A new method for the treatment of the longitudinal and scalar photons

Transcription

1 A new method for the treatment of the longitudinal and scalar photons K. Bleuler (Dated: 10. VI. 1950) Summary: Gupta has introduced an alternative method of quantization for the Maxwell field which differs from the usual one in that the scalar part of the field is quantized by means of the indefinite metric of Dirac. It is shown that this method can be extended into a general and consistent theory, including the case of interaction with electrons. Some of the advantages of the new method are the following: The well known difficulty of normalizing a state vector satisfying the Lorentz condition no longer occurs. For processes taking place within long time intervals the photon vacuum can consistently be stated in the form A + µ (x)ψ 0 = 0, a condition which could not be fulfilled in the ordinary theory. Gauge invariance is exhibited in a peculiar direct way. It is shown, by a canonical transformation, that the theory is equivalent with the reduced theory where the longitudinal field is eliminated, and replaced by the static Coulomb interaction. All physical results are therefore identical with those of the ordinary theory. Lorentz invariance is exhibited in a simple way. Original Ref.: K. Bleuler, Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen, Helv. Phys. Acta 23, (1950). This version of the article was typeset in 2009 using L A TEX 2e and REVTEX v4.0. Translated by Dale Alan Woodside in 1995 using German Assistant TM for Windows TM Vers. 1.0B. Translation is Copyright 1995, 2009 by Dr. Dale Alan Woodside, all rights reserved. German Assistant is a trademark of MicroTac Software, Inc. Windows is a registered trademark of Microsoft Corporation. I. INTRODUCTION. In quantum electrodynamics it has proved very convenient to calculate with all four components of the vector potentials A µ (x) in a symmetrical manner; i.e., one avoids the elimination of the longitudinal and scalar part with a corresponding substitution of the Coulomb potential, (the reduced theory ). Then, however, one has an auxiliary condition (Lorentz condition) to use; according to Fermi [1] one writes in the case without interaction: Ω(x) ψ A µ(x) ψ = 0 (1.1) (x stands for the general space-time (x 1,..., x 4 ); repeated indices µ = 1,..., 4 should be summed over; ψ signifies the time-independent state vector). So that leads one however to a first difficulty: With the usual definition of the operators A µ, fulfilling the condition (1.1) includes in ψ an infinite number of scalar and longitudinal photons.[2] Apart from the unphysical fact, that these photons must also be existing in the vacuum, this also leads to mathematical difficulties: ψ is not normalizable, i.e., (1.1) is not strictly realizable. Belinfante has also shown that this ambiguous circumstance leads to the same outcome.[2] Further, it is very disturbing that despite the symmetrical form of the theory, the vacuum could only be defined through taking the transverse part. To be sure Schwinger [3] has suggested introducing the following vacuum definition: A + µ (x) ψ 0 = 0. (1.2) (A + signifies the positive frequency part.) Formally, if the condition is taken as Lorentz invariant and put in simplest form, one would take the condition on it for which the expectation value of the energy becomes a minimum, (without using the auxiliary condition). In the hitherto existing theory (1.2) is however unrealizable: If A 4 is quantized by the ordinary method, A + 4 is the emission operator and there is no condition apparent to prevent the emission of scalar photons. Furthermore the Lorentz condition speaks against (1.2). In spite of these contradictions, the formal use of (1.2) in the calculation of vacuum expectation values (anyway for processes in long time intervals) leads in the simplest manner to right outcomes. (See also section 2). The justness of the outcomes thereby proves either, that they are identical with the outcomes of the reduced theory (longitudinal and scalar fields eliminated) [4], or through their establishment, that the so obtained vacuum expectation values are the same, even if through their absence the transverse photons alone define the vacuum [5]. In both cases they essentially reduce to the same requirement, that the non-invariantly formulated theory serves as the criterion for justness. This most strange situation leads one to suspect, that the 4 photon type symmetrical treatment gives a non-free-field theory, and that (1.2) has defined the vacuum.

