III. Illustration
The proof of the main result is quite simple. Before giving it, however, a number of illustrations may serve to put it in perspective.
Firstly, there is no difficulty in giving a hidden variable account of spin measurements on a single particle. Suppose we have a spin half particle in a pure spin state with polarization denoted by a unit vector . Let the hidden variable be (for example) a unit vector with uniform probability distribution over the hemisphere > 0. Specify that the result of measurement of a component · is
(4)
where is a unit vector depending on and in a way to be specified, and the sign function is +1 or -1 according to the sign of its argument. Actually this leaves the result undetermined when λ · aʹ = 0, but as the probability of this is zero we will not make special prescriptions for it. Averaging over the expectation value is
(5)
where θʹ is the angle between and . Suppose then that is obtained from by rotation towards until
(6)
where θ is the angle between and . Then we have the desired result
(7)
So in this simple case there is no difficulty in the view that the result of every measurement is determined by the value of an extra variable, and that the statistical features of quantum mechanics arise because the value of this variable is unknown in individual instances.
Secondly, there is no difficulty in reproducing, in the form (2), the only features of (3) commonly used in verbal discussions of this problem:
(8)
For example, let λ now be unit vector , with uniform probability distribution over all directions, and take
(9)
This gives
(10)
where θ is the angle between a and b, and (10) has the properties (8). For comparison, consider the result of a modified theory [6] in which the pure singlet state is replaced in the course of time by an isotropic mixture of product states; this gives the correlation function
(11)
It is probably less easy, experimentally, to distinguish (10) from (3), than (11) from (3).
Unlike (3), the function (10) is not stationary at the minimum value -1(at θ = 0). It will be seen that this is characteristic of functions of type (2).
Thirdly, and finally, there is no difficulty in reproducing the quantum mechanical correlation (3) if the results A and B in (2) are allowed to depend on and respectively as well as on and . For example, replace in (9) by , obtained from by rotation towards until
where θʹ is the angle between and . However, for given values of the hidden variables, the results of measurements with one magnet now depend on the setting of the distant magnet, which is just what we would wish to avoid.
IV. Contradiction
The main result will now be proved. Because ρ is a normalized probability distribution,
(12)
and because of the properties (1), P in (2) cannot be less than -1. It can reach -1 at only if
(13)
except at a set of points λ of zero probability. Assuming this, (2) can be rewritten
(14)
It follows that is another unit vector
using (1), whence
The second term on the right is P(, ), whence
(15)
Unless P is constant, the right hand side is in general of order for small . Thus P (,) cannot be stationary at the minimum value (-1 at ) and cannot equal the quantum mechanical value (3).
Nor can the quantum mechanical correlation (3) be arbitrarily closely approximated by the form (2). The formal proof of this may be set out as follows. We would not worry about failure of the approximation at isolated points, so let us consider instead of (2) and (3) the functions
wherethe bar denotes independent averaging of P (, ) and - . overvectors and within specified small angles of and . Suppose that for all and the difference is bounded by ε:
(16)
Then it will be shown that ε cannot be made arbitrarily small.
Suppose that for all a and b
(17)
Then from (16)
(18)
From (2)
(19)
where
(20)
From (18) and (19), with ,
(21)
From (19)
Using (20) then
Then using (19) and 21)
Finally, using (18),
or
(22)
Take for example Then
Therefore, for small finite δ, ε cannot be arbitrarily small.
Thus, the quantum mechanical expectation value cannot be represented, either accurately or arbitrarily closely, in the form (2).
V. Generalization
The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made. In a more formal way, assuming [7] that any Hermitian operator with a complete set of eigenstates is an “observable”, the result is easily extended to other systems. If the two systems have state spaces of dimensionality greater than 2 we can always consider two dimensional subspaces and define, in their direct product, operators and formally analogous to those used above and which are zero for states outside the product subspace. Then for at least one quantum mechanical state, the “singlet” state in the combined subspaces, the statistical predictions of quantum mechanics are incompatible with separable predetermination.
VI. Conclusion
In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.
Of course, the situation is different if the quantum mechanical predictions are of limited validity. Conceivably they might apply only to experiments in which the settings of the instruments are made sufficiently in advance to allow them to reach some mutual rapport by exchange of signals with velocity less than or equal to that of light. In that connection, experiments of the type proposed by Bohm and Aharonov [6], in which the settings are changed during the flight of the particles, are crucial.
