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Null plane quantum electrodynamics
Volume 46B, number 1
PHYSICS LETTERS
3 September 1973
NULL PLANE QUANTUM ELECTRODYNAMICS F. ROHRLICH and J.H. TEN EYCK Department o f Physics, Syracuse University, Syracuse, New York 13210, USA
Received 6 June 1973 Co~ariant Feynman rules are given which permit calculations in NGED as easily as in the usual theory. The equivalence between the theories can be seen explicitly when all diagrams are.summed. It holds already prior to renormalization..
Null plane coordinates in Minkowski space for relativistic quantum dynamics were first suggested by Dirac [ 1]. The simple transformation u -- ( t - z ) / x / ~ , o = (t +z)/x/~ changes the conventional (t, x, y, z) to the null plane coordinates (u, x, y, v). A four-vector A u then has components Au='(A°+ A z ) / V ~ , A x, Ay and A o = (A ° - Az)/X/~. Later work by Weinberg [2], Susskind, and others [3] expressed this transformation as the limit of a Lorentz transformation. The corresponding misleading name "infinite momentum frame" for the null plane coordinate system is used in the many applications of null plane field theory to high energy physics. Independent of this work null plane quantum elec, trodynamics was developed by the Syracuse group since 1968 motivated by applications to lasers [4, 5]. Parallel work by the Stanford group differed among other things in their exploitation of the two independent degrees of freedom of the spinor field for a twodimensional rather than four-dimensional spinor formulation and in their perturbationexpansion with "non-relativistic" denominators. The latter exploited the null plane subgroup of the Poincar6 group which is isomorphic to the two-dimensional Galilean group. While very well aware of these features [7], the Syracuse group stuck to the conventional covariant perturbation expansion. As a consequence our formulation of null plane quantum electrodynamics (NQED) can be compared relatively easily with the standard Feynman-Dyson formulation. Questions, to what extent the two theodes are equivalent were raised by the Stanford group [6] and later by Bouchiat, Fayet, and Sourlas [8], who were apparently not acquainted with our work. They claim equivalence in second order but only for suitable regulated theories. Similarly, Chang and collaborators recently published null plane field theories for 102
various interactions [9], using also a covariant formulation as we do, but nevertheless arriving at the result that equivalence exists only after renormalizationt 1. In any case, no author has given an equivalence proof for NQED, This theory has the additional difficulty that a special gauge, the null plane gauge, is almost mandatory here. This adds to the problem of comparison with the customary theory. We have given a general equivalence proof between NQED and the usual QED [10]. Here we want to give covariant Feynman rules by which explicit calculation in NQED can be carried out with no greater difficulty than in the usual theory. In particular~ these rules perrnit one to verify that the unrenormalized S-matrix elements of NQED already give the same result as the usual theory. For purposes of covariance which no longer singles out the z-direction as a preferred spatial direction we span Minkowski space by the two orthonormal spacelike vectors ~ and ~ and the two null vectors m u and nU, orthogonal to them, and interrelated by m.:n = - 1 (our metric has trace +2). A vector is then characterized by the four componentsA u = - m . A , A~ = A . e i (j = 1, 2) and Ao = - n "A. We choose u = x o as null time, the null planes of interest being u = const. NQED has been formulated in this way in ref. [5]. In the Dirac picture (interaction picture) it involves the free Dirac field ff and the free electromagnetic potential a u; the latter is restricted by the gauge condition %= 0 and the null plane constraint au = a~ 1 a.a, a 1 and a 2 being the independent componentst2. They satisfy the following commutation relations: t I See especiaUyref. [9], p. 1158. 1"2 We use a for the two-vector (al, a2); the symbol a~,1 is def'med by a~lf(v) -- -½f_~ e(v-v)f(v')dv'. Correspondingly aV2f(V) = ½f'~lv-v'f(v')dv'. Note that av=-~n.a = - a/av.
Volume 46B, number 1
PHYSICS LETTERS
[aU(x), d' (x') ] = -ihUUD(x-x')
(1)
h~ = ~ + (nUav + nVaU)i~-~1
(2)
{~0(x), ~(x')} = iSc(x-x').
