The properties of wave tensorsrspa.royalsocietypublishing.org/content/royprsa/133/821/...311 The Properties of ave Tensors. By Sir Arthur Eddington, F.R.S. (Received July 4, 1931.)

311

The Properties of ave Tensors.By Sir Arthur E ddington, F.R.S.

(Received July 4, 1931.)

1. In wave-mechanics there occur in addition to the ordinary physical vectors and tensors the four-valued quantities 4> 4* introduced by Dirac. In this combination certain relations of invariance in regard to transformation appear which had escaped the ordinary tensor calculus. If we call the new type of quantity involving 4 and its combinations a ^-tensor, the position is that we cannot reach ^-tensors from the calculus of ordinary space tensors, but we can reach space tensors from a calculus of ^-tensors. I have shown that although 4-vectors cannot be expressed in terms of space vectors, mixed 4-tensors can be immediately resolved into space vectors.f

The present paper contains a simplification and systematisation of my earlier work on the ^-tensor calculus as well as a number of new results. The particular point round which the new results centre is as follows. The 4~ tensors occurring in wave-mechanics are the product of two 4-vectors (4, 4*)> and apparently the primary reason for introducing the unobservable quantity 4 rather than working with physical tensors throughout is to impose this condition. I therefore examine the question: If we impose the condition that a wrave tensor is the product of two 4'Vectors, what is the corresponding restriction on the physical tensors equivalent to it 1 The answer is (§7) that whatever is described by two 4-vectors 4> 4* can equivalently be described by two space vectors of equal length at right angles to one another. One of these is the momentum vector; the other (generally ignored in current investigations) presumably represents positional relations (co-ordinates or distance); or rather I would regard it as the source of positional relations, which can only become explicit in more complicated developments involving many particles. The ordinary wave equation for one particle is obtained as an identity.

This investigation was made in connection with the theory of the charge and masses of electrons and protons which I have been trying to develop ; but for the most part I refrain from comment on this application, as I am not yet ready to treat it definitively. I do, however, deal at some length with the

t The earlier papers are ‘ Proc. Roy. Soc.,’ A, vol. 121, p. 524; vol. 122, p. 358 ; vol. 126, p. 696 (1928-30). These are hereinafter quoted as I, II and III.

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312 Sir A rth u r Eddington.

question why the complete five-dimensional relativity admitted in the analytical formulae and believed to hold good in general dynamical theory is cut down to four dimensions in applications to practical problems in our actual universe.

I have found it advisable to change my definition of the matrices EM, so that now EM2 = — 1, instead of -f 1 in my former papers.

2. We take four anticommuting matrices Et, E 2, E3, E4, which satisfy

Em2 = — 1 EmE„ = - E„EM. (2.1)Let

E5 = iE1E 2E 3E4. (2.2)

Then it is easily verified that E5 also satisfies (2.1). We introduce 10 more matrices by the definition

Em„ = EmE„ = — E„M ({x, v = 1, 2, 3, 4, 5 ; p ^ v)

The properties of these 15 matrices can be exhibited more symmetrically by introducing an alternative designation of the original 5 matrices, viz.,

Enii = E„

so that they may all be treated as double-suffixed and antisymmetrical. following rules of multiplication are found to hold

E 2 = — 1EmvEmo. = — EM<rEMV = Ew EM(,E„T = E JE ,, = i ^EAp

1The

(2.3)

where g, v, a, t, X, p form any permutation of the suffixes 0, 1, 2, 3, 4, 5. Matrices anticommute or commute according as they have or have not one suffix in common.

The pentads, i.e.,sets of five mutally anticommuting matrices, are obtained by fixing one of the two suffixes, e.g., E30, E31, E 32, E34, E33. Each matrix is a member of two pentads corresponding to its two suffixes.

We introduce a sixteenth matrix E i6 = i, i.e., i times the unit matrix. Then E1? E 2, . . , E16 is called a complete set or in physical applications a frame of matrices. Any fourfold matrix can be expressed in one and only one way as a linear function of E1} E 2, . . , E16.

Let Em' be another complete set in one-to-one correspondence wTith EM and fulfilling the same equations. (We now use a single suffix notation 1, 2, . . , 16.) I t can be shown that there exists a transformation.*

E f = P 'EmP (2.4)* Cf. G. Temple, * Proc. Roy. Soc.,’ A, vol. 127, p. 342 (1930).



