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Electromagnetic theory treated in analogy to the theory of gravitation
Nuclear Physics B92 (1975) 541-546 © North-Holland Publishing Company
ELECTROMAGNETIC THEORY TREATED IN ANALOGY TO THE THEORY OF GRAVITATION O. KLEIN Ringen 21, Stocksund 182 74, Sweden Received 13 January 1975
As a further contribution to the program of Klein the present paper treats electromagnetic theory as an analogue to that of gravitation, introducing two extra dimensions not belonging to space and time, one space-like and one time-like. The difficulty concerning unitarity mentioned there is surmounted by a rational generalization of Bargmann's a needed for the same purpose for the Lagrangian density belonging to the Dirac equation. Moreover, the treatment of the inversion relations, using one extra dimension, is extended to two such dimensions. These two dimensions lead to the possibility of a natural avoidance of the difficulty present in the so-called five-dimensional representation of electromagnetism of an enormously large mass term, at least for states of so-far known masses. Finally, these two dimensions lead to the introduction of two sets of electromagnetic potentials corresponding to the equations given by Schwinger in developing Dirac's idea of magnetic monopoles, obtaining thereby a further background. It should be stressed that what has been said here about particle states belongs to the primary empty particles without normalization. Hence it is probably premature yet to try to explain these most interesting symmetries, which belong to "non-empty" states.
1. Introduction The general background o f program [1 ] is the extension o f Einstein's idea o f the equivalence o f inertial forces and gravitational forces, used in [3] for the derivation o f his field equations and intended also for a generalization to other fields, in the first place that o f electromagnetism. A preparation for this has long existed in the so-called five-dimensional representation o f electromagnetism, where the elementary quantum of electricity was tentatively explained by assuming the fifth coordinate to be closed, the space being so-to-say cylindrical in that direction, the corresponding momentum, to which the change would be proportional, its quantization being thereby a consequence o f the structure o f space [4]. Leaving this hypothesis aside, we can postpone the decision whether this or the other possibility (i.e. limiting the charge states to 1 and 0 for the particles, and 0 and - 1 for the antiparticles) has to be chosen. The first alternative may, however, have a certain advantage, because it would allow a transformation group analogous to that o f the general coordinate transformations belonging to the space-time dimensions.
542
o. Klein/Electromagnetic theory
As for the extension of the number of extra dimensions from one to two, there is in the first place a formal reason, namely that already one extra dimension requires 8-row matrices, while six Clifford numbers is just what is needed for general 8-row matrices. In fact, as is well-known, 2n such numbers are needed for general 2n-row matrices, where n is an integer, i.e. 1 for spin matrices, 2 for Dirac matrices and 3 for eight-row matrices. Denoting the six basic matrices by P 0, r 1 , 1"2, 1'3, i-4, 1'5, (in the first place in an inertial frame, later, when no confusion will occur, the same notation in an arbitrary frame) all of them anticommuting, 1'0 and p5 having squares equal to - 1 , while those of the rest are +I, those with positive squares being hermitian and those with negative squares antihermitian. 2. Unitarity Denoting the eight-row wave functions by ff and ff T and putting ~ equal to qjta, then apart from a numerical factor depending on the units, the Lagrangian density L of the extended Dirac equation in an intertial frame is given by L = ~Ot~,
O = PUp u ,
(1)
where the Pu denote the six momentum components. For the a in an inertial frame we assume now a = p0p5 .
(2)
Thereby, no mass term is added, since the real masses have to be the result of a more deepgoing development of the theory. From the properties of the 1'u it follows that (aD) t = - a D ,
(3)
the Pu being Hermitean. The proof follows easily from the properties of the 1'u in noticing that a commutes with those having positive squares but anticommutes with those having negative squares (1'0 and 1'5), while at = -a. Thus, finally Lt = - L .
(4)
Hence, L as defined here, is antihermitian in agreement with its role in the exponent used in functional integration. Replacing the Pu by the derivative operators ~u it becomes Hermitian.