(3)
The S-operator is defined by
s = U, exp(-ifTu(X)dX)
(4)
with U+ denoting positive null-rime ordering including sign changes for odd permutations of ~. The interaction Hamiltonian density 9u(X) is given in ref. [5] and can be written as t a
9u(X) = -]'a +½(avl]v)2 +½e2ff .r.a'rvavl('r.a~ ). (5) Here/u - ½[~;, 7u¢]" 9u (x) has vanishing vacuum expectation value, What matters for the S-operator (4) is only the U+ ordered expression for : 9u(x)d4x. Since ordering of products of operators at equal null-time are not uniquely defined it is convenient to adopt the conventionA(x)B(x)-- U+(A(x)B(x))-~ [A(x),B(x)] ±indicating an anticommutator for bosom and a commutator for fermions. Then (5) becomes, with ~ ' - ~k(x'),
u+f
(x)ax --f axax'tie
4(x-x')
+ ½e2 ~3'v ~ ~ ")'01~'a:264(X-x')
(6)
+ ½e2 ~ 7"a'roT'a'g/'aff2a4(x-x')].
(7)
igc(X_X' ) _=o
= iSe(X-X' ) + ½iTvao154(x~-x'):
(ieTU)x ½n ny a ~-25 4 ( x - x ' ) (ieTV)x,
(8)
The last equation is equivalent to one derived in ref. [9]. The Feynman diagrams contain three types of ver. ~-3 We are grateful to Dr. Roskies for pointing out to us that the normal ordering given in ref. [5l is incorrect. It s h o u l d extend only over the currents]~. This footnote is numbered to indicate this unfortunate error.
(9)
while the third term gives the double vertex (ieTU)x( -½7o)av 154(x-x') (ie3~')x ,.
(10)
The similarity between the internal propagators in the double vertex and the second terms of the propagators (7) and (8) are obvious. It is exactly the cancellation of these terms that yields S-matrix elements which differ from those of conventional theory only by the factor hUV in the photon propagator, characteristic of the null plane gauger s. As a simple example we consider the second order electron self-energy. The second term in (6) to first order ( " e 2 !) gives the S-operator (11)
It combines with the second order contribution of the first term in (6) to e 2df ~--T Sc(X-X)Tv~ , ,h ~ Dc(X-x ,)dxdx ,
-e2f~%¢'D~(x_x3a~-la4(x_x3dxdx'.
(x-x') ---o
= -ih~De(x-x') +,inUn~,av264(x_x, )
rices, a Yukawa vertex, an electron-electron, and an electron-photon vertex, corresponding to the three terms in (6). The first is the usual vertex, but the other two are best considered as double vertices corresponding to a nonlocal interaction. Thus, one arrives at the following Feynman rules in x-space: 1) Each internal photon and electron line is associated with a propagator (7) and (8), respectively. 2) Each single vertex has a factor ieyu; each double vertex consists of two single vertices (factors ie7 u) and "propagators" connecting them. For the second term the double vertex thus becomest 4
e 2f ~--, v S c ( x - x ~,o ~ a , - °2 8 4 ( x - x ,~ d x . ,
N o t e t h a t ( 6 ) contains no normal ordering at all. Insertion of (6) into (4) and use of Wick's theorem leads in the usual way to the S-matrix elements for any desired process. The relevant contractions are given in terms of the causal functions D c and S c
-i~
3 September 1973
(12)
Finally, the third term in (6) to first order (~-e2) exactly cancels the second term in (12) to leave exactly the Feynman-Dyson form of the electron-self-energy but with the null plane gauge factor h ~ . This factor re. duces to guy since the a u terms in (2) do not contribute, as can be proven in complete generality by a wellI"4 Our n o t a t i o n (TU)xmeans that in the S-operator this is ~(xh-% (x). ~-s In the Coulomb gauge a~ satisfiesa~ = 0 and a .a,~ = 0; it is reared to a/a in the nu~ plane gauge by ala = hta~aCwith I/ h ~ given in (2). This gauge transformation transforms the commutation relations (1) into those for the Coulomb gauge 103
Volume 46B, number 1
PHYSICS LETTERS
known argument for all processes when one sums over all relevant diagrams. Thus, all vestiges o f preferred directions (vectors mU and nU) disappear. Finally, we note the Feynman rules in m o m e n t u m space: (1) F o r each internal electron line a propagatort 6
iT"p-m + _7o _p2 +m2_ie
2ipo
i3"~-m p2+
(13)
m2_ie
and for each internal photon line a propagator
hUV(k)
1 ÷ nUnV= hUV(~-) 1 k2_ie k 20 k2_ie
(14)
/~""(k) = g~" ~
IcUnv + kVn~
(15)
ko
(2) For each single vertex a factor t7 i.e. JeT~
64 (p-k-p ~ for each electron-electron double vertex 1
(7°) (P°+q°)2 X
(16)
('[o)~4(p+q-p'-q')('yo)~4(p+q-p'--q')
the internal propagator going from the two ingoing to the two outgoing electron lines. One notes the similarity o f (16) with the Coulomb term in the Coulomb gauge. For the electron-photon vertex
1-6 Following the notation of ref. [9] we denote momenta p which axe on the mass shell by ft. t 7 We denote momenta which go into a vertex by unprimed symbols, those which go out by primed ones.