Properties o f Wave Tensors. 313

where16 16

P = P E mEm' P' = iS E M'EM (2.5)i i

and PP' = P 'P = 1. Since by (2.5) P is a matrix, (2.4) is the ordinary transformation of a mixed tensor (III, equation (3.13)). Thus all possible complete sets of matrices are obtained by applying the general transformation of a mixed tensor to an original set EM. We therefore take the EM to be mixed tensors, and thereby avoid reference to any particular set.

I denote the contracted tensor, i.e., the diagonal sum of the matrix, by (EM). This is an invariant for tensor transformations. I have found a complete set consisting of four-point matrices in I, §§ 7, 8 ; in these it is evident by inspection that

' {Em} = 0 except {E16} = 4 (2.6)

This invariant equation must be true for all complete sets.3. When a matrix T is expressed as a linear function of a complete set

16T = 2 « ,E , (3.1)

1we call the numerical coefficients tM the matrix components of T. The formulafor determining was obtained in my earlier paper (I, § 14), but the following is a simpler derivation. Multiply both sides of (3.1) by E„ and contract

{TE„} = S U E m E J.

For the 16 values of jjl, EmE„ gives the original matrices EM in a different order by (2.3), except that there may be a factor —1, or By (2.6) all except one vanish on contraction. The exception is {E,,EJ = {—1} = — 4. Hence {TE„} = a - 4 ) , or

<„ = - H T E „}. (3.2)

An important case is when the matrix is the product of two vectors, be the matrix components of J where

By (3.2)j = 44*

= Em}.

Let

(3.3)

Inserting row-and-column suffixes 44*Em means (44*)aie(EM)/sy = 4a4'/s*(E/*W To contract we set y = a, so that {44*E,u} means 4«4'0*(EM)j8a — 4/3*(E(U)18a4«* The suffixes, being now in chain order, can be omitted according to the usual convention of matrix notation. Hence

in = — H * E^ - (3.4)



314 Sir A rth u r E ddington .

It may be explained that the asterisk is used to distinguish an “ initial vector,” i.e., one which can start but cannot close a chain-product. With this understanding there is no ambiguity as to whether inner or outer multiplication is intended. Thus 4*4 is an inner product, 44* is an outer product. I use the summation convention for row-and-column suffixes, but not for any other suffixes.

4. When an exponential involves matrices it is understood to be defined by the exponential series. It follows from the definition that since EM2 = — 1

— cos 0 -f- sin 0 . EM. (4.1)

Exponentials involving two or more non-commuting non-infinitesimal matrices do not as a rule admit of simple interpretation.

Considering the general transformation of a mixed tensor

T' = j Y , (4.2)where q is any matrix and q' its reciprocal, let q — eiEl'9; then by (3.1)

S E X = (S E X e ~ iJi"= £ a EM*M+ (4.3)

where denotes summation over the eight terms EM which commute with E12 and over the eight terms which anticommute. The eight terms which anticommute are the remaining members of the two pentads which contain E12. As an example of the latter take E ^ -f- E 2£2 which contributes to (4.3).

(E^j -f- E 2£2)(cos 0 — sin 0 . E12)= (E^j -f- E 2£2) cos 0 — (— E 2£x -f Ex£2) sin 0

= .E j (t± cos 0 — t2 sin 0) -j- E 2 sin 0 + ^2 cos ®)-Since the EM cannot satisfy any linear identity'}' we may equate coefficients

of Em on both sides of (4.3). Hence

A = tx cos 0 — 1 t2 sin 0 t sin 0 -f- cos 0. (4.4)

Three other pairs are similarly rotated. The remaining eight terms (comprised in 2 a) are unchanged.

Consider five components tv t2, t3, tA, t5 corresponding to a pentad. The above transformation leaves t3, f4, t5 unchanged ; so that, regarding (tv t3, h ’ h) as an ordinary vector in five-dimensional space, the transformation q = ekKiQ gives a simple rotation of the axes through an angle 0 in the plane xv x2. Similar rotations in all 10 co-ordinate planes are obtained by using the appropriate matrix in q. We therefore call tv t3, t4, th a space vector, since all

t See I, § 3. A general proof has been given by Temple, loc. cit.



