Table Of ContentTorsion and Gauge Invariance in Maxwell-Dirac Electrodynamics
∗
H.T. Nieh
Institute for Advanced Study, Tsinghua University, Beijing 100084, China
(Dated: January 27, 2017)
Ithasbeenknownforalongtimethatthepresenceoftorsionisinconflictwithgaugeinvariance
of the the electromagnetic field in curved Riemann-Cartan space if the Maxwell field is minimally
coupled to thecurved gravitational space through thecovariant derivative. In search for a possible
solution, weconsiderin thisnotethesystemofMaxwell-Dirac electrodynamics inRiemann-Cartan
space. Through investigating consistency of the field equations, and taking cue from the scale
invariance properties of the system, we come up with a solution that satisfies gauge invariance
without having to dispense with torsion in the coupled Maxwell-Dirac system. This is achieved
by modifying the connection that appears in the covariant derivative for the Maxwell field. The
7 modifiedconnectionturnsouttobeintheform ofaWeylconnection,withthetorsion tracevector
1 playingtheeffectiveroleofaWeylgaugefield. Withthismodifiedconnection,whichissymmetrical,
0 theLorentz-spincurrentofthephotonfieldisseentovanish. Inaddition,exceptfortheDiracmass
2 term,thesystemexhibitslocalscaleinvariance. Thesameconsiderationappliestoallgaugetheories,
abelian or non-ableian, in thestandard model of particle physics.
n
a
J PACSnumbers: 04.20.Cv,04.63.+v
8
2
I. INTRODUCTION namics. To achieve consistency of the field equations of
the Maxwell-Dirac system, we shall see that the under-
]
c lying connection for the Maxwell field can not be the
q
Curvatureandtorsionarethetwofundamentaltensors symmetric Christoffel connection unless torsion also dis-
-
r in the Riemann-Cartan space. The basic variables de- appearsintheDiracsector,namely,torsioniscompletely
g scribingtheRiemann-Cartanspacearethevierbeinfields dispensed with in the entire Maxwell-Dirac system. A
[ eaµ and the Lorentz-spin connection fields ωabµ, which, clue in finding a potential candidate for a suitable con-
2 respectively, represent the translational gauge fields and nection to be adopted in the covariantderivative for the
v the Lorentz-rotationgaugefields inthe Einstein-Cartan- Maxwell field comes from the scale invariance properties
2 Sciama-Kibble theory of gravitation [1, 2]. While the of the field equation for the Dirac field, which explicitly
3
energy-momentum tensor is defined as the response of show that the torsion trace vector effectively plays the
1
the physical system to variations of the vierbein field, role of the Weyl gauge field for local scale transforma-
7
the Lorentz-spin current is defined as response to vari- tions [14]. This suggests that the connection should be
0
. ations of the Lorentz-spin connection field [1, 2]. That modifiedsothatthecovariantderivativefortheMaxwell
1
torsion is not compatible with gauge invariance was al- fieldpossessestransparentscalecovarianttransformation
0
ready noted by Kibble [1] in his original paper, and property. Byimposingconsistencyofthefieldequations,
7
1 discussed in the influential review paper of Hehl et al the new connectionfor the Maxwellfield turns out to be
: [3]. Gauge invariance can refer to the abelian U(1) intheformofaWeylconnection,whichissymmetricand
v
gauge theory of electrodynamics as well as to the non- compatible with gauge invariance. The resulting scheme
i
X abelian gauge theories in the standard model of particle isthusseentopossessnotonlygaugeinvariancebutalso
r physics. This non-compatibility problem has since been desirable scale invariance properties.
a discussed by various authors [4–9]. One approach, and
perhaps the consensus approach by now, is to postulate
[3] that the Maxwell tensor F takes the form of the
µν
flat-space curl A −A , without, however,specifying II. MAXWELL-DIRAC SYSTEM IN
µ,ν ν,µ
RIEMANN-CARTAN SPACE
thenon-minimalcovariantderivativethatshouldbeused
to achieve this goal. Another is to bring into the system
additionalspecifictorsionsources,likeintheworkofHo- We consider the Maxwell-Dirac system of electrody-
jman et al [10], which, in addition to being ad hoc and namics in the background of curved Riemann-Cartan
incompatiblewithexperimentalevidence[11],islikelyto space, which is described by the vierbein field ea , their
µ
bring complications to renormalizability [12, 13] of the inverse e µ, and the Lorentz-spin connection field ωab .