3. Inversion theorems
Starting from the behaviour of the pu we have, in the first place, six theorems,
o. Klein/Electromagnetic theory
543
each including one 1-u, and, then also different combinations of several of them, of which the parity transformation, belonging to the product 1-1 . 1-2 . 1-3 , is particularly interesting on account of its lack of validity when particle-antiparticle inversion plays a role. We shall here only treat the inversions in an inertial frame, those in a general frame of reference being a consequence of these, due to the principle of equivalence. Beginning with the inversions, where just one of the 1-u changes its sign, we define six matrices. s " = A I~ ,
(S)
where A is the product of all six 1'u, which is seen to anticommute with any of them. Thus, S u commutes with any r 'v except pu itself, with which it anticommutes, i.e. ( W, v ~ / l , ( s . ) - 1 FvS"
--~t_1'u, u =/l.
(6)
Changing the sign ofpu will then restore the sign of the Lagrangian density. Before coming to the transformations including particle-antiparticle inversions we shall introduce the following notations: 1'= 1'0plF2p3,
/3= 1-'4P5, A =/3r.
a = upS,
(7)
As is seen, a and/3 commute with 1'0,1'1, i-2, 1-3 and, thus, also with 1-. Further a and/3 anticommute with one another, their squares being equal to unity. They are thus Clifford numbers for two-row matrices but regarded here as eight-row matrices, namely a=
,
/3=
/
,
0-1
0
(8)
where the numbers 1, -1 and 1, 1 stand for four-row matrices. Using this representation we put for ff and ~:
l~= ~Ob '
,
,
where in the ordinary theory fib is equal to if*, while (9) may also permit a more general situation. Still, we shall call the states belonging to a and b, respectively, particle and antiparticle states. Thereby/3 is seen to be the operator for particleantiparticle inversion. Using relation (7) we shall now consider the parity inversion of the Lagrangian density (1), namely
544
O. KleoT/Electromagnetic theory L' = - ~ ( r
• r 1 • r 2 • r 3) D r • r 1 • p2 . r 3/3 ~ ,
(10)
where (ppl p2p3)t, as follows easily, is equal to (P • pl . p2 . p 3 ) - l . Thereby letting the two/3 act on ~- and ~ respectively. Now, P anticommutes with r 0, p l , p2, p3 and commutes with 1-4 and F 5. On the other hand, pl . p2 . p3 commutes with all these matrices, but anticommutes with p0, 1-,4, pS, implying that p l , p2, p3, p4, p5 change their sign in the transformation (10), while I '0 is unchanged. Applying/3 on ~, we get
~a ' and similarly for f , thus undergoing a particle-antiparticle inversion. At the same time the sign change of F4 and p5, being multiplied with P0 in the Lagrangian density appears as a factor to the electromagnetic charge. All this is seen to correspond to the PC inversion as considered by Lee and Yang and in this connection especially by Landau.
4. Connection of the present theory to the possibility of enlarging electromagnetism so as to contain free magnetic charges
It is well-known that the idea of free magnetic charges due to Dirac has been further developed in some interesting papers by Schwinger. Since this theory may be represented by two electromagnetic potential vectors instead of the usual simple vector, my original doubts concerning this extension being no longer valid, it occurred to me that the introduction of two non-space-time dimensions, based on reasons presented above, may give further background to the Dirac-Schwinger theory. I think that the following considerations will show that this is in fact the case.
Defining thus the P 4 and P5 in the usual way by means of the six pu, being now regarded as depending on the four space-time coordinates we put P4 = A ,
P5 =iB,
(12)
both A and B being thus Hermitian, we define two four-vectors, in analogy to the gravitational gik, by A k = 1] { A , P k } , B k = ~1 {B, Pk} , k = 0 , 1,2,3 .
(13)
We shall use Schwinger's elementary notations in his Science article [5], his equation being (with the velocity c put equal to unity):
O. Klein/Electromagnetic theory aE
V X H - ~ - = 47rJe , 3H -v xE-~/-=
47rim,
545
~7 "E = 4fro e , V.H=4rrp m
(14)
Using space-vector notations for A k , B k with k = 1,2, 3, we put ¢bo = Ao + iBo ,
(15)
~ =A + iB ,
and, moreover F=E+iH,
j=]e+iJm ,
p=pe+iPm
,
(16)
it follows that ao F = - V • qb0 - ~ - - + iV X ~ .
