Repeated application of (12) gives us (11) and repeated application of the corresponding law for g.t.f.’s will enable us in a similar way to connect the g.t.f. for any region with the g.t.f.’s for the very small subregions into which that region may be divided. This connection will contain the quantum analogue of the action principle applied to fields.
The square of the modulus of the transformation function (qt|qT) can be interpreted as the probability of an observation of the coordinates at the later time t giving the result qt for a state for which an observation of the coordinates at the earlier time T is certain to give the result qT. A corresponding meaning for the square of the modulus of the g.t.f. will exist only when the g.t.f. refers to a region of space-time bounded by two separate (three-dimensional) surfaces, each extending to infinity in the space directions and lying entirely outside any light-cone having its vertex on the surface. The square of the modulus of the g. t. f. then gives the probability of the coordinates having specified values at all points on the later surface for a state for which they are given to have definite values at all points on the earlier surface. The g.t.f. may in this case be considered as a transformation function connecting the values of the coordinates and momenta on one of the surfaces with their values on the other.
We can alternatively consider |(qt |qT)|2 as giving the relative a priori probability of any state yielding the results qT and qt when observations of the q’s are made at time T and at time t (account being taken of the fact that the earlier observation will alter the state and affect the later observation). Correspondingly we can consider the square of the modulus of the g.t.f. for any space-time region as giving the relative a priori probability of specified results being obtained when observations are made of the coordinates at all points on the boundary. This interpretation is more general than the preceding one, since it does not require a restriction on the shape of the space-time region.
ST JOHN’S COLLEGE, CAMBRIDGE.
ON QUANTUM ELECTRODYNAMICS
BY
PAUL A.M. DIRAC, V.A. FLOCK, AND BORIS PODOLSKY
Proceedings of the Royal Society of London, Series A, Vol. 114, p. 243 (1927)
In the first part of this paper the equivalence of the new form of relativistic Quantum Mechanicsko to that of Heisenberg and Paulikp is proved in a new way which has the advantage of showing their physical relation and serves to suggest further development considered in the second part.
PART I. EQUIVALENCE OF DIRAC’S AND HEISENBERG PAULI’S THEORIES.
§ 1. Recently Rosenfeld showedkq that the new form of relativistic Quantum Mechanicsko is equivalent to that of Heisenberg and Pauli.kp Rosenfeld’s proof is, however, obscure and. does not bring out some features of the relation of the two theories. To assist in the further development of the theory we give here a simplified proof of the equivalence.
Consider a system, with a Hamiltonian H, consisting of two parts A and B with their respective Hamiltonians Ha and Hb and the interaction V. We have
(1)
where
and T is the time for the entire system. The wave function for the entire system will satisfy the equationkr
(2)
and will be a function of the variables indicated.
Now, upon performing the canonical transformation
(3)
by which dynamical variables, say F, transform as follows
(4)
Eq. (2) takes the form
(5)
Since Ha commutes with Hb, . On the other hand, since the functional relation between variables is not disturbed by the canonical transformation (3), V kr is the same function of the transformed variables pkr, qkr as V is of p, q. But pa and qa commute with Hb so that , . Therefore
(6)
where
(7)
It will be shown in § 7, after suitable notation is developed, that Eqs. (7) are equivalent to
(8)
where t is the separate time of the part B.
These, however, are just the equations of motion for the part B alone, unperturbed by the presence of part A.
§ 2. Now let part B correspond to the field and part A to the particles present. Eqs. (8) must then be equivalent to Maxwell’s equations for empty space. Eq. (2) is then the wave equation of Heisenberg-Pauli’s theory, while Eq. (5), in which the perturbation is expressed in terms of potentials corresponding to empty space, is the wave equation of the new theory. Thus, this theory corresponds to treating separately a part of the system, which is in some problems more convenient.ks
Now, Ha can be represented as a sum of the Hamiltonians for the separate particles. The interaction between the particles is not included in Ha for this is taken to be the result of interaction between the particles and the field. Similarly, V is the sum of interactions between the field and the particles. Thus, we may write
(9)
where rs are the coordinates of the s-th particle and n is the number of particles.