104
1
-~ "),u'yoT~,i(po+ko) ~4(P+ k-p'-k'),
(17)
again containing an internal propagator from the ingoing to the outgoing momenta, p, p ' referring to electrons, k, k' to photons. Thus, we have succeeded in casting NGED into a form no longer orders o f magnitude more complex than the usual QED and the equivalence between the two formulations becomes explicitly apparent for each process and in each order of covariant perturbation expansion.
References
where
e2
e2
3 September 1973
[1] P.A.M. Dime. Rev. Mod. Phys. 21 (1949) 392. [2] S. Weinberg, Phys. Rev. 150 (1966) 1313. [3] L. Susskind, Phys. Rev. 165 (1968) 1535; L. Susskind and M.B. Halpern, Phys. Rev. 176 (1968) 1686; K. Bardacki and G. Segr~, Phys. Rev. 159 (1967) 1263. [4] R.A. Neville, Ph.D. thesis, Syracuse University (1968); R.A. Neville and F. Rohrlich, Nuovo Cim. 1A (1971) 625 and Phys. Rev. D3 (1971) 1692. [5] F. Rohrlieh, Aeta Phys. Austriaca 32 (1970) 87 and (Suppl.) VIII, 277 (1971). [6] J.B. Kogut and D.E. Soper, Phys. Rev. D1 (1970) 2901; J.D. Bjorken, J.B. Kogut and D.E. Soper, Phys. Rev. D15 (1971) 1382. [7] F. Rohrlich and L. Streit, Nuovo Cim. 7B (1972) 166. [8] C. Bouchiat, P. Foyet and N. Sourlas, Lett. Nuovo Cim. 4 (1972) 9. [9] S.-J. Chang, R.G. Root and T.-M. Yan, Phys. Rev. D7 (1973) 1133; S.-J. Chang and T.-M. Yah, Phys. Rev. D7 (1973) 1147. [10] J.H. Ten Eyck, Ph.D. thesis, Syracuse University (1973); J.H. Ten Eyck and F. Rohrlich, to be published.
PHYSICS LETTERS
3 September 1973
NULL PLANE QUANTUM ELECTRODYNAMICS F. ROHRLICH and J.H. TEN EYCK Department o f Physics, Syracuse University, Syracuse, New York 13210, USA
Received 6 June 1973 Co~ariant Feynman rules are given which permit calculations in NGED as easily as in the usual theory. The equivalence between the theories can be seen explicitly when all diagrams are.summed. It holds already prior to renormalization..
Null plane coordinates in Minkowski space for relativistic quantum dynamics were first suggested by Dirac [ 1]. The simple transformation u -- ( t - z ) / x / ~ , o = (t +z)/x/~ changes the conventional (t, x, y, z) to the null plane coordinates (u, x, y, v). A four-vector A u then has components Au='(A°+ A z ) / V ~ , A x, Ay and A o = (A ° - Az)/X/~. Later work by Weinberg [2], Susskind, and others [3] expressed this transformation as the limit of a Lorentz transformation. The corresponding misleading name "infinite momentum frame" for the null plane coordinate system is used in the many applications of null plane field theory to high energy physics. Independent of this work null plane quantum elec, trodynamics was developed by the Syracuse group since 1968 motivated by applications to lasers [4, 5]. Parallel work by the Stanford group differed among other things in their exploitation of the two independent degrees of freedom of the spinor field for a twodimensional rather than four-dimensional spinor formulation and in their perturbationexpansion with "non-relativistic" denominators. The latter exploited the null plane subgroup of the Poincar6 group which is isomorphic to the two-dimensional Galilean group. While very well aware of these features [7], the Syracuse group stuck to the conventional covariant perturbation expansion. As a consequence our formulation of null plane quantum electrodynamics (NQED) can be compared relatively easily with the standard Feynman-Dyson formulation. Questions, to what extent the two theodes are equivalent were raised by the Stanford group [6] and later by Bouchiat, Fayet, and Sourlas [8], who were apparently not acquainted with our work. They claim equivalence in second order but only for suitable regulated theories. Similarly, Chang and collaborators recently published null plane field theories for 102
