a classical realization of quantum mechanics

Foundations o f Physics, Vol. 8, Nos. 5/6, 1978

A Classical Realization of Quantum Mechanics

M a r k D a v i d s o n 1

Received May 26, 1977

A mechanism is presented by which a classical system couM be described by the laws of quantum theory. Conflict with yon Neumann's no-go theorem is avoided. Experimental predictions are made.

1. I N T R O D U C T I O N

In this paper a mechanism is presented by which a classical mechanical system could be described by the laws of quantum mechanics in an appropriate limit. I t avoids most of the objections leveled against hidden variable theories, as discussed, for example, by Jammer. (1~ I t does not rely on hidden variables to give a classical system a noncausal appearance. Rather, it is proposed that in the limit of infinite energy (or energy density for a field) the causal deterministic nature of classical mechanics is lost. This is because coordinates and momenta cease to exist as mathematical functions of time. The methods used here to study this limit have their roots in a school of thought led by Born, which took the position that classical mechanics has as high a degree of indeterminacy as quantum mechanics due to the unstable nature of many classical systems and the inevitable lack of complete knowledge about initial conditions, c2-4) I t was proposed that this essential indeterminacy be incorporated into the formalism of classical mechanics.

In this paper, a temporal resolution is associated with each dynamic variable f rom the start. Statistical ensemble averages of these variables are considered. When the ensemble energies are taken to infinity, new pheno- mena can be described with this approach which cannot be described by traditional statistical mechanics, and which cannot occur at finite energies. In particular, in this limit, dispersion-free ensembles in the sense of yon Neumann ~ need not exist. As a consequence, the theorem of

1 Lawrence Berkeley Laboratory.

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0015-9018178106000481505,0010 © 1978 Plenum Publishing Corporation

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yon Neumann regarding the incompatibility of dispersion-free ensembles with quantum mechanics does not speak against the theory presented here.

In the model presented, uncertainty bounds are satisfied for coordinates and momenta. Their origins are similar to the conceptualizations of Bohm, (6) who argued that the uncertainty rules may originate from subquantum classical fluctuations. When the resolution times for coordinates and momenta are taken to zero in an ensemble average, the result may depend on the manner in which this process is carried out. In particular, it may depend on the order in which different resolution times are taken to zero. I f this occurs, it can be interpreted that certain variables cannot be defined (or measured) simultaneously with infinite precision. The "subquantum" fluctuations correspond to the high-frequency oscillations of coordinates and momenta. The resulting theory is noncommutative. After constructing a vector space with a Hermitian inner product, classical variables become noncommuting operators on this space. These operators are postulated to satisfy the canonical quantization rules. Considerable effort failed to provide a rigorous derivation of these, but plausibility arguments support this postulate.

The resulting theory appears to be equivalent to quantum mechanics. It makes new predictions about classical systems. These are asymptotic relations to be satisfied by classical correlations, and they can be used to look for examples of this theory. A classical fluid in homogeneous turbulence is discussed as a potential example. Other candidates are mentioned.

2. THE MODEL

Consider a classical system of coordinates and momenturn qi, P~: (i = 1 ..... N). Treat the motion statistically. Let f denote an ensemble average of a function f of the p's and q's. Imagine that the energy of the ensemble is varied to oe. The p's and q's will generally become ill-defined functions of time in this limit. The softened functions

q~(~-, t) = (I/T) dt' qi(t T t') (1)

p~(% t) = (1/~-) j d t 'p i ( t + t') (2) 0

will be less singular. It shall be assumed that these functions exist even in the limit E -+ oe. The following notation will be used:

x~(-c, t) - = q,(r, t) (3)

xN+i('c, t) = pi('r, t) (4)

A Classical Realization of Quantum Mechanics 483

Ensemble averages for the x~ can be formed:

x~(~-~, t) ... x~-.~0~, t)

After averaging, the -r~ can be taken to zero. In general, it is expected that the order in which the ~-'s -~ 0 will affect the result, because the energy has been taken to ~ . This possibility will be allowed for. Denote.

x h ( t ) "'" x j~( t ) ---- lim xh ( r l , t) ---x~(%~, t) (5) 7t/~'t+l-~O

where there is an order to the correlation on the left which may be interpreted as the parameter -r for the leftmost term being taken to zero first, followed by the ~- for the next term to the right, etc. Thus in general

x ~ ( t ) ... x~o(t) ~ P (x j l ( t ) ... x ~ ( t ) ) (6)

where P denotes an arbitrary permutation of the order. None of the laws of classical mechanics, to the author's knowledge, prevent this noncommutative circumstance from occurring. Note that the expectations in (6) are real numbers. At this point, as a matter of convenience, define the following complex correlations (dropping the time argument)