Eq. (5) takes the form
(10)
J stands for the variables describing the field. Besides the common time T and the field time t an individual time ts = t1, t2, . . . tn is introduced for each particle. Eq. (10) is satisfied by the common solution of the set of equations
(11)
andψks =ψks (r1 r2 . . . rn; t1 t2 . . . tn; J), when all the t’s are put equal to the common time T.
Now, Eqs. (11) are the equations of Dirac’s theory. They are obviously relativistically invariant and form a generalization of Eq. (10). This obvious relativistic invariance is achieved by the introduction of separate time for each particle.
§ 3. For further development we shall need some formulas of quantization of electromagnetic fields and shall use amplitudes F(k) and F+(k) are introduced by the equation
(17)
where r = (xyz) is the position vector, k = (kx ky kz) is the wave vector having the magnitude |k| = 2π/λ, dk = dkx dky dkz, the integration being performed for each component of k from −∞ to ∞. In terms of the amplitudes equations of motion can be written
(18)
the other two equations being algebraic consequences of these.
The commutation rules for the potentials are
(19)
all other combinations of amplitudes commuting.
PART II. THE MAXWELLIAN CASE.
§ 4. For the Maxwellian case the following additional considerations are necessary. In obtaining the field variables, besides the regular equations of motion of the electromagnetic field one must use the additional condition P0 = 0, or −cP0 = div . This condition cannot be regarded as aquantum mechanical equation, but rather as a condition on permissible ψ functions. This can be seen, for example, from the fact that, when regarded as a quantum mechanical equation, div contradicts the commutation rules. Thus, only those ψ’s should be regarded as physically permissible which satisfy the condition
(20)
for this purpose some formulas obtained by Fock and Podolsky.kt Starting with the Lagrangian function
(12)
taking as coordinates (Q0 Q1 Q2 Q3) the potentials (Φ A1 A2 A3), and retaining the usual relations
(13)
one obtains
(14)
and the Hamiltonian
(15)
The equations of motion areku
(16)
On elimination of P and P0, Eqs. (16) give the D’Alembert equations for the potentials Φ and A. To obtain Maxwell’s equation for empty space one must set P0 = 0. The quantization rules are expressed in terms of the amplitudes of the Fourier’s integral. Thus, for every F = F(xyst),
Condition (20), expressed in terms of amplitudes by the use of Eq. (18), takes the form
(20’)
To these must, of course, be added the wave equation
(21)
where Hb is the Hamiltonian for the field
(22)
as in 1. c.
If a number of equations Aψ = 0, Bψ = 0, etc., are simultaneousl
y satisfied, then ABψ = 0, BAψ = 0, etc.; and therefore (AB − BA)ψ = 0, etc. All such new equations must be consequences of the old, i.e. must not give any new conditions on ψ . This may be regarded as, a test of consistency of the original equations. Applying this to our Eqs. (20’) and (21) we have
(23)
since A’s commute with Φ’s. Applying now the commutation rules of Eq. (19), we obtain
(24)
Eq. (24) is satisfied in consequence of quantum-mechanical equations, hence
is not a condition on ψ . Thus, conditions (20’) are consistent. Since P0(k) and commute with ∂/∂t, to test the consistence of condition (20) with (21) one must test the condition
(25)
However, since Ṗ 0 = (i|h)(Hb P0 − P0 Hb), Eq. (25) takes the form Ṗ0 ψ = 0, or in Fourier’s componentsṖ0 (k) ψ = −ic|k|P0 (k)ψ = 0
and
But these are just the conditions (20’). Thus, conditions (20) and (21) are consistent.
§ 5. The extra condition of Eq. (20) is not an equation of motion, but is a “constraint” imposed on the initial coordinates and velocities, which the equations of motion then preserve for all time. The existence of this constraint for the Maxwellian case is the reason for the additional considerations, mentioned at the beginning of § 4. It turns out that we must modify this constraint when particles are present, in order to get something which the equations of motion will preserve for all time.