various interactions [9], using also a covariant formulation as we do, but nevertheless arriving at the result that equivalence exists only after renormalizationt 1. In any case, no author has given an equivalence proof for NQED, This theory has the additional difficulty that a special gauge, the null plane gauge, is almost mandatory here. This adds to the problem of comparison with the customary theory. We have given a general equivalence proof between NQED and the usual QED [10]. Here we want to give covariant Feynman rules by which explicit calculation in NQED can be carried out with no greater difficulty than in the usual theory. In particular~ these rules perrnit one to verify that the unrenormalized S-matrix elements of NQED already give the same result as the usual theory. For purposes of covariance which no longer singles out the z-direction as a preferred spatial direction we span Minkowski space by the two orthonormal spacelike vectors ~ and ~ and the two null vectors m u and nU, orthogonal to them, and interrelated by m.:n = - 1 (our metric has trace +2). A vector is then characterized by the four componentsA u = - m . A , A~ = A . e i (j = 1, 2) and Ao = - n "A. We choose u = x o as null time, the null planes of interest being u = const. NQED has been formulated in this way in ref. [5]. In the Dirac picture (interaction picture) it involves the free Dirac field ff and the free electromagnetic potential a u; the latter is restricted by the gauge condition %= 0 and the null plane constraint au = a~ 1 a.a, a 1 and a 2 being the independent componentst2. They satisfy the following commutation relations: t I See especiaUyref. [9], p. 1158. 1"2 We use a for the two-vector (al, a2); the symbol a~,1 is def'med by a~lf(v) -- -½f_~ e(v-v)f(v')dv'. Correspondingly aV2f(V) = ½f'~lv-v'f(v')dv'. Note that av=-~n.a = - a/av.
Volume 46B, number 1
PHYSICS LETTERS
[aU(x), d' (x') ] = -ihUUD(x-x')
(1)
h~ = ~ + (nUav + nVaU)i~-~1
(2)
{~0(x), ~(x')} = iSc(x-x').
(3)
The S-operator is defined by
s = U, exp(-ifTu(X)dX)
(4)
with U+ denoting positive null-rime ordering including sign changes for odd permutations of ~. The interaction Hamiltonian density 9u(X) is given in ref. [5] and can be written as t a
9u(X) = -]'a +½(avl]v)2 +½e2ff .r.a'rvavl('r.a~ ). (5) Here/u - ½[~;, 7u¢]" 9u (x) has vanishing vacuum expectation value, What matters for the S-operator (4) is only the U+ ordered expression for : 9u(x)d4x. Since ordering of products of operators at equal null-time are not uniquely defined it is convenient to adopt the conventionA(x)B(x)-- U+(A(x)B(x))-~ [A(x),B(x)] ±indicating an anticommutator for bosom and a commutator for fermions. Then (5) becomes, with ~ ' - ~k(x'),
u+f
(x)ax --f axax'tie
4(x-x')
+ ½e2 ~3'v ~ ~ ")'01~'a:264(X-x')
(6)
+ ½e2 ~ 7"a'roT'a'g/'aff2a4(x-x')].
(7)
igc(X_X' ) _=
= iSe(X-X' ) + ½iTvao154(x~-x'):
(ieTU)x ½n ny a ~-25 4 ( x - x ' ) (ieTV)x,
(8)
The last equation is equivalent to one derived in ref. [9]. The Feynman diagrams contain three types of ver. ~-3 We are grateful to Dr. Roskies for pointing out to us that the normal ordering given in ref. [5l is incorrect. It s h o u l d extend only over the currents]~. This footnote is numbered to indicate this unfortunate error.
(9)
while the third term gives the double vertex (ieTU)x( -½7o)av 154(x-x') (ie3~')x ,.
(10)
The similarity between the internal propagators in the double vertex and the second terms of the propagators (7) and (8) are obvious. It is exactly the cancellation of these terms that yields S-matrix elements which differ from those of conventional theory only by the factor hUV in the photon propagator, characteristic of the null plane gauger s. As a simple example we consider the second order electron self-energy. The second term in (6) to first order ( " e 2 !) gives the S-operator (11)
It combines with the second order contribution of the first term in (6) to e 2df ~--T Sc(X-X)Tv~ , ,h ~ Dc(X-x ,)dxdx ,
-e2f~%¢'D~(x_x3a~-la4(x_x3dxdx'.
