Mach like principle from conserved charges

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by  Eduardo Guendelman

Mach like principle from conserved charges
Eduardo Guendelman
Roee Steiner
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
January 26, 2012
2
1
0
2 Abstract

n
We study models where the gaugecoupling constants,massesandthe gravitationalconstant
a
J arefunctions ofsomeconservedchargeinthe universe,andfurthermoreacosmologicalconstant

5 that depends on the total charge of the universe. We first consider the standard Dirac action,
2 but where the mass and the electromagnetic coupling constant are a function of the charge
in the universe and afterwards extend this to curved spacetime and consider gauge coupling
]
h constants, the gravitation constant and the mass as a function of the charge of the universe,
t which represent a sort of Mach principle for all the constants of nature. In the flat space
-
p formulation, the formalism is not manifestly Lorentz invariant, however Lorentz invariance can
e be restored by performing a phase transformation of the Dirac field, while in the curved space
h
time formulation, there is the additional feature that some of the equations of motion break
[
the general coordinate invariance also, but in a way that can be understood as a coordinate

1 choice only, so the equations are still of the General Relativity type, but with a certain natural
v
coordinate choice, where there is no current of the charge. We have generalized what we have
7
done and also constructed a cosmological constant which depends on the total charge of the
5
2 universe. If we were to use some only approximately conserved charge for these constructions,
5 like say baryon number (in the context of the standard model), this will lead to corresponding
. violationsofLorentzsymmetryintheearlyuniverseforexample. Wealsobrieflydiscussanother
1
0 nolocalformulationswherethecouplingconstantsarefunctionsofthePontryaginindexofsome
2 non abelian gauge field configurations. The construction of charge dependent contributions can
1 alsobe motivatedfrom the structure of the ”infra-redcounter terms” needed to cancelinfra red
:
v divergences for example in three dimension.
i
X
r 1 Introduction
a
Mach’sprincipleiswellknown,asaprinciplethatrelatesalocalproblemtoanonlocalproblem.
The original Mach principle [1] is based on the claim that the inertial frames are influenced by
the other celestial bodies. In other words’ every mass in the universe is influenced by all the
othermassesintheuniverse. TheMachprincipleisstillinadebate. Inourarticle,wewillshow
that there is a possibility to precisely formulate a Mach principle for electromagnetic coupling
constantandindeedanyotherconstantofnature,wherewetaketheseconstantstobeafunction
of the total charge. This breaks the locality of the problem (in the original Mach principle the
mass broke the locality). We briefly discuss also the possibility of these constants depending on
the Pontryaginindex of some non abelian gauge field configurations.
Inref.[9],inordertocancelinfrareddivergencein3-dimensiongaugetheories,infraredcounter
terms are introduced. This is shown in ref.[9] to be equivalent to the procedure developed in
ref[11]tocancelinfrareddivergencebyintroducingzeroenergymomentumphotons. Theseinfra
red divergences are related to the super renormalizabiltyof the theory in 3- dimensions [10] We
generalize this idea and define the coupling constant to be proportionalto this term (and latter
to be an arbitrary function of (55)), we latter connect this procedure with our treatment.
1
2 The electromagnetic coupling constant as a function of
all the charge in the universe
We begin by considering the action for the Dirac equation (see for example ref [2])
i
S =Z d4xψ¯(2γµ ↔∂µ −eAµγµ−m)ψ (1)
