N
r j 0 12 𝚺¹𝑡º
d^3 r
N
k j 0 𝚺 ^1
ln N
k j 0 2 𝚺 ^1
d^3 k = 3 ¹ 1 ¸ ln πº ¸ 12 ln det
I ¸ 𝑡^2 ¹𝚺 ^1 H ¹ k 0 ºº^2
As det I ¸ 𝑡^2 ¹𝚺 ^1 H ¹ k 0 ºº^2 ^ ¡ 0 , the entropy increases over time.
The theorem suggests that quantum physics has an inherent mechanism to in- crease entropy for free particles, due to the spatial dispersion property of the Hamil- tonian. Note that at 𝑡 = 0 a coherent state (13) reaches the minimum possible
Theorem 7 (Time Reflection). Consider a CPT invariant quantum field theory (QFT) with energy conservation, such as Standard Model or Wightman axiomatic QFT [13]. Let 𝑒 0 = ¹𝜓¹ r 0 º 𝑈 ¹𝑡º » 0 δ𝑡¼º be a QCurve solution to such QFT. Then, 𝑒 1 = 𝑄CPTδ ¹𝑒 0 º is (i) a solution to such QFT, (ii) if 𝑒 0 is in C , D , O , I then 𝑒 1 is respectively in C , I , O , D , making C , I , O , D reflections of C , D , O , I , respectively.
Proof. Let 𝑡^0 = 𝑡 ¸ δ𝑡. The QCurve 𝑒 1 describes the evolution 𝜓CPTδ^ ¹ r 𝑡^0 º during the period » 0 δ𝑡¼. Since 𝑒 0 is a solution to a QFT that is CPT-invariant and time-translation invari- ant, 𝑒 1 is also a solution to the QFT, proving (i). The time evolution of 𝜓CPTδ^ ¹ r 0 º from 0 to δ𝑡 is described by 𝜓CPTδ^ ¹ r 𝑡^0 º, and by (15) 𝜓CPTδ^ ¹ r 𝑡^0 º = 𝜂 𝛾^5 ¹𝛹 yºT^ ¹ r 𝑡^0 ¸ δ𝑡º = 𝜂 𝛾^5 𝛹 ^ ¹ r δ𝑡 𝑡^0 º. Thus, the evolution of 𝜓CPTδ^ ¹ r 𝑡^0 º as 𝑡^0 evolves from 0 to δ𝑡, by Theorem 3, has the same entropies as 𝜓¹ r δ𝑡 𝑡^0 º. Since 𝜓¹ r δ𝑡 𝑡^0 º traverses the same path as 𝜓¹ r 𝑡^0 º but in the opposite time direction, we conclude that 𝑒 1 produces the time evolution states 𝜓CPTδ^ ¹ r 𝑡^0 º in the time interval » 0 δ𝑡¼ traversing the same path and with the same entropies as 𝜓¹ r 𝑡^0 º, but in the opposite time directions. Applying the above to a QCurve respectively in I, D, C, O, results in a QCurve respectively in D, I, C, O. Thus, we conclude the proof of (ii).
For a visualization see Figure 1, which we repeat here from [9] for the reader’s convenience.
Entropy Oscillations
Theorem 8 (Coefficients for two states). Consider a particle in an eigenstate 𝜓𝐸 1 of a Hamiltonian 𝐻 that has only two eigenstates 𝜓𝐸 1 and 𝜓𝐸 2 with eigenvalues 𝐸 1 = ℏ𝜔 1 and 𝐸 2 = ℏ𝜔 2 , respectively. Let this particle interact with an external field (such as the impact of a Gauge Field), requiring an additional Hamiltonian
, ,
D
I ^
(^) , ,
Figure 1. A visualization of the Time Reflection Theorem. (i) Axis 𝑡: A QCurve 𝑒 1 = 𝜓 0 ¹ r º e i𝐻 𝑡^ δ𝑡. (ii) Axis 𝑡^0 = δ𝑡 𝑡: The antiparticle QCurve is created as 𝑒 2 = 𝑄CPTδ ¹𝑒 1 º = 𝜓CPTδ^ ¹ r 𝑡^0 = 0 º e i𝐻 𝑡^0 δ𝑡. Axis 𝑡^0 shows the evolution as going forward in time 𝑡^0. The evolution of 𝜓CPTδ^ ¹ r 𝑡^0 º = 𝜂𝛾^5 ¹𝛹 yºT^ ¹ r δ𝑡 𝑡^0 º is mirroring the evolution of 𝜓¹ r 𝑡º, with 𝑡 = 𝑡^0 evolving from 0 to δ𝑡. If 𝑒 1 2 D, then 𝑒 2 2 I.
term 𝐻I^ to describe the evolution of this system.