(17)
The Schwinger equations being identical with 0F
-aT-iAXF
=4zrj,
A*F=47rp,
(18)
we get from (17) a2----~- A O + V " \ ~ at 2
+7 "
= 47rj,
a2% -- Aq~ 0 = 4frO, at 2
(19)
which by means of the Lorentz condition aq~0 at
--+
v • ~, = 0
(20)
take the ordinary form of the electromagnetic equations for the potentials in the presence of currents and charges, but with the difference that there are not only electric charges but also magnetic ones.
5. Charged fermions and bosons According to the program in question there ought to be charged bosons as well as charged fermions corresponding to the different values of m o m e n t u m components of the space-time theory. As mentioned in sect. 1 we shall leave it open, whether the charged states should belong to all positive and negative integers, or be limited
546
O. Klein/Electromagnetic theory
to the values -+1. At present we shall just consider the latter ones. Thereby we shall not here try to understand why the square of the electrical unit charge is not equal to the natural unit 7~c but smaller by the famous number 137, nor shall we try to understand why the muon, being so like the electron, has a much heavier mass. These, and many other things demand certainly a far more deepgoing study of the interactions than that of this paper, although according to the program, they should be given in principle. However, an interesting feature of the present approach is that it demands two fermions of unit charge corresponding to the two extra non space-time-like dimensions implying 8-row ff matrices. In a similar way we shall expect the existence of charged bosons corresponding to the photons, i.e. of vector character. Before one had discovered the Yukawa particles responsible for the nuclear force I tried to consider them as being the charged particles belonging to the photons. When they were discovered and called pions it became soon evident that they could not be the counterpart of the photons, being not vectors and having their own counterpart in the neutral pion. Could the charged counterparts to the photons be the heavy electrically charged particles supposed to be responsible for the weak interactions, and hence the recently discovered neutral current be just photons? If so, this would mean a further support to the program under consideration.
References [1] [2] [3] [4] [5]
O. Klein, Nucl. Phys. B21 (1970) 253. O. Klein, Nucl. Phys. 4 (1957) 677. O. Klein, Physica Scripta 9 (1974) 69. O. Klein, Nature 118 (1926) 516. .1. Schwinger, Science 165 (1969) 757.
ELECTROMAGNETIC THEORY TREATED IN ANALOGY TO THE THEORY OF GRAVITATION O. KLEIN Ringen 21, Stocksund 182 74, Sweden Received 13 January 1975
As a further contribution to the program of Klein the present paper treats electromagnetic theory as an analogue to that of gravitation, introducing two extra dimensions not belonging to space and time, one space-like and one time-like. The difficulty concerning unitarity mentioned there is surmounted by a rational generalization of Bargmann's a needed for the same purpose for the Lagrangian density belonging to the Dirac equation. Moreover, the treatment of the inversion relations, using one extra dimension, is extended to two such dimensions. These two dimensions lead to the possibility of a natural avoidance of the difficulty present in the so-called five-dimensional representation of electromagnetism of an enormously large mass term, at least for states of so-far known masses. Finally, these two dimensions lead to the introduction of two sets of electromagnetic potentials corresponding to the equations given by Schwinger in developing Dirac's idea of magnetic monopoles, obtaining thereby a further background. It should be stressed that what has been said here about particle states belongs to the primary empty particles without normalization. Hence it is probably premature yet to try to explain these most interesting symmetries, which belong to "non-empty" states.
1. Introduction The general background o f program [1 ] is the extension o f Einstein's idea o f the equivalence o f inertial forces and gravitational forces, used in [3] for the derivation o f his field equations and intended also for a generalization to other fields, in the first place that o f electromagnetism. A preparation for this has long existed in the so-called five-dimensional representation o f electromagnetism, where the elementary quantum of electricity was tentatively explained by assuming the fifth coordinate to be closed, the space being so-to-say cylindrical in that direction, the corresponding momentum, to which the change would be proportional, its quantization being thereby a consequence o f the structure o f space [4]. Leaving this hypothesis aside, we can postpone the decision whether this or the other possibility (i.e. limiting the charge states to 1 and 0 for the particles, and 0 and - 1 for the antiparticles) has to be chosen. The first alternative may, however, have a certain advantage, because it would allow a transformation group analogous to that o f the general coordinate transformations belonging to the space-time dimensions.