The conditions (20’) as they stand, when applied to ψ, are not consistent with Eqs. (11). It is, however, not difficult to see that they can be replaced by a somewhat different set of conditionskv
(26’)
where
(27’)
Terms in C(k) not contained in—cP0(k) are functions of the coordinates and the time for the particles. They commute with Hb —ih ∂/∂t, with P0(k) and with each other. Therefore Eqs. (26’) are consistent with each other and with Eq. (21). It remains to show that Eqs. (26’) are consistent with Eqs. (11). In fact C(k) and C+(k) commute with Rs—ih ∂/∂ts . We shall show this for C(k).
Designating, in the usual way, AB—BA as [A, B], we see that it is sufficient to show that
(28)
and
(29)
By considering the form of C(k), these become respectively
(30)
and
(31)
Now
by Eq. (17) and because A(k) commutes with A(k’). Using the commutation formulas and performing the integration it becomes
(32)
On the other hand
(33)
Thus, Eq. (30) is satisfied. Similarly Eq. (31) is satisfied because
(34)
and
(35)
Thus, conditions (26’) satisfy all the requirements of consistence. It can be shown that these requirements determine C(k) uniquely up to an additive constant, if it is taken to have the form i[k·A(k) − |k|Φ(k)] + f(rsts).
§ 6. We shall now show that the introduction of separate time for the field and for each particle allows the use of the entire vacuum electrodynamics of § 3 and 1. c., except for the change discussed in § 5. In fact, we shall show that Maxwell’s equations of electrodynamics, in which enter current or charge densities, become conditions on ψ function.
For convenience we collect together our fundamental equations.
The equations of vacuum electrodynamics are
(13)
(36)
The wave equations are
(11)
The additional conditions on ψ function are
(26’)
where
(27’)
We transform the last two equations by passing from the amplitudes C(k) and C+(k) to C(r, t) by means of Eq. (17). Thus we obtain
(26)
and
(27)
where X and Xs are four dimensional vectors X = (xyzt), Xs = (xs ys zs ts) and Δ is the so-called invariant delta functionkw
(37)
From Eqs. (13) follows immediately
(38)
so that these remain as quantum-mechanical equations. Using Eqs. (13) and (36) and condition (26) we obtain by direct calculation
(39)
and
(40)
Now, let us consider what becomes of these equations when we put t = t1 = t2 = ... = tn = T, which is implied in Maxwell’s equations and which we shall write for short as ts = T.
For any quantity f = f (tt1 t2 . . . tn )
(41)
and for each of the n derivatives ∂/∂ts we have an equation of motion
(42)
If we put f = A(r, t) or f = Φ (r, t), then, since both commute with Rs , ∂f/∂ts = 0 and we get
(43)
It follows that
(44)
so that the form of the connection between the field and the potentials is preserved. Remembering that for t = ts we have Δ(X − Xs) = 0 and hence grad Δ (X − Xs) = 0, and using Eqs. (26), (39) and (40) we obtain
(45)
(46)
and
(47)
For further reduction of Eq. (46) we must use Eqs. (41) and (42), from which follows
(48)
and is easily calculated, because the only term in Rs which does not commute with is −εs αs ·A(rs ts), and is the momentum conjugate to A. In this way we obtain
(49)
For the reduction of Eq. (47) we need only rememberkx that
(50)
Thus, Eqs. (46) and (47) become
(51)
and
(52)
which are just the remaining Maxwell’s equations appearing as conditions on ψ. Eq. (52) is the additional condition of Heisenberg-Pauli’s theory.
§ 7. We shall now derive Eq. (8) of § 1. For this we need to recall that the transformation (7) is a canonical transformation which preserves the form of the algebraic relations between the variables, as well as the equations of motion. These will be, in the exact notation now developed,
(53)
As we have seen in the discussion following Eq. (5)
(54)
and qb and pb commute with Ha, und commute with and hence with Ha. Therefore Eqs. (53) become
(55)
On the other hand, we have from Eqs. (41) and (42)
(56)
Now the only term in Rs which does not commute with and is so that
(57)
Since , Eqs. (56) become
(58)
Comparison of Eqs. (55) with (58) finally gives
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