I am indebted to Drs. M. Bander and J. K. Perring for very useful discussions of this problem. The first draft of the paper was written during a stay at Brandeis University; I am indebted to colleagues there and at the University of Wisconsin for their interest and hospitality.
REFERENCES
1. A. EINSTEIN, N. ROSEN and B. PODOLSKY, Phys. Rev. 47. 777 (1935); see also N. BOHR, Ibid. 48, 696 (1935), W. H. FURRY, Ibid. 49, 393 and 476 (1936), and D. R. INGLIS, Rev. Mod. Phys. 33, 1 (1961).
2. “But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former.” A. EINSTEIN in Albert Einstein, Philosopher Scientist, (Edited by P. A. SCHILP) p. 85, Library of Living Philosophers, Evanston, Illinois (1949).
3. J. VON NEUMANN, Mathematishe Grundlagen der Quanten-mechanik. Verlag Julius-Springer, Berlin (1932), [English translation: Princeton University Press (1955)]; J. M. JAUCH and C. PIRON, Helv. Phys. Acta 36, 827 (1963).
4. J. S. BELL, to be published.
5. D. BOHM, Phys. Rev. 85, 166 and 180 (1952).
6. D. BOHM and Y. AHARONOV, Phys. Rev. 108, 1070 (1957).
7. P. A. M. DIRAC, The Principles of Quantum Mechanics (3rd Ed.) p. 37. The Clarendon Press, Oxford (1947).
Chapter Six
The formulations of quantum mechanics by Werner Heisenberg and Erwin Schrodinger and the relativistic extension by Paul Dirac provide a way to calculate the dynamics of physical systems under the influence of any instantaneous force. This is very important but is not the complete picture. Electrons, or any charged particle, when accelerated will produce electromagnetic
radiation, and equivalently, when electromagnetic radiation impacts a charged particle, the particle will accelerate. Understanding how quantum particles interact with electromagnetic fields, how they emit and absorb radiation, requires a quantum theory of electrodynamics, and the founders of quantum mechanics immediately began to develop this theory.
Dirac first attempted to tackle this problem in a 1927 paper entited “The Quantum Theory of the Emission and Absorption of Radiation.” In order to develop a complete theory of quantum electrodynamics, Dirac looked for inspiration from the classical theory. There are at least two independent formulations of classical mechanics, known as the Lagrangian and Hamiltonian methods. Quantum mechanics had been built by analogy with the classical Hamiltonian, but it was known that the Lagrangian method was easier to reformulate into a relativistic theory. Since electromagnetics is the theory upon which relativity is built, Dirac’s hope was that by finding a Lagrangian method for quantum mechanics it would be easier to develop a theory of quantum electrodynamics. In 1932, Dirac published “The Langrangian in Quantum Mechanics” in which he applied the Lagrangian method to quantum mechanics. His hope proved correct, and Dirac’s paper helped form the basis of both Richard Feynman’s and Julian Schwinger’s approaches to quantum electrodynamics.
Also in 1932, Dirac published, with the collaborators Vladimir Fock and Boris Podolsky, “On Quantum Electrodynamics,” in which they provided a formulation of quantum electrodynamics and then proved that it was equivalent to an earlier theory produced in 1930 by Heisenberg and Pauli. It was quickly realized that both the Dirac and Heisenberg-Pauli formulations of quantum electrodynamics suffered from a serious problem. Many of the computations required in these theories led to predictions of infinite energy. For example, when calculating the energy of an electron a term must be included that accounts for the self-energy stored in its electric field. For a point particle, the mathematical integral representing this term is infinite. This is clearly not physical, and it indicates a weakness in the theory. These infinities were deeply troubling, and for the next decade the elite of quantum theory struggled to find a way to eliminate them. The progress of this work is summarized in Robert Oppenheimer’s 1947 paper “Electron Theory.”
