ij Just this circumstance, together with the relativistic invariance of the uncertainty relations of quantum mechanics, ensures the compatibility between the argumentation outlined in the present article and all exigencies of relativity theory. This question will be treated in greater detail in a paper under preparation, where the writer will in particular discuss a very interesting paradox suggested by Einstein concerning the application of gravitation theory to energy measurements, and the solution of which offers an especially instructive illustration of the generality of the argument of complementarity. On the same occasion a more thorough discussion of space-time measurements in quantum theory will be given with all necessary mathematical developments and diagrams of experimental arrangements, which had to be left out of this article, where the main stress is laid on the dialectic aspect of the question at issue.
ik Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1933).
il D. Bohm, Quantum Theory (Prentice-Hall, Inc., New York, 1951), see p. 611.
im N. Bohr, Phys. Rev. 48, 696 (1935).
in W. Furry, Phys. Rev. 49, 393, 476 (1936).
io Paul Arthur Schilp, editor, Albert Einstein, Philosopher-Scientist (Library of Living Philosophers, Evanston, Ilinois, 1949). This book contains a thorough summary of the entire controversy.
ip At distances of the order of 10−13 cm or smaller and for times of the order of this distance divided by the velocity of light or smaller, present theories becomes so inadequate that it is generally believed that they are probably not applicable, except perhaps in a very crude sense. Thus, it is generally expected that in connection with phenomena associated with this so-called “fundamental length,” a totally new theory will probably be needed. It is hoped that this theory could not only deal precisely with such processes as meson production and scattering of elementary particles, but that it would also systematically predict the masses, charges, spins, etc., of the large number of so-called “elementary” particles that have already been found, as well as those of new particles which might be found in the future.
iq L. de Broglie, An Introduction to the Study of Wave Mechanics (E. P. Dutton and Company, Inc., New York, 1930), see Chapters 6, 9, and 10. See also Compt. rend. 183, 447 (1926); 184, 273 (1927); 185, 380 (1927).
ir Reports on the Solvay Congress (Gauthiers-Villars et Cie., Paris, 1928), see p. 280.
is Note added in proof.—Madelung has also proposed a similar interpretation of the quantum theory, but like de Broglie he did not carry this interpretation to a logical conclusion. See E. Madelung, Z. 1. Physik 40, 332(1926), also G. Temple, Introduction to Quantum Theory (London, 1931).
it In Paper II, Sec. 9, we also discuss von Neumann’s proof [see J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Verlag, Julius Springer, Berlin, 1932)] that quantum theory cannot be understood In terms of a statistical distribution of “hidden” causal parameters. We shall show this his conclusions do not apply to our interpretation, because he implicitly assumes that the hidden parameters must be associated only with the observed system, whereas, as will become evident in these papers, our interpretation requires that the hidden parameters shall also be associated with the measuring apparatus.
iu See reference 2, Chapter 5.
iv N. Bohr, Atomic Theory and the Description of Nature (Cambridge University Press, London, 1934).
iw This consistency is guaranteed by the conservation Eq. (7). The questions of why an arbitrary statistical ensemble tends to decay into an ensemble with a probability density equal to ψ*ψ will be discussed in Paper II, Sec. 7.
ix This experiment is discussed in some detail in reference 2, Chapter 6, Sec. 2.
iy N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Clarendon Press, Oxford, 1933).
iz See reference 2, Chapter 18, Sec. 19.
ja Note that in the usual interpretation one assumes that nothing determines the precise outcome of an individual scattering process. Instead, one assumes that all descriptions are inherently and unavoidably statistical (see Sec. 2).
jb See reference 2, Chapter 22, Sec. 11, for a treatment of a similar problem.
jc It should be noted that exactly the same problem arises in the usual interpretation of the quantum theory for (reference 16), for whenever two packets overlap, then even in the usual interpretation, the system must be regarded as, in some sense, covering the states corresponding to both packets simultaneously. See reference 2, Chapter 6 and Chapter 16, Sec. 25. Once two packets have obtained classically describable separations, then, both in the usual interpretation and in our interpretation the probability that there will be significant interference between them is so overwhelmingly small that it may be compared to the probability that a tea kettle placed on a fire will happen to freeze instead of boil. Thus, we may for all practical purposes neglect the possibility of interference between packets corresponding to the different possible energy states in which the hydrogen atom may be left.