(x-x') ---
= -ih~De(x-x') +,inUn~,av264(x_x, )
rices, a Yukawa vertex, an electron-electron, and an electron-photon vertex, corresponding to the three terms in (6). The first is the usual vertex, but the other two are best considered as double vertices corresponding to a nonlocal interaction. Thus, one arrives at the following Feynman rules in x-space: 1) Each internal photon and electron line is associated with a propagator (7) and (8), respectively. 2) Each single vertex has a factor ieyu; each double vertex consists of two single vertices (factors ie7 u) and "propagators" connecting them. For the second term the double vertex thus becomest 4
e 2f ~--, v S c ( x - x ~,o ~ a , - °2 8 4 ( x - x ,~ d x . ,
N o t e t h a t ( 6 ) contains no normal ordering at all. Insertion of (6) into (4) and use of Wick's theorem leads in the usual way to the S-matrix elements for any desired process. The relevant contractions are given in terms of the causal functions D c and S c
-i~
3 September 1973
(12)
Finally, the third term in (6) to first order (~-e2) exactly cancels the second term in (12) to leave exactly the Feynman-Dyson form of the electron-self-energy but with the null plane gauge factor h ~ . This factor re. duces to guy since the a u terms in (2) do not contribute, as can be proven in complete generality by a wellI"4 Our n o t a t i o n (TU)xmeans that in the S-operator this is ~(xh-% (x). ~-s In the Coulomb gauge a~ satisfiesa~ = 0 and a .a,~ = 0; it is reared to a/a in the nu~ plane gauge by ala = hta~aCwith I/ h ~ given in (2). This gauge transformation transforms the commutation relations (1) into those for the Coulomb gauge 103
Volume 46B, number 1
PHYSICS LETTERS
known argument for all processes when one sums over all relevant diagrams. Thus, all vestiges o f preferred directions (vectors mU and nU) disappear. Finally, we note the Feynman rules in m o m e n t u m space: (1) F o r each internal electron line a propagatort 6
iT"p-m + _7o _p2 +m2_ie
2ipo
i3"~-m p2+
(13)
m2_ie
and for each internal photon line a propagator
hUV(k)
1 ÷ nUnV= hUV(~-) 1 k2_ie k 20 k2_ie
(14)
/~""(k) = g~" ~
IcUnv + kVn~
(15)
ko
(2) For each single vertex a factor t7 i.e. JeT~
64 (p-k-p ~ for each electron-electron double vertex 1
(7°) (P°+q°)2 X
(16)
('[o)~4(p+q-p'-q')('yo)~4(p+q-p'--q')
the internal propagator going from the two ingoing to the two outgoing electron lines. One notes the similarity o f (16) with the Coulomb term in the Coulomb gauge. For the electron-photon vertex
1-6 Following the notation of ref. [9] we denote momenta p which axe on the mass shell by ft. t 7 We denote momenta which go into a vertex by unprimed symbols, those which go out by primed ones.
104
1
-~ "),u'yoT~,i(po+ko) ~4(P+ k-p'-k'),
(17)
again containing an internal propagator from the ingoing to the outgoing momenta, p, p ' referring to electrons, k, k' to photons. Thus, we have succeeded in casting NGED into a form no longer orders o f magnitude more complex than the usual QED and the equivalence between the two formulations becomes explicitly apparent for each process and in each order of covariant perturbation expansion.
References
where
e2
e2
3 September 1973
[1] P.A.M. Dime. Rev. Mod. Phys. 21 (1949) 392. [2] S. Weinberg, Phys. Rev. 150 (1966) 1313. [3] L. Susskind, Phys. Rev. 165 (1968) 1535; L. Susskind and M.B. Halpern, Phys. Rev. 176 (1968) 1686; K. Bardacki and G. Segr~, Phys. Rev. 159 (1967) 1263. [4] R.A. Neville, Ph.D. thesis, Syracuse University (1968); R.A. Neville and F. Rohrlich, Nuovo Cim. 1A (1971) 625 and Phys. Rev. D3 (1971) 1692. [5] F. Rohrlieh, Aeta Phys. Austriaca 32 (1970) 87 and (Suppl.) VIII, 277 (1971). [6] J.B. Kogut and D.E. Soper, Phys. Rev. D1 (1970) 2901; J.D. Bjorken, J.B. Kogut and D.E. Soper, Phys. Rev. D15 (1971) 1382. [7] F. Rohrlich and L. Streit, Nuovo Cim. 7B (1972) 166. [8] C. Bouchiat, P. Foyet and N. Sourlas, Lett. Nuovo Cim. 4 (1972) 9. [9] S.-J. Chang, R.G. Root and T.-M. Yan, Phys. Rev. D7 (1973) 1133; S.-J. Chang and T.-M. Yah, Phys. Rev. D7 (1973) 1147. [10] J.H. Ten Eyck, Ph.D. thesis, Syracuse University (1973); J.H. Ten Eyck and F. Rohrlich, to be published.