The Classical Limit

4 The Classical Limit

..classical mechanics may be regarded as the limiting case of quantum mechanics when $ℏ$ tends to zero.
P. A. M. Dirac (1902-1984), The Principles of Quantum Mechanics, 4ed. p.88.

...they...speak about Newton's laws being an "approximation" for quantum mechanics, valid when the system size is large - even though no legitimate approximation scheme has ever been found.
Robert B. Laughlin (1950-), A Different Universe, p.31.

In this section the classical limit of QT will be studied. This point is very important for the interpretation of QT. The CI assumes that QT is a complete theory for individual particles. This implies that QT agrees in the limit $ℏ → 0$ , (a limit that may be defined in different ways) with classical mechanics. This limiting behavior plays therefore the role of a necessary condition for the validity of the CI. We discuss various meanings of the term ’classical limit’ and show in subsection 4.3 that this condition cannot be fulfilled.

Note that the limit of large particle number, or the ’macroscopic limit’, of QT is not studied here. We are interested in the relation between two physical theories, which can be characterized by two different basic equations⁵⁰ . The form of these equations depends on the number of particles. The single particle (ensemble) seems not only the simplest but also the most significant way to study this question .

The above quotations [41], [91] (excerpt), expressing diverging opinions of two nobelprice winners from different fields of physics, indicate that the question is a nontrivial one. Thus, some preliminary considerations may be useful. Let us start by considering two examples for pairs of classical theories where the relation between the ’covering theory’ and the ’limit theory’ can be said to be completely understood.

4.1 Two kinds of limit relations between classical theories

Take as a first example relativistic mechanics (RM) as covering theory and Newton’s mechanics (NM) as limiting theory. In RM a new physical constant, the speed of light $c$ , appears, which is absent (infinitely large) in NM. Otherwise, the structures of RM and NM are similar; the relation between these theories may be described as follows:

RM and NM have the same mathematical structure (ordinary differential equations for trajectories) but differ in detail because a new constant $c$ occurs in RM which is absent in NM.
The limit $1 ∕c → 0$ transforms the basic equations of RM in the basic equations of NM (similar remarks apply to the solutions of RM and NM respectively).
As a consequence of the last two properties, NM may be considered as ’limiting theory’ of RM (for $1 ∕c → 0$ ) or RM may be considered as a ’covering theory’ for NM.
In ’reality’ $1 ∕c$ is different from zero, thus RM is superior to NM (or NM is an approximation⁵¹ to RM).

Thus, the relation between theory RM and theory NM follows a simple hierarchical scheme. It is certainly justified to call RM the covering theory of NM (or NM the limit theory of RM) because the basic equations of RM go over to the basic equations of NM if $1 ∕c → 0$ . This is the essential point. As a consequence, the solutions of RM go over to the solutions of NM in this limit. We will refer to the kind of limit relation found in this first example as ’standard limit’.

Let us consider as a second example the relation between a probabilistic version of NM, referred to as PNM, and NM. Being basically interested in QT, we want to study probabilistic theories for single particles. Thus, let the basic equation of theory SP be a Liouville equation in six-dimensional phase space,
$∂-ρ-- -⃗p--∂-ρ-- ⃗ ∂-ρ-- + + F = 0. ∂ t m ∂ ⃗r ∂ ⃗p$ (6)

Here, $⃗r$ and $⃗p$ occur as independent variables in this partial differential equation for the probability density $ρ (r, p, t)$ . The concept of particle trajectories plays an essential role in this probabilistic theory, since (6) can be derived using the constancy of $ρ$ along trajectories and the validity of Newton’s equations in the following form (introducing the definition of momentum as a separate equation):
$⃗p ⃗r˙ = ----, ⃗p˙ = ⃗F . m$ (7)

An original study of various probabilistic theories of this kind has been reported by Bosanac (chapter 1 of [26]).