a µ
physicalgaugetheoriesinthe standardmodelofparticle The metric is defined by
physics. In this paper, we examine this question within
the realisticphysicalsystem ofMaxwell-Diracelectrody- g =η ea eb , (1)
µν ab µ ν
and the affine connection by
∗Electronicaddress: [email protected] Γλ =e λ(aa +ωa eb ), (2)
µν a µ,ν bν µ
2
where η = (1,−1,−1,−1). The covariant derivatives m=0), the latter being defined, with the proper scale
ab
with respect to both local Lorentz transformations and weights for the various fields, by
general coordinate transformations are defined, such as
e µ →e−Λ(x)e µ,
a a
∇ χ λ =χ λ −ωb χ λ+Γλ χ ν, (3)
µ a a ,µ aµ b νµ a
ea →eΛ(x)ea ,
µ µ
∇µχaν =χaν,µ+ωabµχbν −Γλνµχaλ. (4) ψ(x)→e−23Λ(x)ψ(x),
It can be easily verified that ∇ ea = 0 and ∇ e µ = 0
so that ν µ ν a Aµ(x)→Aµ,
∇λgµν =0, (5) ωab (x)→ωab (x).
µ µ
However, the Lagrangian is not gauge invariant in the
∇λgµν =0. (6) presence of torsion because, as is well known, Fµν =
A −A +Cλ A is not.
The connection Γλµν defined by (2) is thus metric- νT,µheEuµl,eνr-LagrµaνngλeequationfortheDiracfieldcanbe
compatible. In general, it is not symmetric, and the obtained straightforwardly. On account of
anti-symmetric part is the torsion tensor:
ǫ−1ǫ =Γλ =Γλ +Cλ , (13)
,µ λµ µλ λµ
Cλ =Γλ −Γλ . (7)
µν µν νµ andthecommutationpropertiesoftheDiracgammama-
trices [15], we obtain [14]
In the presence of torsion, the metric compatibility re-
lations (5) and (6) imply that the connection is of the 1
[iγaeµ(D +iA + Cλ )−m]ψ =0, (14)
general form: a µ µ 2 λµ
Γλ = 1gλρ(g +gνρ,µ−gµν,ρ)+Yλ , (8) whereDµ isgivenin(11). We knowthatthe Lagrangian
µν 2 ρµ,ν µν intheaction(10)isscaleinvariantwhenm=0. TheDirac
equation(14)isthusexpectedtobescaleinvariantexcept
where the contortion tensor Yλ is given by the mass term. We have, by its construction according
µν
to (2), the connectionΓλ has the followingscale trans-
µν
Yλ = 1(Cλ +C λ+C λ). (9) formation property
µν 2 µν µν νµ
Γλ →Γλ +δλ Λ , (15)
µν µν µ ,ν
The basic field variables of the Maxwell-Dirac electro-
which implies
dynamics are the Maxwell field A and Dirac field ψ.
µ
The action for the system is of the form Cλ →Cλ +3Λ . (16)
λµ λµ ,µ
1 We denote
W = d4xǫ[− FµνF
Z 4 µν (10) 1
+21(ψ¯iγaeaµDµψ−ψ¯D¯µiγaeaµψ)−mψ¯ψ], Bµ = 3Cλλµ. (17)
where It transforms as a Weyl gauge field for local scale trans-
formations [14]
i
Dµ =∂µ− 4σabωabµ, (11) Bµ →Bµ+Λ,µ. (18)
The Dirac equation (13) is then expressed as
i
D¯µ =∂¯µ+ 4σabωabµ, (12) [iγaeµa(Dµ+iAµ+ 32Bµ)−m]ψ =0. (19)
and ǫ =detea . The partial ∂¯ in (10) is understood to So,indeed, exceptfor the mass term, the Dirac equation
µ µ
operate on ψ¯ on the left, and σ = i[γ ,γ ] [15]. The written in this form shows explicit scale invariance, and
Maxwell field strength Fµν is defiabned a2s a b with the proper scale weight 32 for the Dirac field ψ.