8i~r /4 c--i~r /~ <0 1 x 6 "'" xj.~ t 0> = ~/---~ x~. "'" x~,~ -t- - - ~ xs "'" x~ 1 (7)

t f we extend this definition by the rules

<0 I(A + B)I 0> = <0 [A t0> + <0I B I0> (8)

(0 I CoA i 0> = Co<0 I A 10> (9)

with Co a number, then correlations of the form

2N 2N

m=l ~i=l jm=i

are determined in terms of the real expectations in Eq. (5). Equation (5) may be~inverted to yield

x h ... x~,, = (Re -k Im)<0 [ x h --- xj~ I 0) (11)

At this point a vector space can be formed by letting an arbitrary vector be denoted by

If> = f l 0> (t2)

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where f i s any series of products of the x~. as in (10). Since such series form a ring, we define addition of two vectors by

I f ) + t g> = ( f - t - g)! 0> (13)

Multiplication by a series can be thought of as an operator on this space with the definition

g If) =gf[ O> (14)

where multiplication of two series is defined in the usual way with the order of the xj preserved in reducing the product to a sum. The adjoint of such operators may be defined by

Jl=l Jm=l

2N 2N

~=I Jl=l Jm=l

With this definition, we have

x / - - - xj (16)

so that the q's and p's are Hermitian operators. We may define a Hermitian product on this space by

<f[g>= <0 [ f ig [ 0> (17)

which is determined in terms of the real expectations in (5). The following results follow from these definitions:

(010> = 1 (18)

<0 [/{ 0>* = <0 l/~ [ 0> (19)

Thus we see that as a result of the order which can occur in a classical system in the limit of infinite energy as defined in (5), it is possible to form. a vector space with a Hermitian inner product and operators on this space. I f f a n d g are two operators, then in general

If, g] # 0 (20)

because of Eq. (6). The author is convinced that no theorems concerning classical mechanics can rule this possibility out.

In order to proceed, more precise knowledge about the algebra of operators on this space is required. So far, no real assumptions have been


made in this paper. A more general situation than is usually considered in classical systems has been allowed for. In order to proceed, the following postulates shall be made.

1. The operators on the vector space above satisfy the algebra of quantum mechanics.

2. I f the operators xjl ..- xj,~ are mutually commuting, then

x h . . . . xj,~ lim~0 xj~('r~) "" x~,~('r,~) (21)

irrespective of the order that the ~'i --~ 0 (including the possibility that they approach zero simultaneously).

It is possible to argue plausibly for the first postulate by paralleling the arguments for the canonical quantization prescription as presented, for example, by Dirac. ~7) The justification for this postdate is essentially that the canonical quantization rules offer the only known consistent nontrivial solution to the problem which we have. The second postulate is reasonable and helps to ensure that our vector space is in fact a Hilbert space.

Thus we have (for equal times)

[qi, qj] ---= [p~, pj] = 0 (22)

[q~, pj] = iJ3ij, J a number (23)

It follows from Eq. (21) and the quantization rules that

[ f(q)j2 = (Orf*(q)f(q)EO) >~ 0 (24)

In order to ensure that the vector space is a Hilbert space, it is necessary finally to assume that equality holds in (24) only i f f = 0, otherwise there are null states. It follows from the above assumptions that for any state I ~b)

<~I ~) > 0 (25)

so that the space is indeed a Hilbert space. At this point, the present theory is mathematically identical to quantum

theory. The state l 0) is not meant to be the ground state. J is an undeter- mined constant of angular momentum. The interpretation of the present theory is similar to quantum mechanics. The following results follow easily:

(01 O(q) l O) .... ig(q) (26)

(O ] O(p) l O) .... O(p) (27)

so that expectations of functions of the q's only or the p's only are equal to ensemble averages. In general, however,

(O I O(p, q) 10) -? O(p, q) (28)

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(recall that order matters for both sides of this inequality), so that in general the ensemble average of a function of the p's and q's is not equal to the expectation of that function. By using the commutation rules, an arbitrary operator O can be written in the form

2N 2N

O(p, q) = ~ ~ ... ~ a,,wl ..... ~-,,~[xh ." xj,, ' -- x~, . ' . xh] (29) ~)~=1 o41=1 9m=l

Rewriting O this way will not change (0 ! O 10), but may change O. When the operator is Hermitian, the a's are real. When O is written in this form, it follows that

(01 O I0) = 0 (30)

so that the quantum expectation is equal to a suitably defined ensemble average. Because of this, it is the author's opinion that this theory is equivalent to quantum theory in statistical assertion.