We called PNM the ’covering theory’ of NM, and NM the ’limit theory’ of PNM. But the relation between these two theories is very different from the corresponding relation in the first example. It can be characterized as follows:

No new constant appears in PNM. PNM’s mathematical structure (partial differential equations) differs radically from that of NM (ordinary differential equations). There is no obvious way (e. g. by performing a limiting process for a parameter as in the last example) to tranform the basic equations of PNM into the basic equations of NM. Thus, NM is not the limit of PNM as far as the form of the basic equations is concerned.
The physics of PNM is rather different from that of NM. For example, in PNM the selection of physically sensible initial conditions $ρ (r, p, t ) 0 0$ at $t = t 0$ becomes a nontrivial and interesting physical problem [26]. No such problem exists in NM, where arbitrary numbers $r0, p0$ can be assigned at $t = t0$ .
While NM is not the limit of PNM as far as the basic equations are concerned, one may allow for singular (delta-function like) initial values for $ρ$ . In this way PNM can be completed to formally include deterministic motion along trajectories. The thus completed theory $----------- P N M$ contains all solutions of NM. Thus, it can be considered in a different sense as a ’covering theory’ of NM (see however the next point).
A closer look shows, however, that this kind of covering relation is of a rather formal nature. Theory $----------- P N M$ is nothing but the union (or ’fusion’ if you prefer) of the purely probabilistic theory A and the deterministic theory NM. Theory PNM was originally constructed as a probabilistic version (for true random initial values only) of NM. If we extend PNM by allowing for deterministic initial conditions we construct the union $----------- P N M$ of the theories PNM and NM, and the statement that PNM is the covering theory of NM is formally correct but trivial - and more or less empty from a physical point. The conclusion is that a comparison of NM with $----------- P N M$ is of a somewhat formal nature⁵² .

Summarizing, the relation between the theories PNM and NM of our second example is more complex than in the first. Theory NM is not the limit of PNM but a formal covering relation (between $----------- P N M$ and NM) may nevertheless be established despite the fact that both theories belong to fundamental different cognitive categories. We will refer to the kind of limit relation found in this second example as ’deterministic limit’.

4.2 What is the ’standard’ limit $ℏ → 0$ of Schrödinger’s equation ?

Let us study the limit $ℏ → 0$ of QT in the light of the above two classical examples. For our problem its simplest form,
$3 ( )2 ℏ ∂ 1 ∑ ℏ ∂ -------ψ + ------ --------- ψ + V ψ = 0, i ∂ t 2m i ∂ x k=1 k$ (8)

describing a single particle (for followers of the CI) - or a one-particle ensemble (for followers of the SI) - without spin in an external mechanical potential, will be sufficient. The corresponding set of equations of Newtonian mechanics is given by
$d p (t ) d dV (x ) ---x (t ) = ------, ----p (t) = - ----------, dt m dt dx$ (9)

Is it possible to derive the ordinary differential equations (9) from the partial differential equation (8) by means of a limiting process $ℏ → 0$ ? We see immediately that the relation between (8) as virtual ’covering theory’ QT and (9) as virtual ’limit theory’ NM is (if it exists) more complex than in each one of the above classical examples. In fact, the present relation shows characteristic features from both previous relations: QT contains not only a new constant - namely $ℏ$ - which is absent in NM (as in the first example) but the mathematical structures of QT and NM are also different (as in the second example). We expect that a two-step process - if any - is required to manage the transition from QT to NM: one has to perform the limit $ℏ → 0$ of (8) (in analogy to the first example) and at the same time specify singular initial values (in analogy to the second example) to ’transform’ a partial differential equation into an ordinary differential equation.

Let us start with the limit $ℏ → 0$ of Eq. (8); this limiting procedure will obviously not change the character of the resulting relation(s) as partial differential equation(s). Use of the relation (8), for the complex variable $ψ$ , does not lead to a sensible result for the limit $ℏ → 0$ . However, if $ψ$ is replaced by two real variables $ρ$ and $S$ , defined by
$√ -- S- ψ = ρei ℏ ,$ (10)

then (8) may be replaced by two coupled nonlinear differential equations for $ρ$ and $S$ . The first is a continuity equation
$∂--ρ- --∂----ρ---∂-S--- + = 0, ∂ t ∂ xk m ∂ xk$ (11)

which does not depend on $ℏ$ . The second depends on $ℏ$ and is given by
$( )2 √ -- ∂ S 1 ∑ ∂ S ℏ2 △ ρ ----- + ------ ------- + V - -------√------- = 0. ∂ t 2m ∂ x 2m ρ k k$ (12)