Properties of Wave Tensors. 315

possible transformations which a space vector possesses in the elementary- vector theory are included in the general tensor transformation (4.2). This would not be true of components which do not form a pentad, a mutual rotation of txand t23 does not correspond to any transformation included in (4.2). I t is easily shown that when the five-dimensional space is rotated the components t12, t13, . ti5 transform as a 10-vector (analogous to a 6-vector in four dimensions); t1G is always an invariant. Thus the tensor T is replaceable by a space vector, a 10-vector and an invariant in a five-dimensional space.

5. In practice we limit space vectors to four dimensions and this leads to a somewhat different grouping of the terms The reason for this distinction of the fifth dimension (proper time) from the other four is considered in § 13.

In four dimensions the 16 components constitute

(a) Two invariants t5 and tVi ^

(b) Two space vectors tv t 2, t3, txand t2-, t35, t45 j>. (5.1)

(c) A 6 -vector ^35 *31, *12> *14, *24’ *34 JThat is to say, when any of the six tensor transformations which correspond to the possible rotations of four-dimensional space are applied, each of these behaves (independently of the others) as a space tensor of the class stated. The two vectors in (6) are of indistinguishable character so long as we confine the transformations to four-dimensional rotations.

I have shown that when the EM are four-point matrices three members of any pentad are imaginary and two are real.* I have since been able to show that this is true generally, but the proof is too long to insert here. (I admit only matrices which are wholly real or wholly imaginary.) We shall take Ej, E 2, E 3 to be the imaginary matrices and E4, E5 real. Assuming that for real phenomena the primitive tensor T is real, we see by (3.2) that the two space vectors have components

tvt2, t3 imaginary, t4real "1. y • (°-2)

*15, *25’ *35 imaginary, real j

Thus in the real world one dimension will be different from the other three in the same way that time is differentiated from space, viz., imaginary time must be associated with real space in order to obtain an isotropic rotatable continuum of four dimensions.

If we had taken E- to be one of the imaginary matrices we should have been

* I, § 8 ; since the EM were there defined so that E^2 = 1, real and imaginary matrices are interchanged.



316 Sir A rth u r E dding ton .

left with two real and two imaginary matrices to form a continuum with two space-like and two time-like dimensions. I cannot say why this alternative is rejected, but it evidently depends on the theory of the singling out of the fifth dimension discussed in § 13.

As in Paper III, I will call the original tensors of our calculus (such as T and Em) ^-tensors in order to distinguish them from the space vectors and tensors which are derived from them in the way here described.

6. Consider a ^-tensor T with the following special properties :—

(a) I t is the product of two ^-vectors.(b) Its diagonal sum vanishes.

Both properties persist when T undergoes any tensor transformation; they are therefore an absolute specialisation not referring to any particular frame of matrices. I call a tensor which satisfies these two conditions a wave tensor and denote it by J.

The question now arises, What is the corresponding specialisation of the space tensors (§ 5 (a),( b), (c) ) into which T can be resolved ? So far ascondition (6) is concerned the answer is immediate ; by (3.2) it requires that

i i . = 0. (6.1)

We shall defer imposing this condition since it is easily introduced at any stage,- and find the conditions imposed by (a).

I have shown (I, § 8) that the following matrices constitute a pentad

0 i 0 0 0 0 0 0 0 0 i

i 0 0 0 0 —i 0 0 0 0 —i 0

0 0 0 i 0 0 i 0 0 -i 0 0

0 0 i 0 0 0 0 —i i 0 0 0

0 1 0 0 0 0 0 1

- 1 0 0 0 0 0 -1 0

0 0 0 —1 0 1 0 0

0 0 1 0 -1 0 0 0

(This is the pentad iSa, iDp, iS7I)v, SJD7, S^D ̂ in the notation of Paper I). Denoting them by Et, E 2, E 3, E4, E5, it is not difficult to prove by straightforward verification that, if is any set of four numbers,

(Ei4»)a(Ei^)j3 + (E24»)a(E24')/3 + "f~ (E440a(E4 )̂|8+ - (Ej.+ME,,*), - 0. (6.3>



Properties o f Wave Tensors. 317

By using the transformation (2.4) we can deduce that any other matrices forming a pentad satisfy the same identity, so that (6.3) is true generally.