The arguments of von Neumann ~5) regarding the difficulties with hidden-variable descriptions of quantum mechanics do not rule out these ideas. First of all, the ensemble described by the state ] 0) is not dispersion free. No conflict with von Neumann here. Any limiting ensemble to which our theory applies would have dispersion. There are, of course, ensembles that are dispersion free. These are the usual classical trajectories, at finite energy. However, our theory does not apply to these. Their behavior is governed by the usual laws of classical mechanics. Once again no conflict. All ensembles to which our theory applies satisfy the uncertainty principle

Aq~Ap~ >~ ½1J] ~,j (31)

These uncertainty rules are brought about by the high-frequency fluctuations o f p and q, and not by the existence of hidden variables which are averaged over. Critics of this paper will have to go beyond the arguments of von Neumann and existing similar works to rule out this theory, as our ideas are completely consistent with these. These uncertainties may very crudely be pictured in the following way. Consider the time interval t to t + A t. In this interval, q~ and p~ must fluctuate so violently as to essentially fill up an area in the q-p plane equal to or greater than J, and no matter how small At is. This singular behavior is possible only in a limiting description as has been used here.

In ordinary quantum mechanics, the ordering of operators in a quantum expectation has no physical interpretation. It is part of the mathematical formalism which ultimately leads to statistical predictions. In this paper, this ordering has a precise physical interpretation. It corresponds to different ways of taking the zero-7 limit. Although in the ultimate limit, these two


theories appear identical, there are tests which may veri~ the claims made here, and these will be discussed in the next section.

Consider systems whose Hamiltonian is of the form

H ( p , q ) = t - I i ( p ) + H 2 ( q ) (32)

Then for these

(0 [HI 0) = g (33)

so that the ensemble average of H is equal to the "quantum" expectation. Our postulates are possible only if

~7 = ~ (34)

Thus we must restrict application of this theory to quantum states with infinite energy. Field theories, when quantized, always have infinite energies. Moreover, the only consistent interpretation of nature at present is in terms of quantized fields. It is natural, therefore, to look for examples of this type of behavior in classical field theories taken to a limit of infinite energy. This shall be discussed, for one special case, in the next section.

At this point it would be possible to examine time deppendence of operators, or to introduce Schr6dinger's equation or any other formalism that is used in quantum mechanics, as the present theory is mathematically identical to it. Besides (11), (26), (27), and (30), the following results for ensemble averages follow:

q i P j - - P j q i = 8 i J (35)

qiqJ - - q~q~ = P ~ P 5 - - P ~ P ~ = 0 (36)

x h "'" x h ( q i q J - - q jq i ) xh+~ "'" x~,~ = 0 (37)

with a similar result for commutators ofpi and p j . Another result is

x h "'" x h ( q l P J - - Psq i ) x~+l "'" x~ m : J.x'j,~ "" x h ~i , j (38)

(note the reversal of order). Numerous other relations can be derived. In the next section methods of testing these for suitable classical systems will be described.

3. E X P E R I M E N T A L I M P L I C A T I O N S

Fields always have infinite energy density when quantized. This infinity comes from violent spatial and temporal fluctuations. The present theory is most naturally applied to fields. Although a discrete index was given to the

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p's and q's no special problems arise in applying the theory to continuous systems. To look for examples, experiments must be performed on classical systems with large energy densities, in which case the relations for ensemble averages would only be approximate, but should become exact as the energy density increases. The limit must be carried out in such a way that a finite parameter J, with dimensions of angular momentum, is still obtainable (dimensionally) from the parameters of the motion. The motion of the fields must also be highly turbulent in this limit.

