Submitted to Physical Review Letters, Oct 1996 (Revised, Feb, 1997)
Quantum Theory and Irreversibility Huaiyu Zhu
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (March 11, 1997) The standard quantum theory is mathematically incompatible with irreversible macroscopic theories. An alternative irreversible quantum theory is proposed, which also provides an explanation of measurement as a physical process. It is based on the postulate that physical world is governed by unitary evolution (wave propagation) punctuated by random discontinuity (particle interaction). The time, space and identities of particles are in superposition, while the types and numbers of interactions are in mixture. 03.65.Bz (found), 05.30.-d (q.sm), 12.20.-m (qed), 89.70.+c (info)
I. INTRODUCTION Currently all the major microscopic physical theories are time-reversible, yet the macroscopic world is irreversible. Many mathematical and physical theories were proposed to reconcile this apparent disharmony. Among them the most inuential are those using statistical ideas to show that microscopic reversible processes may produce irreversible macroscopic phenomena 1]. No such proofs can possibly be correct, as was recognized by Loschmidt and Burbury soon after Boltzmann's proof of his H -theorem 2]. In essence, these proofs fall into two categories: (1) \Reciprocity proof". They assume that the probability of state transition from n to m is the same as from m to n, and prove macroscopic irreversibility. However, such microscopic assumptions are \detailed balances" (symmetry among the states), not \reversibilities" (symmetry between forward and backward time). (2) \Reversibility proof". They assume a truly reversible dynamics, and prove that the present knowledge will decay in the future. However, such proofs also show that the present knowledge \will" decay in the past. They are therefore unrelated to the second law which is concerned with the disorder of the physical system itself: The maximal amount of information available in principle is larger in the past than in the future. Clearly, if all the premises apply to both forward and backward time, so must be all the conclusions. Under such circumstances the most that may be proved about a universal change of entropy is that it is a constant. This paradox is most prominently displayed in the theory of thermal radiation. Planck had the original idea that thermal radiation is the result of equilibrium, and by examining in detail entropy, energy and temperature of thermal radiation he was able to arrive at the Email:
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only energy distribution formula that ts experimental data. To justify his interpolated entropy relation he was forced to make the quantum assumption 3,2,1,4]. It will be shown (xII) that mathematical justication of entropy must always rely on a quantum assumption. Unfortunately, the standard quantum theory subsequently developed was still reversible, save for the measurement process, so the paradoxes became even more acute: (1) The measurement process is irreversible, so it could not, even in principle, be described as part of the physical world. (2) Because of the uncertainty principle the second law could not be attributed to the lack of precision in measurements. The purpose of this letter is to explore the idea that the diculties mentioned above may be overcome by a single postulate, that the random quantum jumps, hitherto conned to measurement alone, if admitted at all, are inherent in the physical world. A full paper giving more details is in preparation. Three views concerning quantum world may be summarized in Einstein's metaphor: (1) God does not play dice (2) God plays dice whenever someone is watching (3) God plays dice on his own accord. The third view is adopted here. Note that the controversy over objectivity versus subjectivity in quantum theory is less concerned with whether the world is deterministic ((1) versus (2) or (3)), than with whether the description of physical reality involves something (observer) which itself does not t in the same description ((1) or (3) versus (2)). The gap between (1) and (3) may be theoretically bridged by a suitable innite-sized random number generator.