542
o. Klein/Electromagnetic theory
As for the extension of the number of extra dimensions from one to two, there is in the first place a formal reason, namely that already one extra dimension requires 8-row matrices, while six Clifford numbers is just what is needed for general 8-row matrices. In fact, as is well-known, 2n such numbers are needed for general 2n-row matrices, where n is an integer, i.e. 1 for spin matrices, 2 for Dirac matrices and 3 for eight-row matrices. Denoting the six basic matrices by P 0, r 1 , 1"2, 1'3, i-4, 1'5, (in the first place in an inertial frame, later, when no confusion will occur, the same notation in an arbitrary frame) all of them anticommuting, 1'0 and p5 having squares equal to - 1 , while those of the rest are +I, those with positive squares being hermitian and those with negative squares antihermitian. 2. Unitarity Denoting the eight-row wave functions by ff and ff T and putting ~ equal to qjta, then apart from a numerical factor depending on the units, the Lagrangian density L of the extended Dirac equation in an intertial frame is given by L = ~Ot~,
O = PUp u ,
(1)
where the Pu denote the six momentum components. For the a in an inertial frame we assume now a = p0p5 .
(2)
Thereby, no mass term is added, since the real masses have to be the result of a more deepgoing development of the theory. From the properties of the 1'u it follows that (aD) t = - a D ,
(3)
the Pu being Hermitean. The proof follows easily from the properties of the 1'u in noticing that a commutes with those having positive squares but anticommutes with those having negative squares (1'0 and 1'5), while at = -a. Thus, finally Lt = - L .
(4)
Hence, L as defined here, is antihermitian in agreement with its role in the exponent used in functional integration. Replacing the Pu by the derivative operators ~u it becomes Hermitian.
3. Inversion theorems
Starting from the behaviour of the pu we have, in the first place, six theorems,
o. Klein/Electromagnetic theory
543
each including one 1-u, and, then also different combinations of several of them, of which the parity transformation, belonging to the product 1-1 . 1-2 . 1-3 , is particularly interesting on account of its lack of validity when particle-antiparticle inversion plays a role. We shall here only treat the inversions in an inertial frame, those in a general frame of reference being a consequence of these, due to the principle of equivalence. Beginning with the inversions, where just one of the 1-u changes its sign, we define six matrices. s " = A I~ ,
(S)
where A is the product of all six 1'u, which is seen to anticommute with any of them. Thus, S u commutes with any r 'v except pu itself, with which it anticommutes, i.e. ( W, v ~ / l , ( s . ) - 1 FvS"
--~t_1'u, u =/l.
(6)
Changing the sign ofpu will then restore the sign of the Lagrangian density. Before coming to the transformations including particle-antiparticle inversions we shall introduce the following notations: 1'= 1'0plF2p3,
/3= 1-'4P5, A =/3r.
a = upS,
(7)
As is seen, a and/3 commute with 1'0,1'1, i-2, 1-3 and, thus, also with 1-. Further a and/3 anticommute with one another, their squares being equal to unity. They are thus Clifford numbers for two-row matrices but regarded here as eight-row matrices, namely a=
,
/3=
/
,
0-1
0
(8)
where the numbers 1, -1 and 1, 1 stand for four-row matrices. Using this representation we put for ff and ~:
l~= ~Ob '
,
,
where in the ordinary theory fib is equal to if*, while (9) may also permit a more general situation. Still, we shall call the states belonging to a and b, respectively, particle and antiparticle states. Thereby/3 is seen to be the operator for particleantiparticle inversion. Using relation (7) we shall now consider the parity inversion of the Lagrangian density (1), namely
544
O. KleoT/Electromagnetic theory L' = - ~ ( r
• r 1 • r 2 • r 3) D r • r 1 • p2 . r 3/3 ~ ,
(10)
where (ppl p2p3)t, as follows easily, is equal to (P • pl . p2 . p 3 ) - l . Thereby letting the two/3 act on ~- and ~ respectively. Now, P anticommutes with r 0, p l , p2, p3 and commutes with 1-4 and F 5. On the other hand, pl . p2 . p3 commutes with all these matrices, but anticommutes with p0, 1-,4, pS, implying that p l , p2, p3, p4, p5 change their sign in the transformation (10), while I '0 is unchanged. Applying/3 on ~, we get
~a ' and similarly for f , thus undergoing a particle-antiparticle inversion. At the same time the sign change of F4 and p5, being multiplied with P0 in the Lagrangian density appears as a factor to the electromagnetic charge. All this is seen to correspond to the PC inversion as considered by Lee and Yang and in this connection especially by Landau.
