Introduction

1 Introduction

In Dirac’s famous book [6] on quantum theory one finds the statement: ”...classical mechanics may be regarded as the limiting case of quantum mechanics when $ℏ$ tends to zero”. A bright student, having read this statement might ask his teacher the following question: ”In quantum mechanics a single particle in an external potential is described by Schrödinger’s equation
$[ 3 ( ) ] ℏ ∂ ℏ2 ∑ ∂ 2 ------- ------ ------- - + V (x, t ) ψ (x, t ) = 0. i ∂ t 2m ∂ xk k=1$ (1)

If Dirac is right then Newton’s equation,
$| d d ∂ V (x, t) | ---mr (t ) = p (t ), ----p (t) = - --------------| . k k k dt dt ∂ xk x=r (t)$ (2)

should follow from Schrödinger’s equation in the limit $ℏ → 0$ . Can you tell us, how this calculation is actually performed ? ” Nobody has ever performed a general exact calculation showing that (1) implies (2) in the limit $ℏ → 0$ . An experienced teacher will nevertheless find an answer to this question. We will not discuss the variety of his possible responses but think that these will probably not include one of the following two statements: (i) Dirac is wrong, (ii) the answer is not known. A typical response is the statement that this limit cannot be understood in such a simple way. A closer look in Dirac’s book shows, however, that, at least as Dirac’s original intentions are concerned, this statement should in fact be interpreted in this simple way.

The attitude of the scientific community with regard to this point - which is extremely important for the interpretation of quantum theory (QT) as well as for more general questions such as the problem of reductionism - is somewhat schizophrenic. On the one hand, Dirac’s dictum - which has been approved by other great physicists - is considered to be true. On the other hand it cannot be verified. Since the beginnings of QT a never ending series of works deal with this question, but the deterministic limit of QT, in the sense of the above general statement, has never been attained. Frequently, isolated ’classical properties’ which indicate asymptotic ’nearness’ of deterministic physics, or structural similarities (such as those between Poisson brackets and commutators) are considered as a substitute for the limit. The point is that in most of these papers (see e.g. [20],[25],[1] to mention only a few) the question ”what is the limit $ℏ → 0$ of quantum theory ?” is not studied. It is taken for granted that the final answer to this question has already been given (by Dirac and others) and the remaining problem is just how to confirm, or illustrate it by concrete calculations. But none of these attempts is satisfying.

In the present paper the question formulated in the title will be studied without knowing the answer. A detailed step-by-step style of exposition has been chosen in order to understand this singular limit. In fact, the paper has been written with the idea in mind to provide an in-depth answer to a student’s question concerning the mathematical details of the transition from (1) to (2). The questions how to perform a limit and how to characterize the relations between different (related) physical theories are closely connected to the basic question how to characterize physical theories themselves. We take a pragmatic position with respect to this question and characterize a physical theory simply by the set of its predictions. It turns out that this leads automatically to a reasonable definition of limit relations between different physical theories. We start by discussing, in section 2, two well-understood concrete limiting relations between two pairs of classical physical theories. These classical limiting relations, referred to as ’standard limit’ and ’deterministic limit’, define possible meanings of the term ”the limit $ℏ → 0$ ” in QT. In section 3 we use the variables introduced by Madelung to obtain the ’standard limit’ of QT, previously found by Rosen [19], Schiller [21] and others. In section 4 we derive the deterministic limit of the ’standard limit’ of QT. We find that Ehrenfest’s relations, which have not been taken into account in previous treatments [19, 3, 13, 18, 8], provide the missing link between the ’standard limit’ (field) theory and the trajectory equations of Newtonian mechanics (NM). In section 5 we investigate the ’deterministic limit’ of QT and conclude that this limit does not exist. In section 6 we try to reconstruct the states of NM from QT by combining the ’deterministic limit’ and the ’standard limit’. In this way we are indeed able to identify a class of (three) potentials and states which allow for a transition from QT to NM in the limit $ℏ → 0$ ; these include among others the coherent states of the harmonic oscillator [22]. In section 7 we try to extend this process to arbitrary potentials. The obtained results are discussed and interpreted in section 8 and the final conclusion is drawn in section 9.

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