2 A suggestion in this direction came from Gupta recently [6] for handling the case without interaction, and it will be shown in the following, that the suggestion can be developed into a general and non-free-field theory. The essentials of Gupta s idea are the following: first of all due to the imaginary character of A 4, it becomes a corresponding operator in the expression in a different manner: positioning the factor i (imaginary unit) via the theory of Dirac and Pauli [7] in their so-called indefinite metric operator η; thereby leads to the conclusion that A 4 is also represented by an absorption operator, i.e., (1.2) becomes a realizable and definite condition in which no photons exist. On the other hand the new Lorentz condition is formulated so that it becomes compatible with the vacuum definition (1.2). Thereby the normilization difficulties of ψ are remedied simultaneously. Now the operator η appears in the normalization, and, in general two different kinds of state vectors appear in the scalar product, and a physically suitable condition is no longer represented through an obvious function ψ : it is rather that always certain additions are possible which deliver no contribution to the scalar product. This has as a consequence, that e.g. the expectation values of the vector potentials are vague up to a certain degree, in fact right within the framework of a gauge transformation. Gauge invariance shows how e.g. the field strengths are obvious. Interpretation difficulties don t exist, because the appearance of negative probabilities is prevented through the new Lorentz condition. In the present work it should now be shown, that one can formulate an integrated theory of the Maxwell s field in interaction with electrons in this manner: Section 2 becomes first of all Gupta s theory of the electromagnetic field alone represented in a somewhat different form; the Lorentz invariance can thereby be inspected easily. Section 3 brings the wording of the interaction to the pertinent expansion of the new Lorentz condition. The connection with the classical theory is surely demonstrated through the corresponding behavior of the expectation values. In section 4 it is shown, with the help of the elimination the longitudinal and scalar part, that one gets the well-known reduced theory; i.e., in a physical sense the new method is wholly equivalent to the usual theory. In the last section the vacuum becomes first of all defined in the old form (no transverse photons): It shows that with a suitable gauge one is led automatically to a condition of the form (1.2). In the case of the existence of sources one gets however an additional term, which has already been obtained by Jauches and Coester [8] in the framework of the usual theory. It follows from the electrostatic field, which must also exist in this case for the photon vacuum. Likewise, procedures that play a role in the finite time interval itself give a contribution, that in simple events represents the instantaneous effect of the Coulomb potentials. This follows also out of a comparison with the reduced theory. For infinite time intervals, this contribution vanishes, so that the vacuum condition takes the form (1.2). 2 II. THE QUANTIZATION OF THE MAXWELL S FIELD. A. The definition of the operators and the expectation values. The field operators A µ fulfill the well-known relations (with = c = 1): A µ (x) = 0, (2.1) i[a µ (x), A ν (x)] = δ µν D(x y); (2.2) (In contrast to the work of Schwinger (loc. cit.) we use here the usual definition of the invariant D-Function.) We have that all four components of A µ are hermitian: A µ = A µ, µ = 1,..., 4; (2.3) (The star should always signify hermitian conjugates here.) i.e., in the usual explicit representation: A µ (x) = 1 2v k (a µ,ke ikx + a µ,ke ikx ) (2.4) k with A + µ (x) + A µ (x), [a µ,k, a ν,e] = δ µν δ ke and a µ,k a µ,k = N µ,k for µ = 1,..., 4. (Translator s note: three-vectors will be written as bold face, e.g., k, instead of with an arrow on top.) (k signifies the spatial part of the four-vector k; k µ k µ = 0. A + µ (respectively A µ ) is the more positive (negative) frequency part

3 of it, i.e., the term e ikx ( e ikx ).) So A + 4 also becomes an absorption operator, i.e., (1.2) is realizable. In the usual theory, however, A 4 through multiplication with i becomes antihermitian; in order for (2.2) to still be fulfilled a 4 and a 4 become interchanged. Because of this reason A + 4 there becomes an emission operator. In place of this method, one now uses the following procedure: The operator η is defined through the following relations: 3 η A r (x) = A r (x) η, r = 1, 2, 3, η A 4 (x) = A 4 (x) η, (2.5) i.e., primarily A 4 becomes changed in sign through the outcome of these relations. It follows first of all, that η 2 commutes with all components; therefore one can write the convenient normalizations as η 2 = I, likewise η = η, (2.6) i.e., η is chosen as hermitian. The explicit representation is designated in the simplified manner: = ( 1) N4 δ NµM µ. (2.7) η NµM µ So that ηa r becomes hermitian, while ηa 4 becomes antihermitian; one writes now the expectation values A in the new form: by which the adjoint condition ψ through µ A µ = (ψ, ηa µ ψ) (ψ, A µ ψ), (2.8) ψ = ψ η (2.9) was defined. A µ thereby get the right reality attributes, especially ia 4 becomes real. Accordingly, the norm N of the state ψ is now taken as: N = (ψ, ψ). (2.10) This value is first of all not positive definite; it shows, however, that all states which fulfill the Lorentz condition result in N 0. B. The Lorentz condition and the Normalizations. Following the proposal of Gupta, one writes, instead of (1.1) for the Lorentz condition, the following: Ω + ψ A+ µ ψ = 0. (2.11) (In the usual form of quantization this condition is equivalent with (1.1); with a representation analogous to (2.13) one verifies easily e.g. that in this case (1.2) and (2.11) determine the same (non-normalized) state functions. I owe this remark to Dr. R. Jost at Princeton.) This condition on the operators A µ is weaker than (1.1); nevertheless, it follows from this that the expectation value of the four-divergence vanishes: The hermitian conjugate equation, because of A µ + = A µ (compare with (2.4)), is then ( ψ A ) r A 4 = 0 ; x r r x 4 upon multiplication with η from the right i.e., together with (2.11) (Ω = Ω + + Ω ) ( A ) ψ µ ψ Ω = 0 ; ( ψ, A ) µ ψ = 0. (2.12)