One important method for removing the infinities from the theory was published in 1934 by Max Born and Leopold Infeld. In “Foundations of the New Field Theory,” they modified Maxwell’s equations, which are the equations that govern the theory of electrodynamics, in such a way as to keep the self-energy of a point particle finite. Just how in Einstein’s theory of relativity there is a cutoff velocity—the speed of light—above which nothing can travel, Born and Infeld postulated that there may be a cutoff above which no electric field is possible. They modified Maxwell’s equations to incorporate this assumption. In the next chapters we will find that such an assumption was not necessary. In the late 1940s a method was developed for regularizing the infinities without having to explicitly modify the classical form of Maxwell’s equations.
THE QUANTUM THEORY OF THE EMISSION AND ABSORPTION OF RADIATION
BY
PAUL A.M. DIRAC
From Proceedings of the Royal Society of London, Series A, Vol. 114, p. 243 (1927)
§ 1. INTRODUCTION AND SUMMARY.
The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is non-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty.
The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if Er is the energy of a component labelled r and θr is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each Er and θr to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian
(1)
equal to the total energy, H0 being the Hamiltonian for the atom alone, since the variables Er , θr obviously satisfy their canonical equations of motion
When there is interaction between the field and the atom, it could be taken into account on the classical theory by the addition of an interaction term to the Hamiltonian (1), which would be a function of the variables of the atom and of the variables Er , θr that describe the field. This interaction term would give the effect of the radiation on the atom, and also the reaction of the atom on the radiation field.
In order that an analogous method may be used on the quantum theory, it is necessary to assume that the variables Er , θr are q-numbers satisfying the standard quantum conditions θr Er − Erθr = ih, etc., where h is (2π )−1 times the usual Planck’s constant, like the other dynamical variables of the problem. This assumption immediately gives light-quantum properties to the radiation.js For if νr is the frequency of the component r, 2π νr θr is an angle variable, so that its canonical conjugate Er /2πνr can only assume a discrete set of values differing by multiples of h, which means that Er can change only by integral multiples of the quantum (2πh)νr . If we now add an interaction term (taken over from the clasical theory) to the Hamiltonian (1), the problem can be solved according to the rules of quantum mechanics, and we would expect to obtain the correct results for the action of the radiation and the atom on one another. It will be shown that we actually get the correct laws for the emission and absorption of radiation, and the correct values for Einstein’s A’s and B’s. In the author’s previous theory,jt where the energies and phases of the components of radiation were c-numbers, only the B’s could be obtained, and the reaction of the atom on the radiation could not be taken into account.
It will also be shown that the Hamiltonian which describes the interaction of the atom and the electromagnetic waves can be made identical with the Hamiltonian for the problem of the interaction of the atom with an assembly of particles moving with the velocity of light and satisfying the Einstein-Bose statistics, by a suitable choice of the interaction energy for the particles. The number of particles having any specified direction of motion and energy, which can be used as a dynamical variable in the Hamiltonian for the particles, is equal to the number of quanta of energy in the corresponding wave in the Hamiltonian for the waves. There is thus a complete harmony between the wa
ve and light-quantum descriptions of the interaction. We shall actually build up the theory from the light-quantum point of view, and show that the Hamiltonian transforms naturally into a form which resembles that for the waves.
The mathematical development of the theory has been made possible by the author’s general transformation theory of the quantum matrices.ju Owing to the fact that we count the time as a c-number, we are allowed to use the notion of the value of any dynamical variable at any instant of time. This value is a q-number, capable of being represented by a generalised “matrix” according to many different matrix schemes, some of which may have continuous ranges of rows and columns, and may require the matrix elements to involve certain kinds of infinities (of the type given by the δ functionsjv). A matrix scheme can be found in which any desired set of constants of integration of the dynamical system that commute are represented by diagonal matrices, or in which a set of variables that commute are represented by matrices that are diagonal at a specified time.jw The values of the diagonal elements of a diagonal matrix representing any q-number are the characteristic values of that q-number. A Cartesian co-ordinate or momentum will in general have all characteristic values from − ∞ to + ∞, while an action variable has only a discrete set of characteristic values. (We shall make it a rule to use unprimed letters to denote the dynamical variables or q-numbers, and the same letters primed or multiply primed to denote their characteristic values. Transformation functions or eigenfunctions are functions of the characteristic values and not of the q-numbers themselves, so they should always be written in terms of primed variables.)
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