Let 𝜔I 𝑖 𝑗 = (^1) ℏ 𝜓𝐸𝑖 𝐻I^ 𝜓𝐸 (^) 𝑗 , 𝜔total 1 = 𝜔 1 ¸ 𝜔I 11 , 𝜔total 2 = 𝜔 2 ¸ 𝜔I 22 ,
𝜂 =
𝜔total 1 𝜔total 2
¸ 4 ¹𝜔I 12 º^2 , and 𝜆 = 𝜔total^1 ¸𝜔 2 total^2 𝜂. The probability of the particle to be in state 𝜓𝐸 2 at time 𝑡 is
4 ¹𝜔I 12 º^2 𝜂^2 sin
2 ¹𝜆¸^ ^ 𝜆 º𝑡
Proof. The Hamiltonians in the basis 𝜓𝐸 1 𝜓𝐸 2 are
and 𝐻I^ = ℏ © «
𝜔I 11 𝜔I 12
𝜔I 12 𝜔I 22
where the real values satisfy 𝜔I 21 = 𝜔I 12 as 𝐻I^ is Hermitian. The eigenvalues of the
golden rule [4, 6]. In [7] we showed that when the probability of two states oscillates as above, the entropy oscillates too. The derivation of 𝛼 2 ¹𝑡º can be extended to multiple states. However, for multiple states, the sum over all the frequencies 𝜆𝑘 𝜆𝑖 may cancel the oscillations unless some frequencies dominate the sum, such as when the transition to the ground state dominates other transitions. Thus, to obtain the entropy oscillation in the presence of multiple transitions may require approximations similar to the ones that are usually used in derivations of Fermi’s golden rule.
An Entropy Law and a Time Arrow
In classical statistical mechanics, the entropy provides a time arrow through the second law of thermodynamics [2]. We have shown that due to the dispersion property of the fermions Hamiltonian some states in quantum mechanics, such as coherent states, already obey such a law. However, current quantum physics is described as time reversible. In [7] we conjectured the following
Law (The Entropy Law). The entropy of a quantum system is an increasing function of time.
The law may help explain why particles are created and/or annihilated in scenarios such as high-speed collision e¸^ ¸ e ^! 2 γ, kaons decay into mesons, and photon creation and emission when the electron in the hydrogen atom transitions from an excited state to the ground state. In those scenarios, while such final states are reachable in a unitary evolution of the initial state, it seems that only those evolutions in which entropy increase are realized. According to the S-matrix formulation [12], similar to Fermi’s golden rule in QM, these final states are among the possible transition states. We note that similarly to Fermi’s golden rule, these are also entropy
oscillation scenarios in which the evolution is blocked from entering a time interval of decreasing entropy. The creation and/or annihilation of a particles seem to occur when the entropy of the evolution from the initial to the final state is oscillating, and but for such events the entropy would decrease, which the conjectured law forbids.
Furthermore, the spin-entropy evolution of system of particles or fields is also subject to this law, which may have implications in all physical scenarios including quantum information and quantum computing.
CONCLUSIONS
The concepts of entropy in quantum phase spaces in [7] were further developed here. We extended the coordinate-entropy in QM to multiple particles. We proved that the coordinate-entropy is invariant under coordinate transformations, Lorentz transformations, and CPT transformations. We analyzed the entropy evolution of coherent states, showing that the Dirac’s Hamiltonian has a mechanism to disperse the information and to increase entropy. We proved that Time Reflection transforms QCurves in C, I, O, D into QCurves in C, D, O, I, respectively. We proved that for a two-state Hamiltonian, the addition of a Hamiltonian term not only causes a state os- cillation (as suggested by Fermi’s golden rule when the appropriate approximations hold) but also causes entropy oscillation. In light of the technical advancements here, we reviewed the conjectured entropy law [7]. According to that law, not only a time arrow would emerge, but should the formation of new particles be triggered by the entropy law, the history of the universe would have to be revised through such a lens. Perhaps, the collapse of a wave function occurs not due to measurements, but instead due to the restrictions posed by the entropy law.