Multiply (6.3) by any four-valued quantity (inner multiplication) and let

<H* = J = (6.4)Then since

*«*(E A)a = = - by (3.4)the result is

( j i^ i + ^ 2 ^ 2 +^3^3 + J 4E4 + J 5E5 — 0. (6.51)We find similarly

+*0i®i 4“ J2̂ 2 4" J/3E 3 4~ J4E4 4~ J5E5 416̂ 16) = 6 . (6.52)

We call (6.51) and (6.52) the wave equations. They are equations to determine the wave vectors <];> 41* (factors of the wave tensor) when certain components of the wave tensor are prescribed. They have an infinity of solutions since the wave tensor is not fully determined by the six components occurring in (6.51) and (6.52).

7. Multiply (6.51) by initial 41* and use (3.4) ; we obtain

j i 4 -j 22 4- i 32 4 -j 42 4 -is2 - i i e 2 = o. l7-1)Since any pentad can be used in (6.3) we have similarly

iis2 4- J252 4-J352 4- i452 4- Jh — Ji62 = 0- (7-2)Again multiply (6.51) by initial '|*E 5 ; this gives

4**( 4lEi5 J2®25 J 3E 35 ^4^45 ^le^s)^ == 0-Hence by (3.4)

JlJlo +J2J254“ ̂ 3̂ 35 4" — (7-3)

Let X = (jv j 2, j 3, j 4) and P = (j15, j 25, j 35, j 45) be the two space vectors contained in the wave tensor. By (7.3) they are at right angles and by (7.1) and (7.2) they have the same length R = \ / ( j 162 —j 2).

Given any two equal vectors X and P at right angles, it is always possible to find a wave-tensor containing them. For taking any solutions of (6.51) and (6.52), (j;']'* will contain the required vector X, but in general (j15, 5,

J 45) will be a vector P' different from the prescribed P. Since P' and P are of the same length (both equal to X) and at right angles to X, P' is transformed into P by a rotation of four-dimensional space about X which leaves X unchanged. Since rotations of space correspond to transformations of ^-tensors, we have only to make the corresponding transformation of <J>* in order to obtain the required wave tensor.



318 Sir A rth u r Eddington .

To prescribe P and X is equivalent to imposing six conditions, the eight components of the two vectors being reduced to six independent quantities by the conditions of equality and perpendicularity. A tensor of the form

contains seven independent constants, one constant being lost because, if ^ is multiplied by a number a and by b, only the product ah is involved. Hence only one further condition can be imposed on This condition maytake the form of assigning j ie,and as stated in our definition we take j u = 0 for a wave tensor.

Hence a wave tensor is completely specified by two equal space vectors at right angles to one another. Any physical system described by can alternatively be described by such a pair of space vectors.

8. The vector fP is generally called the momentum vector. We write

Jzo ’ ^45’ J 5) = P i ’ 2 h ’ P P Jl G 0* (8-1)

Using the other pentad E15, E 25, .. in (6.51), the momentum wave equation is

(®i5Pi + E 25̂ 2 + ^35^3 + b 45p4 + E5(3)h = 0 (8.21)t|j*(Ei5pi -f- E 25p 2 -f- E35p 3 + E45p4 + E5(3) = 0. (8.22)

Multiplying (8.21) by initial E5 and (8.22) by final E5, we obtainH^ = (E jft + E 2p2 + E3?3 + E4p4 - = 0 (8.31)

'!>*H' = r(® iP i + E 2p 2 + E + E4p4 + (3) = 0 ’ (8.32)

which will be taken as the standard form of the wave equation.By straightforward multiplication

HH' = H'H = — (ft2 + f t 2 + + f t 2 + (32) = 0 (8.4)

by (7.2) and (8.1). Writing f t = ip0 so that p 0 is the real energy (or mass) component, we have

(32 = — f t 2 — p i — + (8.5)

so that (3 is the corresponding proper m or proper energy. For real wave- vectors j b is real and therefore (3 is imaginary. Since the particles dealt with in physics have real proper mass some kind of modification of the primitive real wave tensor must have been introduced in connection with the singling out of the fifth dimension from the others. The fact that (3 is real involves the introduction of complex wave tensors.