4 So that (2.12) together with (2.1) produces the connection with the classical theory. Now, to inspect that (2.11) is realizable and compatible with the vacuum condition (1.2), we use for the spatial part A of the field the well-known representation: 3 1 A(x) = e m (k) 2v k (a m,ke ikx + a m,ke ikx ) (2.13) with m=1 k (e m, e n ) = δ mn, (e 1, k) = (e 2, k) = 0, (e 3, k) = k, by which the a m of (2.4) are defined, while A 4 remains unchanged. N 3 and N 4 represent then the numbers of longitudinal and scalar photons. (2.11) results in now for each vector k: (a 3,k + ia 4,k ) ψ = 0. (2.14) (At this point, it is easy to inspect that (1.1) and (2.11) are now no longer equivalent: Out of (1.1), besides (2.14), a second condition with emission operators follows, which is unrealizable.) We write an abbreviation (leaving out the index k) for a state function ψ(n 3, N 4 ), so the ϕ(n 3, N 4 ) include exactly N 3 longitudinal and N 4 scalar photons, i.e., and so fulfills (2.14) through the following states ψ n : ϕ(n 3, N 4 ) = δ N 3N 3 δ N 4N 4, (2.15) ψ 0 = f 0 (N 1, N 2 ) ϕ(0, 0) (2.16) ψ 1 = f 1 {ϕ(1, 0) + iϕ(0, 1)} ψ 2 = f 2 {ϕ(2, 0) + i 2 ϕ(1, 1) ϕ(0, 2)}..... { (n ) } ψ n = f n ϕ(n, 0) + + (i) r ϕ(n r, r) + + (i) n ϕ(0, n) r..... All the f n signify arbitrary functions of N 1 and N 2. One gets the general state ψ, which satisfies (2.14), through the linear combination: 4 ψ = n c n ψ n. (2.17) For the normalizations we now heed (2.7) and (2.10), so that ψ 0 possess a positive norm: (ψ 0, ψ 0 ) = N 1N 2 f 0 f 0 > 0, (2.18) while all different norms and scalar products vanish: (ψ n, ψ m ) = 0 for n 0 or m 0 (2.19) (One notes a simple attribute of the binomial coefficient ( n r) ). From this follows, first of all, that also the general state (2.17) does not possess a negative norm; it vanishes for c 0 = 0. One demands now: (ψ, ψ) = 1. (2.20) With c 0 0 this gives N 1N 2 g 0 g 0 = 1, with g 0 = c 0 f 0. (2.21)