jd See, for example, reference 2, Chapter 11, Sec. 17, and Chapter 12, Sec. 18.
je D. Bohm, Phys. Rev. 84, 166 (1951).
jf For a treatment of how the theory of measurements can be carried out with the usual interpretation, see D. Bohm, Quantum Theory (Prentice-Hall, Inc., New York, 1951), Chapter 22.
jg A similar requirement is obtained in the usual interpretation. See references 2, Chapter 22, Sec. 8.
jh Even in the usual interpretation, an observation must be regarded as yielding a measure of such a potentiality. See reference 2, Chapter 6, Sec. 9.
ji Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1933).
jj J. von Neumann, Mathematics Grundlagen der Quanienmechanik (Verlag. Julius Springer, Berlin, 1932).
jk A leading nineteenth-century exponent of the positivist point of view was Mach. Modern positivists appear to have retreated from this extreme position, but its reflection still remains is the philosophical point of view implicitly adopted by a large number of modern theoretical physicists.
jl See G. Wentsel, Quantum Theory of Fidds (Interscience Publishers, Inc., New York, 1948).
jm L. de Broglie, An Introduction to the Study of Wave Mechanics (E. P. Dutton and Company, Inc., New York, 1930), see Chapters 6, 9, and 10.
jn N. Rosen, J. Elisha Mitchel Sci. Soc. 61, Nos. 1 and 2 (August, 1945).
jo Reports on the 1927 Solvay Congress (Gauthiers-Villars et Cie., Paris, 1928), see, p. 280.1933).
jp N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Clarendon Press, Oxford, 1933).
jq Reference 2, Chapter 16, Sec. 25.
jr Reference 2, Chapter 6, Secs. 3 to 8; Chapter 22, Secs. 8 to 10.
js Similar assumptions have been used by Born and Jordan [‘Z. f. Physik,’ vol. 34, p. 886 (1925)] for the purpose of taking over the classical formula for the emission of radiation by a dipole into the quantum theory, and by Born, Heisenberg and Jordan [‘Z. f. Physik,’ vol. 35, p. 606 (1925)] for calculating the energy fluctuations in a field of black-body radiation.
jt ‘Roy. Soc. Proc.,’ A, vol. 112, p. 661, § 5 (1926). This is quoted later by, loc. cit., I.
ju ‘Roy. Soc. Proc.,’ A, vol. 113, p. 621 (1927). This is quoted later by loc. cit., II. An essentially equivalent theory has been obtained independently by Jordan [‘Z. f. Physik,’ vol. 40, p. 809 (1927)]. See also, F. London, ‘Z. f. Physik,’ vol. 40, p. 193 (1926).
jv Loc. cit. II, § 2.
jw One can have a matrix scheme in which a set of variables that commute are at all times represented by diagonal matrices if one will sacrifice the condition that the matrices must satisfy the equations of motion. The transformation function from such a scheme to one in which the equations of motion are satisfied will involve the time explicitly. See p. 628 in loc. cit., II.
jx Loc. cit., II, §§ 6, 7.
jy Loc. cit. I.
jz The theory has recently been extended by Born [‘Z. f. Physik,’ vol. 40, p. 167 (1926)] so as to take into account the adiabatic changes in t
he stationary states that may be produced by the perturbation as well as the transitions. This extension is not used in the present paper.
ka Loc. cit., I, equation (25).
kb See § 8 of the author’s paper ‘Roy. Soc. Proc.,’ A, vol. 111, p. 281 (1926). What are there called the c-number values that a q-number can take are here given the more precise name of the characteristic values of that q-number.
kc We are supposing for definiteness that the label r of the stationary states takes the values 1, 2, 3,...
kd When s = r, ψ (Nʹ1 , Nʹ2 ... Nʹr − 1 ...Nʹs + 1) is to be taken to mean ψ (Nʹ1Nʹ2 ...Nr ...).
ke Loc. cit., I, § 3.
kf Born, ‘Z. f. Physik,’ vol. 38, p. 803 (1926).
kg The symbol x is used for brevity to denote x, y, z.