Before looking at special cases, let us consider the general question of experimental verification. Let qi(t) and p,(t) again denote classical coordinates and momenta. For finite energy density, these functions will have an approximate high-frequency cuttoff, which we denote by COH. The cutoff COH will tend to infinity as the energy density increases. There may also be a low- frequency cutoff oJr. The theory of the previous section demands that the following relations become more accurate as energy density (and ~OH) approaches infinity:

xh(r l , t) "'" x~ ( r~ , t) ~ (Re + Im)(0 / x jl(t) -.- x~,(t) ', 05

1/'r, >~ 1/r,+,, oJu > 1/ri >~ coL, all i (39)

The right-hand side is a quantum expectation with h replaced by a new parameter aT. The left-hand side is defined for a classical system in the previous section. The values of 1/r~ must be taken well within the limits (wL, wu) because in our previous discussions we took the limit E -~ oo (and therefore co n --+ m) before taking the limit r~ -+ 0. The state I 0) is arbitrary, and will depend on initial conditions. It may not be easy to find. We can expect best agreement for expectations with the fewest xji since the inequalities are easier to satisfy.

The following relations follow, which do not depend on the state [ 0): For r , ~ r2, WH>~ 1/r,>~ ~OL, we have

q,(r 1 , t )p j (%, t) -- q;(%, t ) p j ( r , , t ) ~ JS,,j (40)

qi(r l , t) q,(%, t) -- q,(%, t) qj( 'q, t) ~ 0 (41)

Pi(%, t)ps(r2, t) - -P i (%, t )p j (%, t) ~ 0 (42)

Numerous others can be derived using Eqs. (37) and (38), with suitable inequalities imposed on the r's. These relations constitute scaling laws which would become exact in the asymptotic limit. J is still an unknown parameter which will depend on initial conditions. Once it is determined from, say, Eq. (40), the other relations involving J can be tested. The value of J that best satisfies these relations may change as the limit is taken.


The ideas of this paper might apply to a turbulent classical fluid. The motion of the fluid should have the following properties:

1. Viscous dissipation is ignorable.

2. Energy density is large.

3. The motion is highly turbulent.

4. A finite constant J, with dimensions of angular momentum can be formed from the relevant parameters of a fluid.

Such a situation as this is possible for classical fluids. Let V(x, t) be the velocity field of the fluid. Define

V(k, t) = f dax [exp(ik • x)] V(x, t) (43)

The kinetic energy can be written

E f dax 21-pV2(x)= f dale" 1 pV2(k) (44) (27r) a 2

The inertial range in k space for homogeneous turbulence was first examined by Kolmogorov. (a,u) In this range viscous dissipation is unimportant. Kolmogorov's similarity hypothesis allows only three nongeometric parameters to affect expectations. These are e, the energy per unit time per unit mass added to the fluid by stirring; v, the kinematic viscosity; and p, the mass density. Whether this assumption is exact or not, it indicates a limiting process for the fluid in which our theory might be realized. Define (V is volume)

1 E(k) -- 4rrz V kW~(k) (45)

so that E(k) dk is the energy per unit mass contained in the interval dk. With the similarity hypothesis, dimensional analysis yields

E(k) = e2/ak-5/af(kva/4/e 1/4) (46)

The function f is assumed to be universal and independent of the fluid in question. I f we pass to the limit va/4/d/4 -+ 0, and if f(0) is finite, then we find

E(k) = e2/ak-a/3f(O) (47)

The total energy/mass in the inertial range (Ex) is then approximately

E1 ~ ~E Kmmjtv) (48)

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EI approaches infinity as e2/~ for increasing e. A constant of angular momentum is

Jo = P¢3/4] E~/~ (49)

Constants with dimensions of length and time are

length = l == ],'3/4/el/4 (50)

time = r = (v ie) I/~ (5t)

These can be reexpressed in terms of J0

~. = (]olp)~/~ ~-5/1~, / = ( ] o l p ) ~ / ~ ~-~/t3 (52)

Our limiting theory could be realized as e --~ ~ with J0 fixed. This can be achieved by varying v and p in (49) to hold J0 constant as e --~ ze. In this limit, 1 --> 0 and r -,- O. We must have

k m a x ~- a/ l (53)

c o . = b / ~ (54)

col = ckmin l /7 (55)

for constants a, b, and c. We see that wH - - COL ~ o~ as e --~ 0% if p and oro are fixed. For small kmln,

co~ - - col ~ ea/13(P/Jo)m~ (56)

I f a small rigid body is immersed in this fluid, it will become mechanically coupled to it. The coordinates and momentum of this body, q / t ) and px( t ) ,

should satisfy our statistical relations if our theory applies to the fluid. Equations (40)-(42) should be satisfied by these coordinates with coil and col given in Eqs. (54) and (55). It is clear that no problems in principle prevent these relations from being tested. If successful, then other, more complicated relations can be derived and tested. To the author's knowledge, such experiments have not been carried out.