5 The c n remain arbitrary for n > 0. The function g 0 (N 1, N 2 ) becomes therefore in the usual sense the probability amplitude of interpreting N 1 and respectively N 2 transverse photons, while for the physically not truly observable longitudinal and scalar photons no prescription exists. The general probability declaration is given by w = N 1N 2 g 0 1 g where g 0 1 and g 0 2 are the corresponding shares that the two different states ψ 1 and ψ 2 signify. This value can now be written, because of (2.18,19), in the simple form w = (ψ 1, ψ 2). (2.22) It is for this theory characteristic, that only state functions ψ, which fulfill the Lorentz condition (2.11) in the stated manner, are interpreted as physical. Further, it follows out of this definition, that a submitted state function (it must possess the form (2.17)) doesn t change its physical meaning, if the function c n f n (n > 0) is chosen differently. On the other hand, one can not leave out these admixtures, because they will appear in the case of the interaction with electrons by themselves in the solution of the time-dependent Schrödinger equation, if the initial condition is chosen with c n = 0 (for n > 0). The meaning of this addition lies in the possibility of gauge transforming the potential in an arbitrary manner. This should be illustrated in two simple examples: One predicts the expectation values A µ in the states ψ = ψ 0 + c 1 ψ 1 as a function of c 1. (ψ 1 itself covers only the wavenumber vector k.) With the help of (2.13) one then obtains with A µ = (ψ 0, A µ ψ 0 ) Λ Λ = c 1 g 01 e ikx + c 1g 01e ikx, by which the constant g 01 of k, c 0 f 0 and f 1 depends. In this manner, it can be in any inverse gauge, which fulfills Λ = 0. One sees from this, that the field strengths remain unequivocally unchanged. In general one has only gauge invariant values of R (energy, impulse, etc.) having obvious expectation values: R (ψ, Rψ) = (ψ 0, Rψ 0 ). A typical example which is not gauge invariant and therefore does not have an obvious value is also the operator P {A µ (x), A ν (y)}. (P signifies the temporal order of the product, (i.e., time ordered product.)) The vacuum expectation value comes e.g. before the calculation of the collision between two electrons: U = 1 2 P {j µ (x)j ν (y)} P {A µ (x)a ν (y)} 0 d4 x d 4 y. (2.23) One uses with it the vacuum state ψ = ψ 0 + c 1 ψ 1 (compare with (2.24)), so the c 1 dependent additions are eliminated through the condition for the four-current j µ = 0, i.e., the outcome remains unequivocally unchanged. Out of (2.16) follows the compatibility of the Lorentz condition (2.11) and the normalizations (2.20) with the vacuum definition (1.1). One now gets the state, in which no photons exist: ψ 0 = µ δ Nµ,0, (2.24)

6 6 i.e., c n = 0 for n > 0, g 0 0 = δ N1,0δ N2,0. However, one uses the usual vacuum condition A + (x) ψ 0 = 0, (2.25) by which A represents the transverse part of A (i.e., both the m = 1 and m = 2 terms in (2.13)), so one gets a condition, that is (2.24) that only distinguishes definite admixtures of the form (2.17); i.e., both vacuum definitions are physically equivalent. (Translator s note: A is used here for the transverse part, instead of a special script A with a vector sign on top.) C. The Lorentz invariance. The Lorentz invariance of the stated quantization procedure can be inspected with the usual methods; it shows some differences only in the reality attributes. The submitted transformations are One forms out of the operators A µ the new values x µ(x) = a µν x ν. (2.26) A µ(x ) = a µν A ν ( x(x ) ) (2.27) (x(x ) signifies the inverse transformations to (2.26)), so this fulfills again the same relations (2.1, 2.2); they can be obtained through a similarity transformation of the old values: For infinitesimal transformations one obtains S explicitly: x 4=const A µ(x) = S 1 A µ (x)s. (2.28) a µν = δ µν + ε µν, ε µν = ε νµ (2.29) S = I + ε µν Θ µν. (2.30) (Summation over all indices.) With i(θ µν Θ νµ ) signifying the angular momentum operator; { ( Aµ Aσ A σ Θ µν (x 4 ) = A ν + x µ 1 )} A σ A σ δ x 4 x 4 x ν 2 x λ x 4ν d 3 x. (2.31) λ The integration stretches over the surface x 4 = const.; because of angular momentum conservation, however, Θ µν Θ νµ becomes independent of x 4. From (2.2) one gets the commutation relation (C.R.): [ ] Aµ (x, x 4 ), A x ν (y, x 4 ) = δ(x y) δ µν. (2.32) 4 So, one easily verifies, that now out of (2.28) with (2.30) and (2.31) follows A µ(x) = A µ (x) + ε µν A ν (x) + ε σλ x σ A µ (x) x λ. This is correct with (2.27) for the infinitesimal transformations. Now, however, one must note that S because of (2.3) and the imaginary character of ε k4 = ε 4k is no longer unitary. For it follows now out of (2.30, 31) because of (2.5) and (2.3): i.e., the transformation still gives the norm, or general scalar product invariant: S η = η S 1 ; (2.33) ψ = S ψ (2.34) (ψ, ψ) = (ψ, ψ ). (2.35)