II. INFORMATION AND ENTROPY Despite its prominence, entropy is not the only measure of information. Mathematically any member of the family of \information divergences" (-deviations), Z 1 1; (1) D (p q) = (1 ; ) 1 ; p q 2 0 1] is generally suitable as a measure of amount of information for discrimination between probability distributions p and q 5,6]. Among them the most important are the 1-deviation (Kullback-Leibler divergence or cross entropy) and the 1=2-deviation (Hellinger's distance), Z (2) D1 (p q) = K (p q) = p log pq Z D1=2 (p q) = 2 (pp ; pq)2 : (3) The KL divergence is characterized by its additivity for a system of several independent parts. Furthermore, if the state space is nite and q is the uniform distribution qi = 1=n, we recover entropy plus a constant Z K (p q) = log n + p log p (4) These two assumptions underly Shannon's entropy in communication theory. As emphasized by Jaynes 7], the formal resemblance between Shannon's and Boltzmann's entropy alone does not justify either of them as a correct measure of information in any given situation. 2
Generally for any dynamical process (deterministic or stochastic) which changes x into y, the amount of information (using any of the -deviations) in y cannot be greater than that in x It is invariant if and only if y is a sucient statistic of x 8]. In physics, this means in particular that any causal dynamics increases entropy unless it is reversible. This essentially limits the kinds of fundamental physical laws. For when faced with a choice between an irreversible world and a deterministic world, there is hardly any empirical evidence to favor the latter. Incidentally this also shows that to derive irreversibility from reversible mechanism based on \information leaking" requires the assumption that the observable part of the universe contains more information of the past while the unobserved universe contains more information of the future An assumption entirely equivalent to irreversibility. If the physical world were entirely described by amplitudes evolving under the action of unitary operators, the Hilbert norm (corresponding to the Hellinger distance) would act as a good measure of all the available information, as is apparently the case microscopically 9{11]. The universal and spontaneous increase of entropy in the real world however implies that there exist irreversible physical processes. The fact that only Boltzmann's entropy serves as a good measure of irreversibility indicates that it is the accumulated e ect of a large number of independent events. The fact that entropy is dened relative to phase space volume indicates that the size of such \elementary disorder" (Planck) is proportional to phase space volume. 1 Finally, Planck's formula for thermal radiation revealed the nite size of such \elementary quanta of action". Many mathematical theories can prove the quasiergodicity or the increase of algebraic entropy, based on the idea of \mixing": Two distinct trajectories may move closer together in the future. However, they are not relevant to the irreversible increase of physical entropy without the idea of \smearing": If two trajectories are suciently close in phase space they will be physically indistinguishable forever in the future even if they move apart later. The essential role of smearing is well illustrated by the di erence between folding a sheet of cloth and folding a blob of wet our. In the literature there are essentially three schemes to account for the universal increase of entropy: (1) the inertia of entropy (involving the Big Bang), (2) the collective entropy change (involving the inuence of the whole universe), (3) the local production of entropy. The last view is adopted here, with the specication that this local production of entropy happens exactly at quantum jumps. We shall not consider the view that uncertainty of measurement itself increases with time as this is clearly in conict with the constancy of Planck's constant. Obviously it is also unrelated to the approximation errors in our computations. It is worth mentioning here that two types of discreteness appear in quantum theory: The U -discreteness arises from the discrete spectrum, while the R-discreteness is the random quantum jumps considered in this paper. It is possible to construct mathematical models in which one type of discreteness exists while the other is absent (continuous light beams It may be innitesimal but can not be zero. It corresponds to assumptions such as \molecular disorder" (Boltzmann), \condition A" (Burbury) and \natural radiation" (Planck) 2]. 1
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striking at discrete spectral lines or discrete photons striking at a continuous spectrum). Therefore they cannot be logically explained by each other.