4. Connection of the present theory to the possibility of enlarging electromagnetism so as to contain free magnetic charges
It is well-known that the idea of free magnetic charges due to Dirac has been further developed in some interesting papers by Schwinger. Since this theory may be represented by two electromagnetic potential vectors instead of the usual simple vector, my original doubts concerning this extension being no longer valid, it occurred to me that the introduction of two non-space-time dimensions, based on reasons presented above, may give further background to the Dirac-Schwinger theory. I think that the following considerations will show that this is in fact the case.
Defining thus the P 4 and P5 in the usual way by means of the six pu, being now regarded as depending on the four space-time coordinates we put P4 = A ,
P5 =iB,
(12)
both A and B being thus Hermitian, we define two four-vectors, in analogy to the gravitational gik, by A k = 1] { A , P k } , B k = ~1 {B, Pk} , k = 0 , 1,2,3 .
(13)
We shall use Schwinger's elementary notations in his Science article [5], his equation being (with the velocity c put equal to unity):
O. Klein/Electromagnetic theory aE
V X H - ~ - = 47rJe , 3H -v xE-~/-=
47rim,
545
~7 "E = 4fro e , V.H=4rrp m
(14)
Using space-vector notations for A k , B k with k = 1,2, 3, we put ¢bo = Ao + iBo ,
(15)
~ =A + iB ,
and, moreover F=E+iH,
j=]e+iJm ,
p=pe+iPm
,
(16)
it follows that ao F = - V • qb0 - ~ - - + iV X ~ .
(17)
The Schwinger equations being identical with 0F
-aT-iAXF
=4zrj,
A*F=47rp,
(18)
we get from (17) a2----~- A O + V " \ ~ at 2
+7 "
= 47rj,
a2% -- Aq~ 0 = 4frO, at 2
(19)
which by means of the Lorentz condition aq~0 at
--+
v • ~, = 0
(20)
take the ordinary form of the electromagnetic equations for the potentials in the presence of currents and charges, but with the difference that there are not only electric charges but also magnetic ones.
5. Charged fermions and bosons According to the program in question there ought to be charged bosons as well as charged fermions corresponding to the different values of m o m e n t u m components of the space-time theory. As mentioned in sect. 1 we shall leave it open, whether the charged states should belong to all positive and negative integers, or be limited
546
O. Klein/Electromagnetic theory
to the values -+1. At present we shall just consider the latter ones. Thereby we shall not here try to understand why the square of the electrical unit charge is not equal to the natural unit 7~c but smaller by the famous number 137, nor shall we try to understand why the muon, being so like the electron, has a much heavier mass. These, and many other things demand certainly a far more deepgoing study of the interactions than that of this paper, although according to the program, they should be given in principle. However, an interesting feature of the present approach is that it demands two fermions of unit charge corresponding to the two extra non space-time-like dimensions implying 8-row ff matrices. In a similar way we shall expect the existence of charged bosons corresponding to the photons, i.e. of vector character. Before one had discovered the Yukawa particles responsible for the nuclear force I tried to consider them as being the charged particles belonging to the photons. When they were discovered and called pions it became soon evident that they could not be the counterpart of the photons, being not vectors and having their own counterpart in the neutral pion. Could the charged counterparts to the photons be the heavy electrically charged particles supposed to be responsible for the weak interactions, and hence the recently discovered neutral current be just photons? If so, this would mean a further support to the program under consideration.
References [1] [2] [3] [4] [5]
O. Klein, Nucl. Phys. B21 (1970) 253. O. Klein, Nucl. Phys. 4 (1957) 677. O. Klein, Physica Scripta 9 (1974) 69. O. Klein, Nature 118 (1926) 516. .1. Schwinger, Science 165 (1969) 757.