where ψ¯ = ψ γ0.However here we take the coupling constant e to be proportional to the total
†
charge (we will afterwards generalize and consider an arbitrary function of the total charge).
e=λ ψ (~y,y0 =t )ψ(~y,y0 =t )d3y =λ ρ(~y,y0 =t )d3y (2)
Z † 0 0 Z 0
and we will show that physics does not depend on the time slice y0 =t
0
so after the new definition of ”e” the action will be:
i
S =Z d4xψ¯(x)(2γµ ↔∂µ −m)ψ(x)
λ( d3yψ¯(~y,y0 =t )γ0ψ(~y,y0 =t ))( d4xψ¯(x)A γµψ(x)) (3)
− Z 0 0 Z µ
we can express the three dimensional integral as a four dimensional integral
d3yψ¯(~y,y0 =t )γ0ψ(~y,y0 =t )= d4yψ¯(y)γ0ψ(y)δ(y0 t ) (4)
Z 0 0 Z − 0
so finally the action will be
i
S =Z d4xψ¯(x)(2γµ ↔∂µ −m)ψ(x)−λ(Z d4xψ¯(x)Aµγµψ(x))(Z d4yψ¯(y)γ0ψ(y)δ(y0−t0)) (5)
ifweconsiderthefactthat δψ¯a(x) =δ4(x z)δ and δψ(x) =0wegettheequationofmotion
δψ¯b(z) − ab δψ¯(z)
δS
=0= δ4(x z)(iγµ∂ m)ψ(x)d4x
δψ¯(z) Z − µ−
λ( d4xδ4(x z)A γµψ(x))( d4yψ¯(y)γ0ψ(y)δ(y0 t ))
− Z − µ Z − 0
λ( d4xψ¯(x)A γµψ(x))( d4yδ4(y z)γ0ψ(y)δ(y0 t )) (6)
− Z µ Z − − 0
so to accomplish our goal we need just to perform the integrations in the last equation, and
then the expression will simplified to
δS
=(iγµ∂ m)ψ(z) λ( ψ¯(y)γ0ψ(y)δ(y0 t )d4y)A γµψ(z)
δψ¯(z) µ− − Z − 0 µ
λ( ψ¯(x)A γµψ(x)d4x)γ0ψ(z)δ(z0 t ) (7)
− Z µ − 0
whichcanbesimplifiedmorebytheuseofnewdefinitionb =λ( ψ¯(x)A γµψ(x)d4x)which
e µ
is a constant, and by the definition in equation (2) R
δS
=[iγµ∂ m eA γµ b γ0δ(z0 t )]ψ(z)=0 (8)
δψ¯(z) µ− − µ − e − 0
2
sowe canseethatthe lastterminthe equationofmotion(8)containsAGFγµ whereAGF =
µ µ
∂ Λ and Λ=b θ(z0 t ) is a pure gauge field. so the solution of this equation is
µ e 0
−
ψ =e−ibeθ(z0−t0)ψD (9)
where ψ is the solution of the equation
D
[iγµ∂ m eA γµ]ψ =0 (10)
µ µ D
− −
fromwhichitfollowsthatjµ =ψ¯ γµψ =ψ¯γµψsatisfiesthelocalconservationlaw∂ jµ =0
D D µ
and therefore we obtain that Q= d3xj0 is conserved,so it does not depend on the time slice,
furthermore it also follows that it Ris a scalar as we have proved in the appendix (A).
3 Mass as a function of all the charge in the universe
We will show now that we can do the same thing as in paragraph(2) for the mass. We consider
the action equation (1) where we take the mass to be equal to
m=λ ψ¯γ0ψd3y (11)
Z
wedothesamethinglikeinparagraph(2)soweexpandequation(11)likewedidinequation
(4), so we will get the equation of motion
δS
=[iγµ∂ m eA γµ b γ0δ(z0 t )]ψ(z)=0 (12)
δψ¯(z) µ− − µ − m − 0
whereb =λ ψ¯ψd4x. Soagainwecaneliminateδ(z0 t )termbyaphasetransformation,
m 0
−
with the same conRclusion that there is no violation of Lorentz invariance.
4 Coupling constant as a general function of all the charge
in the universe
For a generalcoupling constant”a” and a generalfunction F(ψ¯,ψ)and a coupling constant”e”
the action of the Dirac equation is
i
S =Z d4xψ¯(2γµ ↔∂µ −eAµγµ−m)ψ+Z d4x[aF(ψ¯,ψ)] (13)
we take the coupling constants ”a” and ”e” as an arbitrary functions g and g of
a e
Q= ψ (~y,y0 =t )ψ(~y,y0 =t )d3y
† 0 0
R
a=ga(Z ψ†(~y,y0 =t0)ψ(~y,y0 =t0)d3y)=ga(Z ρ(~y,y0 =t0)d3y) (14)
e=g ( ψ (~y,y0 =t )ψ(~y,y0 =t )d3y)=g ( ρ(~y,y0 =t )d3y) (15)
e Z † 0 0 e Z 0
we put the last definition in equation (13) and expand equation (14) like we did in equation
(4), we do a variation and use the fact that δψ¯a(x) = δ4(x z)δ and δψ(x) = 0 and do the
δψ¯b(z) − ab δψ¯(z)
integration as like we did in peragraph (2) so we get to the general motion equation
δS ∂F(ψ(¯z),ψ(z))
=[iγµ∂ m eA γµ b γ0δ(z0 t )]ψ(z)+a( )=0 (16)
δψ¯(z) µ− − µ − ae − 0 ∂ψ¯(z)
whereb = ∂ga(Q) d4xF(ψ¯,ψ)+∂ge(Q) d4xψ¯γ ψAµ sowegetthatanycouplingconstant
ae ∂Q ∂Q µ
in this form canbe a fuRnction of the chargeinRthe universe without violatingLorentz invariance
(again after performing the appropriate phase transformation).
3
5 Gravitational coupling constant as a function of all the
charge in the universe
We willshowthatwecandefine the gravitationcouplingconstantasafunctionofallthe charge