7 7 This is necessary in the framework of this theory and is valid for all transformations which come into consideration. (Compare with (2.20, 22).) Through the transformations (2.34), the states now become transformed in the right manner in the expectation values A µ : With (2.27, 28) and (2.33) it follows namely that A µ(x ) (ψ, A µ ψ ( ) = a µν A ν x(x ) ). (2.36) The reversal of the time-axis x r = x r, x 4 = x 4 can be represented here through the transformation ψ (N µ,k ) = ψ (N µ, k ) (2.37) (i.e., for real representation of the operators a, a in (2.4).) The key to this section has been, that for the operator A 0 relationship of the here stated values the following: introduced by Gupta [6] one obtains through the A 0 = ηa 4. (2.38) It is valid then that, because of (2.5) and (2.6), one obtains A 2 0 = A 2 4. And (2.38) also establishes for A 0 the right C.R., so the expectation value becomes real: A 0 (ψ, A 0 ψ) = (ψ, A 4 ψ). III. THE INTERACTION WITH ELECTRONS. In order to describe the interaction, we use the Interaction Representation with flat surface t = const. The time dependence of the state ψ(t) is then given by i ψ { } = Hψ A µ (x) j µ (x) d 3 x ψ(t). (3.1) x 4 =it With it one uses the operators A µ defined in the past sections, while j µ represents the four-current of the Dirac field without interaction: j 4 = iρ, j µ = 0, [j µ, η] = 0. (3.2) The normalization of the state is again given through N = (ψ, ψ) = 1. (3.3) Because of (2.3) and the antihermitian character of j 4, H is not hermitian, for now one has: H η = η H. (3.4) This is necessary, however, for the temporal conservation of the normalizations (3.3). Accordingly, the expectation values are given through A µ (x, t) = ( ψ (t), A µ (x, t)ψ(t) ). (3.5) For the Lorentz condition one now writes, in generalization of (2.11) in the sense of an initial condition, for ψ(t 0 ): abbreviated as: { A + µ (x ) + x µ (x 4 =it 0 ) } D + (x x) ρ(x) d 3 x ψ(t 0 ) = 0, (3.6) Ω + (x ; t 0 ) ψ(t 0 ) = 0.

8 8 With it is D + = 1 2 (D + id1 ), D = 1 2 (D id1 ) = D +. (3.7) (The sign of D + corresponds to the appropriate one of D; it is therefore opposite and changed from the work of Schwinger loc. cit. For D compare with W. Pauli, Rev. Mod. Phys. 13, 203 (1941).) One therefore has to replace, in the usual auxiliary condition, again A with A + as well as D with D +. (3.6) is to be fulfilled for all space-times x ; because of it suffices also to demand that (Ω + ) x 4 =it 0 ψ(t 0 ) = 0 and Ω + (x ; t 0 ) = 0 ( ) x Ω + ψ(t 0 ) = 0. 4 x 4 =it 0 With the usual help of the equation of motion (3.1), it follows that the initial condition (3.6) is valid for all times t: This yields out of the identity: for which the C.R. Ω + (x ; t) Ω + (x ; t) ψ(t) = 0. (3.8) + i [H(t), Ω + (x ; t)] = 0, i [A + µ (x), A ν (y)] = δ µν D + (x y) (3.9) is to be heeded. For through the expectation values defined by (3.5), the classical equations are now valid again: One defines the operators Ω (respectively Ω ) corresponding to (3.6) with A µ and D (respectively A µ and D ) put in, so out of (3.8) with (3.7) and (2.5) follows first of all: So that because of Ω = Ω + + Ω is yielded (ψ (t), Ω(x ; t) ψ(t) ) = 0. ψ (t) Ω (x ; t) = 0. (3.10) One puts herein x 4 = it, so because of the vanishing of the D-function outside of the light-cone, one gets ( ) A µ ψ Aµ (x) (t) ψ(t) = 0. (3.11) x 4 =it Using (3.1) and (3.4), in connection with the C.R. (2.32), yields further for x 4 = it x 4 A µ (x) = A µ x 4 and 2 x 2 4 A µ = 2 A µ x 2 4 together with the corresponding obvious relations for the spatial derivatives, it follows then with (2.1) In order to get the right energy tensor, it is essential that j µ ; A µ = 0 and A µ = j µ. (3.12) (ψ, Ω 2 ψ) = 0 is also fulfilled; (we owe this remark to Prof. Belinfante at Princeton.) This equation follows directly out of (3.8), (3.10) and the C.R. (One notes in addition to this [Ω + (x ; t), Ω (x ; t)] = 0. D + = 0 and [ρ(x, t), ρ(y, t)] = 0.) So this has assured the connection with the classical theory. The Lorentz invariance follows in the conventional manner (one notes the remarks of section of 2, C); moreover the generalization of (3.1) and (3.6) to so-called curved surfaces is obvious.