Eqs. (11) and (12) have been derived by Madelung [94] and are sometimes referred to as hydrodynamical formulation of QT. This nomenclature may be misleading because the physical content of (11) and (12) on the one hand and Schrödinger’s equation (8) on the other hand is identical; both formulations differ only by a variable transformation⁵³

Using (11) and (12) as basic equations the limit $ℏ → 0$ may now easily be performed; the continuity equation remains unchanged while (12) takes the form
$( )2 ∂ S 1 ∑ ∂ S ----- + ------ ------- + V = 0, ∂ t 2m ∂ xk k$ (13)

i.e. the last term in (12) has simply to be omitted . Thus the classical limit $ℏ → 0$ of Schrödinger’s equation (as expressed in terms of $ρ$ and $S$ ⁵⁴ is given by the two differential equations (11) and (13).

The above simple steps are analogous to the first example in subsection 4.1 insofar as the vanishing of the parameter $ℏ$ does not change the mathematical structure of the covering theory QT; both QT and the $ℏ → 0$ limit of quantum theory (LQT) - which is sometimes described more precisely as ’probabilistic Hamilton-Jacobi theory’ - are field theoretic initial value problems. From a physical point of view both QT and LQT are probabilistic theories, as shown by the presence of the continuity equation⁵⁵ .

The classical limit of QT is obviously quite different from classical mechanics (compare the pair of equations (11), (13) with Newton’s trajectory equations (9). The claim, found in numerous text-books and original works, that the classical limit $ℏ → 0$ of QT agrees with classical mechanics can not be supported - at least if this limit is understood in the present (standard) way. In many text-books the fact that (13) agrees with the Hamilton-Jacobi equation is considered as evidence for this claim [the second Eq. (11) is then simply neglected]. This is however, unjustified, because the construction of trajectories with the help of (13) requires the sophisticated formalism of canonical transformations. And such a formalism does not exist in QT or in its $ℏ → 0$ limit and cannot be created by magic (QT should not be magic); the partial differential equations (11), (13) have to be solved as initial value problems for given $S (x, 0 ), ρ (x, 0 )$ .

4.3 The significance of the limit $ℏ → 0$ for the interpretation of quantum theory

"wave mechanics is an extension, not of ordinary Newtonian mechanics, but of statistical mechanics; and this simple observation is enough to explain many of its otherwise puzzling features."
John Clarke Slater, physicist and theoretical chemist (1900-1976).

Despite the vast literature on ’the classical limit of QT’ I was unable to find a clear answer to the question ”What is the limit $ℏ → 0$ of quantum theory?”. Most physicists assume that this question has already been answered by Dirac and others (in the affirmative). I had to analyze this question myself [85] (arXiv:1201.0150, this paper contains also an improved version of the last two subsections) and found that Dirac’s claim - quoted at the beginning of this section - is not justified. The proof of his claim, which is reported in his famous book [42], is based on assumptions which cannot be satisfied in reality. The final conclusion [85] is that a deterministic limit of Schrödinger’s equation exists only in a few special cases (for specific initial conditions in the three potentials $n ~ x , n = 0, 1, 2$ , where Ehrenfest’s relations agree with Newton’s equations) and that consequently, the classical ( $ℏ → 0$ ) limit of QT disagrees with classical mechanics. In all other cases, i.e. for arbitrary (nonsingular) initial conditions and arbitrary potentials, the classical cimit of QT is a classical probabilistic theory (given by the Hamilton-Jacobi theory (13) and the probability conservation law (11)). The latter reduces for special (singular) initial conditions to the deterministic equations of NM. But this limiting behavior can be found in all classical statistical theories, and does not mean that NM can be considered as the limit $ℏ → 0$ of QT. Thus, the classical limit of QT is not NM but a classical statistical theory.

This mathematical result is incompatible with an individuality interpretation of QT (with the CI); how can QT be a theory describing individual particles, if its classical limit is a theory about ensembles ? Thus, this mathematical result represents a strong argument in favor of the SI and against the CI.