III. IRREVERSIBLE QUANTUM THEORY We must now nd a formalism which accommodates both irreversible random jumps and unitary dynamics, bearing in mind the following correspondence: Reversible: wave propagation unitary superposition Irreversible: particle interaction random mixture The standard Feynman diagram may be schematically written as X P (futurejpast) =
Z
interactions paths
2 amplitudes :
(5)
The sum over interactions corresponds to Feynman diagrams with di erent vertex structures. Our new formula is obtained by changing particle interactions into probability mixtures while keeping wave propagation as superpositions, 2 X Z 1 (6) P (futurejpast) = Z paths amplitudes interactions where Z is a normalization factor (partition function). Since identical particles should not be distinguished, the interchange of various paths should remain in superposition as long as the vertices remain the same. This is automatically satised when the occupation number representation is used. Formula (6) constitutes the core of our new theory, which will be called the \mixed theory", in contrast to the standard theory as a \unitary theory". As an illustration, consider the idealized case of a single particle in a constant external eld, where the Feynman diagrams only di er in the number of interactions. Denote the interaction Hamiltonian as V , the scattering matrix as S , the time ordering operator as T , and the wave function as . In the standard quantum electrodynamics, after transforming away the unperturbed Schr odinger propagator, Dyson's formula may be succinctly written as 12] 13, x73]
Z
S = T exp ;i V =
X k
Sk Sk = (;ki!)
k
Z
k
T (V k ):
(7)
The transitional probability from i to f is given as
2
Pf i = f Si = f Si i S f :
(8)
It is relativistically invariant in the sense of Tomonaga 14]. Our corresponding new formula is 4
2 X Pf i = Z1 f Sk i = Z1 f f k
=
Z=
X f
X k
Sk i i Sk
f f = tr = i
X k
!
Sk Sk i
(9) (10) (11)
where is the density matrix. It may appear that Pf i is still symmetric with respect to f and i, but it is not, due to the partition function Z . A transition amplitude is always reversible because it must be unitary, while a transition probability is in general irreversible.
IV. THEORETICAL IMPLICATIONS We outline several features of the mixed theory which also serve as critics of the unitary theory. (1) Our formula contains the partition function, which is of fundamental importance in statistical physics, but is entirely unjustiable in the standard quantum theory. (2) The density matrix now arises naturally, since even if the initial state is a pure state (rank one matrix), the nal state is in general a mixture (full rank matrix). In the unitary theory the density matrix is spurious. For if the initial state is given as a mixture of pure states, the nal state will be a similar mixture,
i =
X
piii =
X
pi = Si :
(12)
They are sucient statistics of each other, and the amount of information in the system is unchanged by either the forward or the backward dynamics. (3) Interactions are now physically meaningful, as they cannot be transformed away like the propagators of free particles. On the other hand, in the unitary theory, the interaction Hamiltonian plays the same role as that of a free particle, so that by a suitable unitary transform we could establish a coordinate system in which the whole universe appears static. Such a non-dissipative world would be silent, smooth, reversible, and transparent, like a condensate at its ground state, behaving like a giant superuid and superconductor. Nothing in it could possibly be regarded as an observer. (4) The clicks in a Geiger counter (likewise the fate of Schr odinger's cat) are in a mixture instead of superposition, because the density matrix does not contain cross terms between di erent interactions, unlike that of the unitary theory
=
X kl
Sk i i Sl :
(13)
(5) For very weak interactions, S0 S , Sk 0 for all k 1, the probability of interaction is small, and the dynamics is described by Schr odinger's equation for a free particle. 5
(6) If the number of interactions is large, it approaches the Poisson distribution with intensity given as in Dirac's theory 15], satisfying the usual rule for composition of probabilities. Z
P20 = P21 P10 : 1
(14)
The overall equation approaches a di usion equation (Kolmogorov's forward equation). It is therefore irreversible since Kolmogorov's backward equation describes \anti-di usion" which moves away from equilibrium. (7) If a system is close to equilibrium, its overall behavior can be represented, ignoring higher order terms, by a few \macroscopic variables". By denition they automatically satisfy the ergodicity hypothesis, the validity of which has been the major theoretical diculty of statistical mechanics. If the interaction between systems A and B is weak in the sense that A can be considered adiabatic for its inuence on B 16], it is called a \classical apparatus", and their interaction is called a \measurement" of A on B . If they are separated after the measurement the equilibrium of A will in general depend on the interaction, thereby