The Euler-Lagrange equation for the Maxwell field is
F =∇ A −∇ A , obtained straightforwardly. It is of the form
µν µ ν ν µ
and Fµν =gµλgνρF . (∇µ+3Bµ)Fµν =Jµ, (20)
λρ
The Lagrangian in the action (10) is invariant under where the current J is given by
µ
local Lorentz transformations, general coordinate trans-
formations as well as local scale transformations (with Jµ =ψ¯γaeaµψ. (21)
3
III. CONSISTENCY OF FIELD EQUATIONS property. ThetorsionvectorB actsasaneffectiveWeyl
µ
scale gauge field, with its coefficient in (19) properly re-
In the presence of torsion, the field equation (20) is flecting the scale weight 32 of the Dirac field ψ. If we
not gauge invariant. We would like to check whether lookatthecurrentconservationequation(22),wenotice
current conservation is valid and whether the system of that the coefficient of the Bµ field is 3, though the cur-
field equations, namely (19) and (20), are mutually con- rent Jµ given by (21) actually carries a scale weight of
sistent. As aconsequenceofthe Dirac equation(19)and 4, which includes the scale weight of eaµ. The mismatch
its hermitian conjugate equation for ψ¯, it is straightfor- is accounted for by the scale transformation property of
ward to verify that the current Jµ is indeed conserved, the connection term in (22). Explicitly, (22) is
(∇µ+3Bµ)Jµ =0. (22) Jµ,µ+ΓµνµJν +3BµJµ =0, (22′)
inwhichtheconnectiontransforms,accordingto(15),as
Consistency of (20) with this current conservation equa-
tion (22) requires that Γµ →Γµ +Λ ,
νµ νµ ,ν
(∇ +3B )(∇ +3B )Fµν =0. (23)
µ µ ν ν which makes up for the missing scale weight. This sug-
gests the consideration of a modified connection
Making use of the anti-symmetry of Fµν, it is straight-
forward, though tedious, to show that Γ˜λ =Γλ −δλ B , (25)
νµ νµ ν µ
(∇ +3B )(∇ +3B )Fµν =−Rµ Fρν
µ µ ν ν ρµν (24) which is invariant under scale transformations. In terms
+21Cµρν∇µFρν + 23Fµν(∇µBν −∇νBµ). of the newly defined connection, current conservation
(22’) takes the form
For the right-hand side of (24) to vanish, it is necessary,
due to its structure, that the second term has to vanish. Jµ +Γ˜µ Jν +4B Jµ =0, (26′)
That is, we have to set Cµ = 0. When torsion van- ,µ νµ µ
ρν
ishes,B alsovanishes,andtheconnectionreducestothe or, equivalently,
µ
Christoffelconnection,implyingthatRµ is symmetric
inρandν. Thethreetermsontheright-ρhµaνndsideof(24) (∇˜µ+4Bµ)Jµ =0, (26)
thus allvanish. Consistency ofthe two field equationsof
which properly accounts for the scale weight 4 of the
the system (19) and (20) is seento require the vanishing
current Jµ, recalling that eµ in Jµ has a weight of 1.
of torsion. And, as a result, the system becomes gauge a
IntheMaxwellequation(20),similarly,thereisanap-
invariantat the same time. However,the two field equa-
parentmismatchofthescaledimensions. Corresponding
tions (19) and (20) are no longer scale invariant (with
to the newly defined connection, the Maxwell tensor is
m=0), even though the action W in (10) remains invari-
defined as
ant. It is also clear that if the Christoffel connection is
adoptedfortheMaxwellsector,consistencyrequiresthat F˜ =∇˜ A −∇˜ A , (27)
torsion is to be dropped from the coupled Dirac sector µν µ ν ν µ
as well. That is, torsion is to be totally dispensed with ItisthisnewlydefinedF˜ thatsubstitutesF intheac-
µν µν
in the system of Maxwell-Dirac electrodynamics. What
tionWin(10). Whiletheresultingfieldequationforthe
we havelearnedhere is that eventhoughcurrentconser-
Diracfieldstaysunchangedasin(19),thecorresponding
vation, being a consequence of the Dirac equation, does
Euler-Lagrange equation for the Maxwell field takes the
not depend on gauge invariance, consistency of the field
form
equations of the coupled system does.
(∇˜ +4B )F˜µν =Jµ, (28)
ν ν
IV. SEARCHING FOR SUITABLE which properly reflects the scale weight 4 of F˜µν.
CONNECTION
We have just seen that gauge invariance is indeed V. GAUGE INVARIANCE AND SCALE
closelytiedupwiththeconsistencyofthefieldequations. INVARIANCE
If, however, torsion is to play any role in the Maxwell-
Dirac electrodynamics, we need to rescue it by finding a WiththeMaxwell-Diracsystemtreatedwiththenewly
suitably modified connection that satisfies gauge invari- definedconnection,weagainchecktheconsistencyofthe
ance as well as achieves consistency of the field equa- correspondingfieldequations(19)and(28). Consistency
tions,without,however,requiringavanishingtorsion. A requires that (19) is compatible with the current conser-
cluecomesfromobservingthebehaviorofthefieldequa- vation (26), namely,
tions under scale transformations. We have noticed that
the Dirac equation (19) has a clean scale transformation (∇˜ +4B )(∇˜ +4B )F˜µν =0. (29)
µ µ ν ν
4
The left-hand side of the equation can be straightfor- functional derivativeof the actionW with respectto the
wardly calculated. The result is spin-connection ωab :
µ
1 1
−R˜µ F˜ρν+ C˜µ ∇˜ F˜ρν+2F˜µν(∇ B −∇ B ). (30) δW = d4xh S µδωab .