Solutions of the wave equation are found as follows. If / is an arbitrary four-valued quantity, H H '/ = 0 by (8.4). Hence ^ = H '/ is a solution of H^ = 0. Inserting row-and-column suffixes, = H'y gives

— H'aiXi + H 'n2/ 2 -f- H 'a3/ 3 + H'a4X4-




Since y is arbitrary this is an arbitrary linear combination of four elementary solutions of the wave equation H 'al, H 'a2, H 'o3, H 'a4. If the second suffix corresponds to rows, the elementary solutions are simply the four columns of H / ; similarly the elementary solutions for tj;* are simply the four rows of H. Actually only three of the elementary solutions are independent.

Using the pentad (6.2) we have by (8.31)

— iH = jj2-f- ifi, p x — 0 ,

Pi + iPi > - l h + i& —P3’ 00, —p 3, + ip a

Ps> °> — —P2 + i$-To obtain H' we reverse the sign of (3. Thus, taking the first column of H' and the first row of H, particular solutions are

^ = (Pz - » Pi + iPi , 0 , p3) = (p2 + i% p1 ip4, 0, p 3).

Although these are formally conjugate complex quantities they are not actually so, since in real problems p4 is imaginary. If it is desired to employ genuinely conjugate wave vectors this can only be managed by making a non-relativity transformation of <\j. For our purposes there is nothing to be gained by insisting on conjugate wave vectors, or even by pairing the solutions for ^ and in one-to-one correspondence. To require that ^ and <j>* shall be conjugate does not ensure that the physically interesting quantities

shall be non-complex, and the reality conditions for these physical vectors are best introduced directly.

9. Equations (8.31) and (8.32) are not quite identical with Dirac’s wave equation ; he gives (3 the same sign in both. We should have obtained his form if we had set j 5 — 0 instead of j 16 — 0 in (6.51) and (6.52) and identified the coefficients correspondingly. We did not do this because j 6 = 0 is not an invariant condition (except for a limited class of transformations), and it involves choosing a particular frame of reference. Or we can obtain Dirac’s form by setting

tl; = y (9.1)

in (8.21) and (8.22) and multiplying the equations by initial and final respectively. The equations are then simplified since

E15 = E15 ein^ E5 e — 1.

This is a non-relativity transformation and the quantities y, y* are unsuitable for use in general relativity theory.

I do not know whether there is any particular advantage in using y, y* in



320 Sir A rth u r E ddington .

practical problems rather than <]/*. If so, we can change over at any time by equation (9.1). But if, following Dirac, we use y, /*, the wave equation is relativistic in four dimensions only, and rotations involving the fifth dimension (proper time) are no longer permitted. This has an important bearing on the supposed difficulty with regard to negative energies, but I caimot enter into the question here.

We have

— 4 = x*ei7rEs einB*x= x*Em x if Em and E5 anticommute = y*EME5y if EM and E5 commute

Thus the vectors P and X are the same whether calculated from ^ or */, but the invariants and the 6-vector are changed. Since E12E5 = — fE ^ by (2.2), the “ magnetic ” components j 23, j zl,j12 become interchanged with the “ electric ” components iju , ij^ , iju . I t is for this reason that in Dirac’s theory the spin appears to be purely magnetic, although the corresponding 6-vector in our theory is, as shown in the next section, of purely electric type

ju (8ee (10-2) )•10. We may choose our space-time axes so that of the two space vectors P

is along xA and X is along xv If their common length is unity I find the following solution

= 1, ♦, - 1 ) * = (», 1, 1) (10.1)

the matrices being as in (6.2). By (3.4) we find

k = 1 k5 = 1 k = i kx = — i (10-2)and all the other components vanish.

This shows the relation of the other space tensors to P and X. The invariant j 5 is i times their common length, as is also apparent from (7.1) or (8.4). If we extend the vector X to five dimensions (jv j 2, j 3, j 4, j b) it is always a null vector. The 6-vector has only one non-vanishing component j 41, so that it lies wholly in the plane x4, xv i.e., the plane of P and X. If S is the 6-vector and I the invariant (j5)we have

IS = [PX], (10.3)

where the bracket indicates vector product. Evidently (10.3) is independent of our special choice of axes.