9 9 IV. THE ELIMINATION OF THE LONGITUDINAL PART. The here given theory differs from the usual formulation through the treatment of the scalar photons and in the wording of the Lorentz condition. For many practical problems, as well as first of all for the definition of the vacuum, one usually uses a formulation in which the longitudinal and scalar photons are replaced through the Coulomb interaction ( reduced theory ). It should now be shown, that also in the new formulation this elimination is possible, with the result that one thereby returns directly to the old, reduced theory; i.e., in a physical sense, (the behavior of the transverse photons), delivers the same outcomes in the new method. In the performance of a suitable transformation, one uses: with ψ(t) = S(t) ψ (t) (4.1) S(t) = e ig(t), (4.2) by which G(t) = Λ(x) ρ(x) d 3 x. (4.3) x 4 =it The longitudinal potential Λ is obtained through or Λ(x, t) 2 Λ(x, t) = A(x, t), (4.4) A(x, t) Λ(x, t) = 4π x x d3 x. (4.5) (Translator s notes: The operator div is replaced by. The definition of the differential operator has been added to (4.4). Commas are added to commutators where they are omitted. The operator grad is replaced by. Three-vector inner products like (A j) are replaced by A j. Finally, there was no equation (4.11) in the original?!) The equation of motion (3.1) results in now for ψ (t): i ψ In the calculation of individual terms, we carry out the development: = S 1 H S ψ i (S 1 ddt ) S ψ. (4.6) S 1 H S = H + i [G, H] + (4.7) S 1 d dg S = i dt dt + 1 [ G, dg ] + (4.8) 2 dt Here the second term results in the Coulomb interaction: first of all one gets from the C.R. (2.32) by operating with (4.5) on both sides: [ ] Λ(x, t) i, Λ(x 1, t) = 4π x x. (4.9) Furthermore observe that from this, especially because of (3.2), also follows [ ρ(x, t), [j µ (x, t), j ν (x, t)] = 0; (4.10) ] ρ(x, t) = 0. (4.12)

10 10 So that one gets with (4.3): i 2 [ G, dg ] = 1 ρ(x, t) ρ(x, t) dt 2 4π x x d 3 x d 3 x H Coul. (4.13) Further, it also follows, that the series (4.8) breaks off at this term; the first term transforms itself with (3.2): dg { Λ(x, t) dt = j x k + Λ } k ρ d 3 x. (4.14) The first commutator in (4.7) already vanishes (compare with (4.10) and (2.2)). Now, in (4.6), allowing the two terms to combine through the introduction of the transverse field A : So that one has the equation of motion (A 4 = iv ) A = A Λ (4.15) ( A = 0 because of (4.4)). i ψ = H ψ { ( A j d 3 x + H Coul. + V + Λ ) } ρ d 3 x ψ (4.16) (H trans. + H Coul. + H long. )ψ. Herein the first two terms give the well-known Hamiltonian operator of the reduced theory; the last term is eliminated now with help of the Lorentz condition (it is at this point that the calculation becomes essentially different from the usual theory). This is now: with With the help of C.R. (3.9) one gets first of all: Ω + (x ; t) ψ (t) = 0 (4.17) Ω + = S 1 Ω + S = Ω + + i [G, Ω + ] + (4.18) i [Λ(x), A + (x )] = D + (x x); (4.19) so that the commutator results in i [G(t), Ω + (x ; t)] = D + (x x) ρ(x) d 3 x, (4.20) x 4 =it i.e., in (4.18) it directly cancels the second part of Ω + (compare with (3.6)); the series (4.18) breaks off at the second term. The Lorentz condition is just the transformation: A + µ (x) ψ = 0 or ( V + + A )ψ + = 0. (4.21) One now has therefore the auxiliary condition of the theory without interaction; so the realizability of (3.6) is also proved. One could now apply the interpretation given in section II; with it one regards, that through the transformations (4.1) the normalizations remain as: (ψ, ψ ) = 1. (4.22) Now, to inspect that the last term in (4.16) could be left out, we note first of all, that the relations (2.1), (4.4), (4.15) are valid also for the corresponding positive frequency parts alone: A + = 2 Λ +, Λ + = 0; (4.23)