kh In a more recent paper (‘Nachr. Gesell. d. Wiss.,’ Gottingen, p. 146 (1926)) Born has obtained a result in agreement with that of the present paper for non-relativity mechanics, by using an interpretation of the analysis based on the conservation theorems. I am indebted to Prof. N. Bohr for seeing an advance copy of this work.
ki See Klein and Rosseland, ‘Z. f. Physik,’ vol. 4, p. 46, equation (4) (1921).
kj The ratio of stimulated to spontaneous emission in the present theory is just twice its value in Einstein’s. This is because in the present theory either polarised component of the incident radiation can stimulate only radiation polarised in the same way, while in Einstein’s the two polarised components are treated together. This remark applies also to the scattering process.
kk Pauli, ‘Z. f. Physik,’ vol. 18, p. 272 (1923).
kl Processes for partial differentiation with respect to matrices have been given by Born, Heisenberg and Jordan (ZS. f. Physik 35, 561, 1926) but these processes do not give us means of differentiation with respect to dynamical variables, since they are not independent of the representation chosen. As an example of the difficulties involved in differentiation with respect to quantum dynamical variables, consider the three components of an angular momentum, satisfying
mx my − my mx = ih mz.
We have here mz expressed explicitly as a function of mx and my , but we can give no meaning to its partial derivative with respect to mx or my.
km Jordan, ZS. f. Phys, 38, 513, 1926.
kn It is customary in field dynamics to regard the values of a field quantity for two different values of (x, y, z) but the same value of t as two different coordinates, instead of as two values of the same coordinate-for two different points in the domain of independent variables, and in this way to keep to the idea of a single independent variable t. This point of view is necessary for the Hamiltonian treatment, but for the Lagrangian treatment the point of view adopted in the text seems preferable on account of its greater space-time symmetry.
ko Dirac, Proc, Roy. Soc. A 136, 453, 1932.
kp Heisenberg and Pauli, ZS. f. Physik, 56, 1, 1929 and 59, 168, 1930.
kq Rosenfeld, ZS. f. Physik 76, 729, 1932.
kr h is Planck’s constant divided by 2π.
ks This is somewhat analogous to Frenkel’s method of treating incomplete systems, see Frenkel, Sow. Phys. 1, 99, 1932.
kt Fock and Podolsky, Sow. Phys. 1, 801, 1932, later quoted as 1. c. For other treatments see Jordan and Pauli, ZS. f. Physik, 47, 151, 1928 or Fermi, Rend. Lincei, 9, 881, 1929. The Lagrangian (12) differs from that of Fermi only by a four-dimensional divergence.
ku A dot over a field quantity will be used to designate a derivative with respect to the field time t.
kv We shall drop the asterisk and in the following use ψ instead ψ*.
kw See Jordan and Pauli, ZS. f. Physik 47, 159, 1928.
kx Heisenberg und Pauli, ZS. f. Physik 56, 34, 1929.
ky Research Fellow of the Rockefeller Foundation. I should like to thank the Rockefeller Foundation for giving me the opportunity to work in Cambridge.
kz This expression has nothing to do with “unitary” field theory due to Einstein, Weyl, Eddington, and others where the problem consists of uniting the theories of gravitational and electro-magnetic fields into a kind of non-Riemannian geometry. Specially some of Eddington’s formulæ, developed in § 101 of his book “The Mathematical Theory of Relativity” (Cambridge), have a remarkable formal analogy to those of this paper, in spite of the entirely different physical interpretation.
la ‘Ann. Physik,’ vol. 37, p. 511 (1912); vol. 39, p. 1 (1912); vol. 40, p. 1 (1913). Also Born, ‘Göttinger Nachr,’ p. 23 (1914).
lb The attempt to avoid this difficulty by a new definition of electric force acting on a particle in a given field, made by Wentzel (‘Z. Physik,’ vol. 86, pp. 479, 635 (1933), vol. 87, p. 726 (1934)), is very ingenious, but rather artificial and leads to new difficulties.
lc Born, ‘Nature,’ vol. 132, p. 282 (1933); ‘Proc. Roy. Soc.,’ A, vol. 143, p. 410 (1934), cited here as I.
ld See Born and Infeld, ‘Nature,’ vol. 132, p. 1004 (1933).
le See Eddington, “The Expanding Universe,” Cambridge, 1933.
lf The adaptation of the function L (1.1) to the general relativity by multiplication with is quite formal. Any expression can be made generally invariant in this way.
lg See Eddington, “The Mathematical Theory of Relativity,” Cambridge, § 48 and 101 (1923).