The additional tensors I and S are redundant as part of a physical description of the system since they are determined by P and X ; but the interaction of




their components with P and X must be taken into account when we apply relativity transformations other than the six rotations of space-time.

The following relativity transformation is important. Let<J/ = eiF̂ 9 < J = (10

As in (4.4) we have

Ji = hcos 6 — is sin 6 in ' = in cos 9 — i« sinOis' = i i sin 0 + is cos 0 h-o in sin 0 + its °os 0

Applying this to (10.2) we have

i i = e~ie 345 = e~id is ' j f = — ie~ie

and the other components remain zero. This result is simply

i i = j t,e~id (10.5)or J ' = Je~ie.

The transformation

<]/ = e^*e ^ (10.6)

also gives the same result (10.5).The generalisation of (10.4) when the space vector X is not along x1 is

4*' — exp ( I 0 (Ei5X i + E 25X 2 4 - E 35X 3 + e 45x 4)/r } 41

exp { - J6 (E16X j + e 25x 2 + e 35x 3 + E45x 4)/R} (10-7)

where R2 = X42 -)- X 22 + X 32 + X42. Since the square of (E15 X4 + .. + E45X4)/R is — 1, its occurrence in the exponential does not create any difficulty of interpretation although it contains non-commuting matrices. As already proved for special axes, (10.7) gives J ' = Je~ie.

11. Provisionally we may suppose that X represents a positional vector of some kind (displacement) as distinguished from momentum represented by P. Then the conjugated quantities, co-ordinate and momentum, will be represented in two distinct domains (as in a phase-space representation), viz., (aq, x 2, x3, x4) and (x15, x25, x35, cc45), which, however, rotate together in the ordinary relativity rotations of space-time. I t is interesting to note that the method of wave vectors provides no way of obtaining positional relations X without a momentum P attached, so that a purely geometrical space without momentum content is unthinkable from its standpoint. The dynamical effects of position (interaction energy) are associated with position at the start.

The fact that X is at right angles to P seems to imply that for a state of definite momentum, i.e., an eigenstate, positional relation is limited to the three-dimensional section of the world at right angles to P. This section may

VOL. c x x x iii.—A. Y



322 Sir A rth u r E dd ing ton .

be called a simultaneous instant for the eigenstate in question. We can thus recognise simultaneity of different parts of a system in a way which does not contradict the principle of relativity, for the simultaneity is relative to a particular eigenstate and not to an absolute direction in the world.

I think it probable (though I have not proved it) that in the wave equation for two electrons (III, equation (16.5)) the co-ordinates (a^, x,/) and the proper distance r of the two electrons must correspond to a simultaneous instant for the eigenstate in question. In particular, if we are seeking solutions in which the centre of gravity of the system is at rest we must set = If this is right it removes the difficulty as to the dual time.

The argument is that positional relation except in a plane at right angles to the momentum vector is not expressible by wave-mechanics and is therefore presumably non-existent; so that positional relation in four dimensions is only possible when the momentum vector is indeterminate. This is evidently connected with the Uncertainty Principle which asserts that when a momentum component is determinate the corresponding co-ordinate is completely indeterminate and therefore meaningless ; but it is not an immediate consequence since it has reference to positional relations internal to the system.

12. A five-dimensional representation of phenomena is inevitable if we adopt Dirac’s conception of dynamical states of a system enduring from t — — oo to t = -J- oo, which are nevertheless continually undergoing change by perturbations due to other systems or subjectively by the observer becoming aware of additional data. This evidently postulates an independent variable outside the four dimensions of the state, to which the changes can be referred. This variable s is .called proper time, and the changes occur throughout the state at instants of s.There can be only one co-ordinate s referring to the whole system, for it is contrary to the conception of a four-dimensional state for the state to change in part and not as a whole. Thus, if the system is described as consisting of a number of particles, each particle will in general have its own co-ordinates (aq, x2, x3, t), but one value of s will apply to them all. If two systems are combined into one their separate proper times must be replaced by one proper time for the combination. I t is for this reason that we cannot combine wave functions (or probabilities) for two systems by a simple multiplicative law ; a transformation is required to eliminate the redundant proper time of one of them and the elimination introduces the terms recognised as interaction energy.