11 11 i.e., So, (4.21) can be written as: ( V + A + = 2 2 Λ+. (4.24) ) Λ+ ψ = 0. (4.25) So that the explicit representation presently shows, however, that this is equivalent with (V + + Λ+ ) ψ = 0. (4.26) Here, therefore, one finds directly the positive frequency part of the corresponding expression in H long.. One calculates now the contribution of H long. in (4.16) due to the time variation of the amplitude w(t): w(t) = ( ϕ, ψ (t) ) (4.27) (compare with (2.22)), by which ϕ signifies an arbitrary state, that however has to have fulfilled (4.26), i.e., also ( ) ϕ V + Λ = 0. (4.28) Now one has i dw dt = ( ϕ, H ψ (t) ) ( compare with (4.16) ). Here, however, H long. gives no contribution to the result obtained from the division into the more positive and more negative frequency parts: ( ( ϕ, V + dλ ) )ρ ψ = 0 (4.29) dt (compare with (4.28) and (4.26)). The same is also valid for all higher order derivatives, if one notes still, that H long. commutes with all different terms of H. One has, according to (2.1), (4.4), and (4.15), [A, Λ] = 0. (4.30) One sees from this, that H long. only changes the admixtures of the states ψ. (4.16) is therefore equivalent with i ψ = (H trans. + H Coul. )ψ. (4.31) One can now leave out the quantum numbers N 3 and N 4 in ψ (the system now remains continually in the state of the form ψ 0 (compare with (2.16)). The normalization (4.22) then means (ψ, ψ ) = 1. (4.32) So that, the reduced theory is recovered. This is also recovered directly through the quantization of a corresponding classical theory in a non-free-field manner. V. THE DEFINITION OF THE VACUUM. Now, one no longer has the usual definition of the vacuum from the reduced theory (no transverse photons). One investigates then, with the help of the inverse transformations (4.1), which condition results from the symmetrical theory. One has therefore for ψ the vacuum condition A + ψ 0 = 0. (5.1)

12 The quantum numbers N 3 and N 4 are retained, so one still has to use the Lorentz condition (4.25), as well. The second section implies, however, that one can apply the more restrictive conditions: V + ψ 0 = 0, (5.2) Λ + ψ 0 = 0. (5.3) There V + includes only absorption operators for scalar photons, while Λ + includes only absorption operators for longitudinal photons, this signifying only, that one is restricted to a condition of the form ϕ(0, 0) (compare with (2.16)). Now one uses the transformation (4.1) to go back to the symmetrical representation: Then, using (5.3) this becomes: or ψ 0 = S 1 ψ 0. (5.4) (S Λ + S 1 ) ψ 0 = 0 (5.5) { Λ + i [G, Λ + ] } ψ 0 = With the same methods used in section IV this results in: { } Λ + (x ) + F + (x x) ρ(x) d 3 x ψ 0 (t 0 ) = 0 (5.6) x 4 =it 0 with the kernel: F + (x x) = x 4 =x 4 D + (x x) 4π x x d3 x. (5.7) (Translator s note: The first vector x in the denominator of (5.7) was incorrectly written as a scalar x in the original. Also, the right curly bracket in (5.6) was missing.) Further, the conditions (5.1) and (5.2) remain unchanged because of Therefore, one still has to add to (5.6) both and Using (4.15) one also gets So that using (5.6), (5.8) and (5.9) yields { A + k (x ) + x k [Λ, V + ] = [Λ, A + ] = 0. A + ψ 0 = 0 (5.8) A + 4 ψ 0 = 0. (5.9) A + = A + + Λ+. } F + (x x) ρ(x) d 3 x ψ 0 (t 0 ) = 0 (5.10) x 4 =it 0 A + 4 (x ) ψ 0 (t 0 ) = 0 (5.11) for all x and k = 1, 2, 3 and where F has been defined through (5.7). Thereby, one has gained the strict vacuum definition of the symmetrical theory. In contrast to (1.2) one has a charge dependent addition; this is stated in a different connection already by Jauches and Coester [8]. One sees immediately, that this addition must exist by reason of the compatibility with the Lorentz condition (3.6): One differentiates namely (5.10, 11) with respect to the corresponding coordinates x µ and adds the four equations, so one directly gets (3.6). In this theory the vacuum definition is therefore nothing different than a tightening of the Lorentz condition. The physical meaning of the