"...Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
Bertrand Russell (1872-1970), in "Mathematics and the Metaphysicians"

One would expect general agreement with regard to the last point, but this is, unfortunately, not the case. The reason is that a particular (simple, physical) way of describing reality has been chosen in my paper. According to this point of view we cannot describe individual particles because we cannot predict their observable properties. The emphasis is on predictions and observations; two concepts which refer to an objective reality. On the other hand, many modern physicists argue, that probabilistic theories are more general than deterministic theories, because their manifold of solutions is larger and contains the subset of deterministic solutions as a special case. Considered from such a (mathematical) point of view, QT can be considered as a covering theory of NM and NM as a special case of QT. Then, in the same way as the difference between probabilistic and deterministic theories becomes unimportant, the fact, that NM is not the limit of QT becomes also unimportant. All that matters is the mathematical fact that the predictions of NM are a subset of the predictions of QT.

It does not seem that such a predominantly mathematical way of reasoning is useful for our understanding of QT or of nature in general. Some modern scientists seem even to believe that all mathematical structures correspond to something real. But mathematics should not be confused with reality. Physics is the task of selecting those mathematical structures which are useful for the description of nature and this task cannot be done by looking at mathematical structures alone - reality must also be taken into consideration.

"There are grounds for cautious optimism that we may now be near the end of the search for the ultimate laws of nature"
Stephen William Hawking, (1942-) British theoretical physicist and author

Bohr constructed QT (more precisely: those parts of it, which were still ambiguous, i.e. its interpretation) as a generalization of classical mechanics. He used the principle of reductionism as a fundamental methodic principle to characterize the relation between classical mechanics and QT.

"Some forms of reduction have been very successful in producing an understanding of the world that could not have been achieved in other ways; Maxwell's electrodynamics and its reduction of optics and electromagnetism is a case in point. Yet many physical systems, entities, and properties seem to defy reductionist explanations."
Margaret Morrison, "Emergence, Reduction, and Theoretical Principles: Rethinking Fundamentalism", Philosophy of Science, 73 (2006) 876

This principle says that a fundamental physical theory exists, from which ”in principle” all other, less fundamental theories and everything else can be derived. This principle is very deeply founded in our thinking. Many physical fields may be found which follow this scheme but many more may be found (different phases and states of matter, like the electron gas, superconductivity, quantum hall effects) where it does not work. See [6], [91], [106] (pdf) for a more detailed discussion.

As a consequence of the principle of reductionism, it was a fact beyond any doubt for Bohr that QT had to be the covering theory (of NM) from which NM could be derived in the classical limit $ℏ → 0$ . Thus, QT had to be a theory describing single particles. A major problem for this interpretation was the different nature of predictions of both theories (statistical predictions of QT versus deterministic predictions of NM), most obviously expressed by the different form of its basic equations. This problem could not be solved in the framework of physics, it was ”solved” in a philosophical rather than physical framework (see section 5 for details), by denying reality of unobserved properties⁵⁶ .

Questions of interpretation are generally difficult (stricly speaking impossible) to answer by purely mathematical means. The classical limit of QT is, however, a problem that can be studied mathematically and the result is highly relevant for the interpretation of QT⁵⁷ . The result that NM is not the classical limit of QT [85], leads to two conclusions, namely

The relation between NM and QT cannot be described by the principle of reductionism.
As a consequence there is no reason to interpret QT as a theory describing single particles.

"Human science fragments everything in order to understand it,..."
Lev Nikolayevich Tolstoy (1828 - 1910), Russian writer

Thus, QT is a theory describing statistical ensembles and not single particles. The limit $ℏ → 0$ transforms QT, a quantum-statistical theory, in a classical-statistical theory. In classical-statistical theories all the uncertainty is in the initial conditions while the particle paths are ruled by deterministic laws. In contrast, in quantum-statistical theories there is even more uncertainty⁵⁸ : there are no deterministic laws for particles any more (the concept of initial conditions becomes meaningless). More details may be found in my papers [84] (url) , and [112] (url).

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4 The Classical Limit

4.1 Two kinds of limit relations between classical theories

4.2 What is the ’standard’ limit of Schrödinger’s equation ?

4.3 The significance of the limit for the interpretation of quantum theory

4.2 What is the ’standard’ limit $ℏ → 0$ of Schrödinger’s equation ?

4.3 The significance of the limit $ℏ → 0$ for the interpretation of quantum theory