ρµν 2 ρν µ µ ν ν µ Z 2 ab µ
For this to vanish, it is necessary that The contributiontothe spincurrentbythe Diracfieldis
well-known [1, 3, 14] and given by
C˜µ =Γ˜µ −Γ˜µ =0, (31)
ρν ρν νρ 1
ε ψ¯γ γdψ.
abcd 5
which means that Γ˜λ is symmetric and implies that 2
µν
The contribution from the Maxwell photon field can be
Cλµν =gλµBν −gλνBµ. (32) obtained by varying −41F˜µνF˜µν in the action W with
respect to ωab . It vanishes identically, since the sym-
µ
The contortiontensorYλµν is calculatedaccordingto (9) metric connection Γ˜λνµ completely drops out from F˜µν.
to yield Thus, the Maxwell photon field has no contribution to
the spin current, and, as a result, neither to torsion. It
Yλµν =gµνBλ−gλνBµ. (33) is clear that gauge invariance of the Maxwell tensor Fµν
requires a symmetric connection, which, in turn, implies
We then obtain from (8) and (25) the results
that photon does not give rise to torsion.
1
Γλ = gλρ(g +g −g )+g Bλ−g B , (34)
µν 2 ρµ,ν νρ,µ µν,ρ µν ν µ
VII. CONCLUDING REMARKS
1 Ever since the advent of the Einstein-Cartan-Sciama-
Γ˜λ = gλρ(g +g −g )+g Bλ−gλ B −gλ B .
µν 2 ρµ,ν νρ,µ µν,ρ µν µ ν ν µ Kibble theory of gravitation, compatibility of torsion
(35) with gauge invariance has been a problem. Gauge in-
With the result (35), we calculate the antisymmetric variance can refer to abelian U(1) gauge theories as well
part of R˜µ in (30) and obtain as to non-abelian gauge theories in the standard model
ρµν
of particle physics. The compatibility problem would be
1 anobstacle for Cartan’storsionto play a role in the real
(R˜µ −R˜µ )=2(B −B ).
2 ρµν νµρ ν,ρ ρ,ν world of physics. As such, it is a hurdle waiting to be
overcome. To maintain gauge invariance, we do need
Thefirsttermandthirdtermin(30)exactlycanceleach a symmetric connection for the Maxwell field. But the
other out. Indeed, with the modified connection Γ˜λ symmetricChristoffelconnectionisnottheanswer,asits
µν
given in (35), consistency of the field equations (19) and adoption would imply that torsion should vanish in the
(28) is established. At the same time, Γ˜λ being sym- coupledDiracsectoraswell,andtheendresultwouldbe
µν
metric, F˜ = A −A , gauge invariance is also re- thattorsionisabsentintheentirecoupledMaxwell-Dirac
µν ν,µ µ,ν
stored. system. We have come up with a solution that exhibits
The Affine connection Γ˜λ given in (35) is seen to transparent gauge invariance, and, in addition, scale in-
µν
have exactly the same structure as a Weyl connection, variancein the system(except for the Dirac mass term),
with the torsion trace vector B = 1Cλ , a geometric withoutdispensewithtorsion. Thissamesolutionapplies
µ 3 λµ
entity in the Riemann-Cartan space, effectively playing as well to other gauge theories, abelian or non-abelian.
the role of the scale gauge field. With this connection The solution is embodied in a modified affine connec-
adoptedfor the Maxwell-Diracelectrodynamics,the sys- tion that possesses the structure of a Weyl connection,
temisselfconsistent,gaugeinvariantandscaleinvariant withthetorsiontracevectoreffectivelyservingtheroleof
(with m=0), without having to dispense with torsion. the Weyl scale gauge potential. For any given Riemann-
In contrast to the purely metric Christoffel connection, Cartan background, torsion is coupled to the Maxwell
theincorporationoftorsionintotheWeyl-likeconnection field Aµ through this connection, though it disappears
enables the system to exhibit good scale transformation in Fµν. Torsion is still effectively disengaged from the
properties, in addition to maintaining gauge invariance. photons, but stays engagedwith the coupled Dirac field.
This connection represents perhaps a minimal extension
of the ”minimal coupling” procedure for photons in the
generalRiemann-Cartanspace. Inasense,itprovidesan
VI. SPIN CURRENT AND TORSION
explicit and specific justification for the ansatz of Hehl
et al [3] in taking the Maxwell field strength in the form
The relationship between torsionand spin is an essen-
of its flat-space curl, without having to dispense with
tialfeatureofthe Einstein-Cartan-Sciama-Kibbletheory
of gravitation. The spin current S µ is defined as the torsion.
ab
5
Acknowledgments
The author would like to thank Professor Friedrich
Hehl for his thoughtful comments.
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