containing a \record" of B . The equilibrium is generally given by the projection onto an eigenspace of a Hermitian operator, which explains the standard denition of measurement. The projection operator describes the result of the irreversible approach to equilibrium which is a many-to-one mapping. (8) The random e ect of A on B cannot be entirely removed from the record of B in A, which gives rise to \uncertainty of measurement", which always concerns the conditional distribution of future given the past but not vice versa. This also explains why it corresponds to non-commutability of operators. (9) Most phenomena treated by the standard theory are either single-interaction (such as two-slit experiment or atomic spectrum) or adiabatic (such as blackbody radiation or specic heat). For single-interaction phenomena the change from to jj2 is attributed to measurement. For adiabatic phenomena a short-cut is used by directly writing down the equilibrium distributions, such as Bose-Einstein or Fermi-Dirac statistics. The contradiction lies in the fact that with unitary dynamics the equilibrium cannot be reached automatically. Many interpretations of standard quantum theory (notably those under the name of many worlds/histories and decoherence) and the majority of applications actually yield irreversible results at some stage. So they cannot be approximations of (5). In fact Dirac's derivation of Einstein's coecients 17,15] is an approximation of (6), by which entropy increase at each absorption and emission of a photon 18]. (10) Because interchange of identical particles is summed in amplitudes while interchange of di erent particles is summed in probability, Gibbs' paradox 1] does not arise. The fact that a free particle is a collection of various excited states yet it is still to be described by the reversible Schr odinger's equation may be reconciled in two ways. The rst is simply to assume that virtual particles are in superposition rather than mixture, thereby retaining the standard theory in this case. The second choice is to consider the real particle as an equilibrium state of all the virtual particles, so that the Schr odinger's equation is only 6
an adiabatic approximation. Future study on this issue is likely to have implications on renormalization. (11) There is a possible implication on \quantum computation" 19]. If uncertainty is present in physical process itself, it will impose an upper limit of 2c2=h" = 1:7 1051=kg s of computational capacity for any machine, although this is still much higher than is achieved by current processor technology (less than 1015 =kg s). However, if uncertainty is only associated with the measurement process then the limit is only applicable to input-output processes. There exist many interesting computational problems whose exact solutions are exponentially more costly than input-output. For example, under our theory it is impossible in principle (based on the current understanding of NP -completeness) to build a 1kg computer which could exactly solve an arbitrary 100-city traveling salesman problem in 1s, while for the standard theory this is well within the theoretical limit. (12) It may be possible to directly test microscopic irreversibility. Construct a large cavity with highly reective walls. Suppose it has 3m diameter, with 99% reectivity. Then the equilibration time for radiation will be about T = 1s. Shine a bright beam of monochromatic light into this cavity for a small fraction of T . Wait for a chosen period t = kT , where k = 1 2 3 : : :, let out the radiation in a small opening for a small fraction of T , and measure its spectrum It should approach the equilibrium distribution with time t independent of the intermediate observations, provided that the opening is very small. (13) There are experiments which may determine whether particle interactions are mixtures or superpositions. Consider an experiment in which electron exhibit wave-like behavior, such as a di raction experiment. Suppose an electromagnetic eld is imposed in the path before the nal recording apparatus. Then electron-photon scattering will occur. According to our theory, this is to be described by mixture rather than superposition. Moreover, if the frequency of the intervening eld is very high, it becomes single electron-photon scattering. Such experiments therefore have the advantage of providing a bridge between microscopic and macroscopic phenomena. In case our current formulation turns out to be incorrect (its exact form is by no means certain at this stage), they will still provide clues leading to a correct irreversible quantum theory.
This work was motivated by 20]. The idea that entropy increases at each quantum jump occurred to me in 1992{93 while studying a type of neural networks with discrete random state jumps and continuously evolving potentials. I thank M. Gell-Mann, R. Rohwer, E. Van Nimwegen and E. Knill for helpful discussions. The actual formulation of the theory drew much inspiration from 3,2,1,4]. My work in SFI is sponsored by TXN, Inc.
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