13 addition is the following: Calculating the expectation values of the field strengths through the photon vacuum defined by (5.10, 11), gives one directly the electrostatic field of the charge distribution ρ (expectation value of the charge): E(x, t0 ) 0 = ρ(x, t 0 ) 4π x x d3 x. (5.12) One describes the significance of the additional term as follows: It puts into the symmetrical theory the instantaneous effect of the Coulomb fields, which explicitly appears in the shortened theory. The term is actually essential for processes that play a role in the finite time interval itself; one sees this e.g. easily with help of the calculating method of Dyson. In example (2.23), one then has to put in the finite time limits of integration ±T : it now uses (5.10, 11) instead of applying (1.2), so it already delivers the first approximation the second order contribution: U T = i +T T d 4 x (y 4 = it ) U 1 = i +T T d 3 y j k (x) F + (x y)ρ(y) i x k A µ (x) j µ (x) d 4 x, +T T d 4 x (y 4 =+it ) 13 d 3 y j k (x) F (x y)ρ(y), (5.13) x k (F is defined similarly to (5.7)). Incidentally, one also verifies easily, that the reduced theory delivers the same outcome: One uses the procedure of Kroll and Karplus, so one has to appraise the expression U = 1 2 +T T j k (x) j l (y) P { A k (x) A l (y) } 0 d4 x d 4 y (5.14) with the help of (5.1): this results in a straight partial integration of the additional term (5.13). Jauches and Coester have shown now, that U T for T = vanishes; actually one also has, following out of (5.10), that the additional term, which only describes the Coulomb field, delivers no contribution for t 0 = ± : It vanishes then for all finite x. In this case, therefore, one could calculate with the vacuum condition (1.2) for the charge free case calculation. One now sees the advantage of our method: (1.2) is realizable. Initial and final states could be used with the methods of the second section as the states of the field without interaction and without longitudinal and scalar photons. There the Lorentz condition is included in the vacuum condition, so it follows furthermore, that for virtual states one must not explicitly use the Lorentz condition, which is what Gupta has done in his examples, as well. This proves that in a trivial manner all could be generalized, so that if real transverse photons occur, they require no further explanation. I might heartily thank Professor W. Heitler for a lot of advice and valuable discussions. I thank Dr. S. N. Gupta and Prof. F. Coester very much for the opportunity of looking at their manuscripts before publication. Zürich, Theoretical seminar of the university. TRANSLATOR S ACKNOWLEDGMENTS I would like to thank Ted Klein, a retired German-born technical person living here locally in Sydney, Australia, for his help with several tricky words and sentences. I am also extremely grateful for his donation of a matched pair of technical German dictionaries that were authored and published by H. Wernicke under the sponsorship of Rohde & Schwarz. The detailed references for the dictionaries are: Harry Wernicke, Dictionary of Electronics, Communications, and Electrical Engineering, Volume I, English- German, (Publisher: H. Wernicke, Printed by Biedermann, Munich, Germany, 1977). Von Harry Wernicke, Lexikon der Elektronik, Nachrichten- und Elektrotechnik, Band II, Deutsch-Englisch, (Verlag: H. Wernicke, Hat gedruckt durch Gerber, München, Deutschland, 1973). Last, but not least, I would like to acknowledge the use of the nice computer software, German Assistant TM for Windows TM, which gave me a rough starting point for the translation. By incorporating many specialist scientific words from the above mentioned technical German dictionaries into the software s auxiliary dictionary, I was able to achieve a reasonable translation.

14 It should be noted that in translating this journal article, I did not try to change the formal technical German structure of the sentences too much. Naturally, a reordering of some of the words of a sentence was inevitable since the software would in many cases translate word for word. Also, some things are just said differently in the two languages. However, I tried to remove as few words as possible, while at the same time I added words only reluctantly. That is, I resisted a complete rewording of what was written. Instead, I tried to maintain the formal German flavor of the original ground breaking paper. On the other hand, to improve readability, I changed some notations to the appropriate modern ones, as noted in the various translator s notes. Dale Alan Woodside, Department of Physics, Macquarie University Sydney, New South Wales 2109, Australia; Electronic mail: dalew@physics.mq.edu.au ; August 1995, September [1] Fermi, Rev. Mod. Phys. 4, 131 (1932). [2] Ma, Phys. Rev. 75, 535 (1949). Belinfante, Phys. Rev. 76, 226 (1949). [3] Schwinger, Phys. Rev. 74, 1439 (1948) and 75, 651 (1949). [4] Karplus and Kroll, Phys. Rev. 77, 536 (1950). [5] Dyson, Phys. Rev. 77, 420 (1950). [6] Gupta, Proc. Phys. Soc. 63, 681 (1950). [7] Dirac, Comm. Dublin Inst. Advanced Studies, A. No. 1, Pauli, Rev. Mod. Phys. 15, 175 (1943). [8] Jauch and Coester, Phys. Rev. 78, 149 (1950).