The proof is simple: by a transformation with the Jacobian is changed into and |akl | into |; for the dxk are contravariant, akl convariant.
lh This assumption has already been considered by Einstein, ‘Berl. Ber.,’ pp. 75/37 (1923) and p. 414 (1925), from the standpoint of the affine field theory.
li See Eddington, loc. cit., § 1 01.
lj Einstein and Mayer, ‘Berl. Ber.,’ p. 3 (1932).
lk In I it has been stated that the two expressions for , obtained with help of L and H, are different; this has turned out to be a mistake.
ll E.g., Jahnke-Emde, “Tables of functions” (Teubner 1933), p. 127.
lm Mie, ‘Ann. Physik,’ vol. 40, p. 1 (1913).
ln Born, ‘Ann. Physik,’ vol. 28, p. 571 (1909); Pauli, “Relativitätstheorie,” p. 642 (Teubner).
lo The method used in I for deriving the equation of motion is not correct. It started from the action principle in the form
δ ∫ L dτ = 0 (instead δ ∫ H dτ = 0) ;
then in the development instead of the coefficients the appear, which become infinite at the centre of the electron. Therefore the transformation of the space integral is not allowed. In the first approximation we have
and not
The mistake in the former derivation is also shown by the wrong result for the mass (the numerical factor was half of that given here).
lp Calculated by Mr. Devonshire.
lq Born, ‘Nature,’ vol. 133, p. 63 (1934).
lr Author’s note, 1956. Approximate solutions of the Heisenberg equations of motion were obtained by Yang and Feldman, Phys. Rev., 79, 972, 1950; and Källén, Arkiv För Fysik, 2, 371, 1950.
ls For a convenient account, see H. E. White, Introduction to Atomic Spectra (McGraw-Hill Book Company, New York, 1934), Chap. 8.
lt J. W. Drinkwater, O. Richardson, and W. E. Williams, Proc. Rov. Soc. 174, 164 (1940).
lu W. V. Houston, Phys. Rev. 51, 446 (1937); R. C. Williams, Phys. Rev. 54, 558 (1938); S. Pasternack, Phys. Rev. 54, 1113 (1938) has analyzed these results in terms of an upward shift of the S level by about 0.03 cm−1 .
lv H. A. Bethe in Handbuch der Physik, Vol. 24/1, §43.
lw G. Breit and E. Teller, Astrophys. J. 91, 215 (1940).
lx Phys. Rev. 72, 241 (1947).
ly W. V. Houston, Phys. Rev. 51, 446 (1937).
lz R. C. Williams, Phys. Rev. 54, 558 (1938).
ma E. C. Kemble and R. D. Present, Phys. Rev. 44, 1031 (1932); S. Pasternack, Phys. Rev. 54, 1113 (1938).
mb E. A. Uehling, Phys. Rev. 48, 55 (1935).
mc It was first suggested by Schwinger and Weisskopf that hole theory must be used to obtain convergence in this problem.
me Jagdish Mehra and Helmut Rechenberg (2001) The Historical Development of Quantum Theory Springer pp. 1099
mf Helge Kragh (1990). Dirac: A Scientific Biography, Cambridge University Press, pp. 184
mg Translated from the paper, Bull. I. P. C. R. (Riken-iho), 22 (1943), 545, appeared originally in Japanese.
mh [A, B] = AB − BA. We assume that the field obeys the Bose statistics. Our considerations apply also to the case of Fermi statistics.
mi We use the square brackets to indicate a functional. Thus ψ [v′ (xyz)] means that ψ is a functional of the variable function v′ (xyz). When we use ordinary parentheses () , as ψ (v′ (xyz)), we consider ψ as an ordinary function of the function v′ (xyz). For example: the energy density is written as H(v(xyz), λ(xyz)) and this is also a function of x, y and z, whereas the total energy H = ∫ H (v(xyz), λ(xyz))dv is a functional of v(xyz) and λ(xyz) and is written as H [v(xyz), λ(xyz)].
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