In non-relativity theory proper time is represented within the fourdimensional world and not as a separate variable ; in particular, if the system




is at rest, it is regarded as identical with co-ordinate time t. Changes of state which occur throughout the system at an instant of s are therefore supposed to occur at an instant of t. This is, of course, a flat contradiction of relativity which cannot allow world-wide instants of t to be distinguished in this way. In comparing our results with Dirac’s formulae we have to remember that he does not generally make any distinction between the fourth and fifth dimensions, so that E4 and E5 are liable to be interchanged. He minimises the discrepancy by choosing a special set of matrices such that E45 is a diagonal matrix (p3) which is unity for two components of ^ and — 1 for the other two ; thus E4 = E5 for two components of ^ and E4 = — E5 for the other two. The failure to discriminate between proper time and co-ordinate time must therefore occasionally lead to an error of sign, and I think this is the root of the supposed difficulty as to negative energies.

13. The complete sets of matrices which form the starting point of our analytical calculus are equivalent; we distinguish them relatively to one another by the transformation P which changes one into the other, but there is no absolute specification of the set tha t we start from. This is perhaps more easily seen in G. Temple’s work, in which the EM are not even identified with matrices but are perfectly general symbols obeying the same laws and transformations as our matrices.* The frames of matrices should, therefore, in physical problems connote systems of reference which are a priori equivalent. In a purely statistical theory, such as the modern quantum theory, this equivalence is necessarily a statistical equivalence. A transformation to an equivalent frame must not alter the a priori probability; that essentially is the form of the general principle of relativity which I adopt in my theory. In particular, rotations of the five-dimensional space found in § 4 do not affect the a priori probability of a configuration.

This principle of relativity is in a sense a truism, for the only meaning that I would attach to a priori probability is that we are (conventionally) allowed to assume that it applies unless information to the contrary is given. It need not apply and the solver of the problem may thus reach wrong conclusions, but in that case he is entitled to complain that he was kept in ignorance of important conditions of the problem ( ., the existence of an electric orgravitational field). I t is, however, the task of the complete theory to find the effects of possible deviations from a priori probability of the unspecified parts of the system on the behaviour of the specified particles, and to indicate what kind of information should be supplied.

* ‘ Proc. Roy. Soc.,’ A, vol. 127, p. 349 (1930).



324 Properties of Wave Tensors.

If a priori probability applied to tbe actual world in accordance with our principle of relativity, all orientations of a system in five dimensions would be statistically equivalent and there would be no way of separating the fifth dimension from the four-dimensional world. The fact that we can separate it uniquely implies that actual probability deviates from a priori probability. I t is well known that for the universe as a whole it does deviate ; and by the second law of thermodynamics the deviation tends to diminish with the lapse of proper time. Since this deviation of actual from a priori probability affects all practical problems, we do not treat it as a special modification but as a permanent transformation of our basis of a priori probability. With this transformation there is no longer statistical equivalence of orientations at different inclinations to the axis of proper time, and rotations in the fifth dimension are inadmissible (unless we make complicated corrections for the non-isotrophy).

Against this view it might be urged that, whilst any reference frame of macroscopic space and time and proper time must be defined by reference to the states of the universe (or at least of a vast number of unspecified particles), and any absoluteness attributed to a particular frame must come from the non-uniform relation of actual to ideal probability of these states, yet the “ states ” here referred to correspond to a broad classification which can scarcely take cognisance of condition in regard to thermodynamic equilibrium. To this there is, I think, the conclusive answer that it is well known that direction of proper time is distinguished by reference to deviation from thermodynamic equilibrium (entropy) and by nothing else.

I conclude that there is not really a misfit between the analytical theory with its five-dimensional relativity and the practical problems in which the fifth dimension is kept distinct and unrotatable. The contrast arises from something characterising the actual universe which is generally acknowledged to be exceedingly improbable a priori. As regards dynamical theory of simple systems and the interactions of electrons and protons, we shall have no such fixity of the fifth dimension and all the relativity transformations will be applicable. This point is important in justifying my enumeration of 136 relativity rotations of a system consisting of two particles, the results of which I have treated in other papers.



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The properties of wave tensorsrspa.royalsocietypublishing.org/content/royprsa/133/821/...311 The Properties of ave Tensors. By Sir Arthur Eddington, F.R.S. (Received July 4, 1931.)