The parameter J in the previous section must, on dimensional grounds, be proportional to J0 for the fluid

J = aso (57)

where • is a dimensionless constant. In keeping with the spirit of Kolmogorov, should be a universal constant, independent of the fluid type. These experi-

ments would be interesting regardless of whether our theory describes them. In particular, if the left-hand side of Eq. (40) is found to be nonvanishing, as

A Classical Realization of Qum~tum NIechanics 491

one would expect, since there is no reason for it to vanish, then some sort of noncommutative theory will have to be developed to describe this pheno- mena. The one presented here is probably the simplest possibility.

Other systems could be considered. A turbulent plasma has many similarities to a turbulent fluid. ~I°~ Some relativistic field theories can be expected to have turbulence-like behavior. For example, the Lagrangian

f L -~ d3x [~d,~h d"~ -- g44] (58)

with initial conditions on

T~=0 = C, a constant independent of x (59)

leads to solutions which are unstable. Small spatial inhomogeneities at t = 0 can be shown to grow with time, leading to a situation similar to turbulence. Solutions to this type of problem could be attempted on a computer and the statistical predictions of this paper could be tested.

Unfortunately, the author has not been able to derive sufficient conditions for the postulates of Section 2 to be satisfied. They might be satisfied by only certain Lagrangians. It is the author's hope that the turbulent fluid will confirm the theory and stimulate further investigation. If the fluid investigations are negative, however, this will not eliminate the possibility that other systems may have the postulated properties in the appropriate limit.

4. CONCLUSION

Quantum mechanics may emerge from a classical theory in the explicit way which has been illustrated here. This is possible because of the singular behavior of classical mechanics in the limit of infinite energy. The uncertainty principle would apply for ensembles in this limit with h replaced by a different constant of angular momentum characteristics of the system. The laws of quantum mechanics would determine suitably defined ensemble averages.

Experimental predictions are made for systems described by this theory. The classical fluid is the most promising possibility. Relativistic field theories could be investigated on computers. The experiments mentioned for fluids would be interesting independent of this paper, but have not to the author's knowledge been performed.

No explicit assumptions were made in this paper about the form of the Lagrangian for the classical system. The ideas here can be generalized to include fermion fields, Thus, the theory is su~ciently versatile to describe nature.

82518/5[6-r2

492 Davidson

Quantum mechanics could be an example of this theory. The fields of nature might be essentially classical, but their motion turbulent such that the ideas of Section 2 apply with J = h. A unification of classical and quantum physics could thus occur which goes far beyond the usual correspondence principle.

I f classical systems could be found that have the properties discussed here, then studying them would certainly lead to a deeper understanding of quantum theory and therefore nature.

Implicit in the model presented here is the nonexistence of a joint probability distribution for position and momentum; otherwise the singu- larity discussed in Section 2 could not occur. Cohen m~ has proved that such a joint distribution cannot reasonably exist if quantum mechanics is exactly correct, and this theorem has been leveled against stochastic interpretations of quantum mechanics. The model presented here, although stochastic in nature, does not suffer from this difficulty.

REFERENCES

1. M. Jammer, The Philosophy of Quantum Mechanics (WiJey. New York, 1974). 2. K. Popper, Br. Phil. Sci. 1, 117-133, 173-195 (1950). 3. M. Born, Math.-Fys. Medd. 30, 1-26 (1955). 4. L. Brillouin, Science and Information Theory (Academic Press, New York, 1956). 5. J. yon Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univer-

sity Press, 1955). 6. D. Bohm, in Observation and Interpretation in the Philosophy of Physics--With Special

Reference to Quantum Mechanics, S. K~Srner and M. H. L. Pryce, eds. (Dover, New York, 1962).

7. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford at the Clarendon Press, 1958).

8. A. N. Kolmogorov, Compt. Rend. Acad. Sci. URSS 30, 301 (1941); 32, 16 (1944). 9. G. K. Bateheler, Theory of Homogeneous Turbulence (Cambridge University Press,

Cambridge, 1953). 10. T. D. Lee, Quart. AppL Math. 10, 69 (1952). 11. L. Cohen, Thesis, Yale Univ. (1966) (available from University Microfilm, Ann Arbor,

Michigan).

a classical realization of quantum mechanics

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