inthe universe. We startwiththe definitionofthe equationsofeachofoursparametersbecause
we are dealing with curved space. The Dirac action in curved space is [3]
i
SD =Z d4x√gψ¯(2γα D↔α −m)ψ (17)
where D = eµ∂ ieµη ωc σαb is the covariant derivative for fermion and eµ is the
α α µ − 4 α αc bµ α
vierbein and σαβ = i[γα,γβ] is the commutator of the Dirac gamma metric, and the spin
2
connection is ωc = ec∂ eν + eceσΓν where Γν is the Christoffel symbol and η is the
bµ ν µ b ν b σµ σµ αc
lorentzian metric. the second definition is the action of the gravity [4]
1
SG =−16πGZ √gR(x)d4x (18)
where we will take gravitationalconstant to be
1
=λ √gψ¯e0γαψd3y (19)
−16πG Z α
the total action is
S =S +S (20)
t G D
we will do now the variation on (20) by the vierbein and by ψ¯:
1 1
δS = √g[Rλ δλR]eαδeµd4x+
t 8πGZ µ− 2 µ λ α
i
λ(Z √gR(x)d4x)Z ψ¯γβψ[eβ0eαµδeαµ+δeβ0]√gd3y+Z d4x√geαµδeαµψ¯(2γβ D↔β −m)ψ
i
+Z d4x√gδeβµψ¯(2γβ D↔µ)ψ
+λ(Z √gδψ¯eα0γαψd3y)(Z √gR(x)d4x)+Z d4x√gδψ¯(iγαDα−m)ψ =0(21)
because we have two different variations, we can produce two equation. The first equation
(from the variation with respect to ψ) is:
λ( √gδψ¯e0γαψd3y)( √gR(x)d4x)+ d4x√gδψ¯(iγαD m)ψ =0 (22)
Z α Z Z α−
we will use equation (4) and the fact that δψ¯a(x) = δ4(x z)δ and δψ(x) = 0 and after
δψ¯b(z) − ab δψ¯(z)
integration we get:
(iγαD m+λe0γαδ(z0 t )( d4xR√g))ψ =0 (23)
α− α − 0 Z
whichmeanthatagainwehaveaphasetransformationofψthateliminatestheδ(z0 t )term.
0
−
Before we proceedto the secondequationwe will notice that the expressionδeβµψ¯(2iγβ D↔µ)ψ in
equation (21) can be modified to:
i i
δeβµψ¯(2γβ D↔µ)ψ = 4δeβµψ¯(γβ D↔µ +γµ D↔β)ψ (24)
4
So the second equation is(from the variation with respect to the vierbein):
1
[Rλ δλR]eα =8πGλ( √gR(x)d4x)ψ¯γβψδ(y0 t )[e0eα +δαδ0]
− µ− 2 µ λ Z − 0 β µ β µ
i i
+8πGeαµψ¯(2γα D↔α −m)ψ+8πG4ψ¯(γα D↔µ +γµ D↔α)ψ (25)
By contracting equation (25) with e =eδ η and using the fact that eαeδ η =g and
αγ γ αδ λ γ αδ λγ
the fact that ψ¯(2iγα D↔α −m)ψ =−λ( √gR(x)d4x)ψ¯eβ0γβψδ(y0−t0) we get the Einstein field
equation with a modification R
1 i i
Rγµ gµγR= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − 4 4
8πλG( √gR(x)d4x)ψ¯γβψδ(y0 t )[eδg δ0] (26)
− Z − 0 β δγ µ
Becauseequation(26)issymmetric(thelasttermisnotsymmetric)thenwehavetoconclude
that
ψ¯γβeδg ψ =0 (27)
β δi
which means that
ψ¯γi(∗)ψ =0 (28)
where γi(∗) =γβeβδgδi , which means that we have imposed a gauge condition for the coordinate
so that spatial part of the Dirac current is equal to zero, so equation (26) will become in these
coordinates,
1 i i
Rγµ gµγR= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − 4 4
8πλG( √gR(x)d4x)ψ¯γβeδg ψδ(y0 t )[δ0δ0] (29)
− Z β δ0 − 0 γ µ
It is important to note that γ and µ in γ and γ in equation (29) are general coordinate
γ µ
indices while β in γβ is a Lorentz index.
The solution of equation (23) is ψ = eiλ(R√gR(x)d4x)θ(z0−t0)ψD where (iγαDα m)ψD = 0.
−
If we expand the term ψ¯(4iγγ D↔µ +4iγµ D↔γ)ψ in equation (29) we will get:
i i i i
ψ¯( γγ D↔µ + γµ D↔γ)ψ =ψ¯D( γγ D↔µ + γµ D↔γ)ψD
4 4 4 4
λ
−2(Z √gR(x)d4x)δ(z0−t0)ψ¯D[γγδµ0 +γµδγ0]ψD (30)
If we use the condition in equation (28) we will get:
i i i i
ψ¯( γγ D↔µ + γµ D↔γ)ψ =ψ¯D( γγ D↔µ + γµ D↔γ)ψD
4 4 4 4
λ( √gR(x)d4x)δ(z0 t )ψ¯ γ [δ0δ0]ψ (31)
− Z − 0 D 0 γ µ D
so equation (29) will become to the ordinary Einstein equation with the conventional form
for the energy momentum tensor for fermions:
1 i i
Rγµ gµγR= 8πGψ¯D( γγ D↔µ + γµ D↔γ)ψD (32)
− 2 − 4 4
since both sides of equation (32) have a nice covariant structure the same form of equation
(32) will be maintained in any coordinate system.
5
6 Gravitational coupling constant as a general function of
all the charge in the universe
We will redefine equation (19) to be:
1
=λF( √gψ¯e0γβψd3y)=λF(Q) (33)
−16πG Z β
so now equation (21) will be:
1 1
δS = √g[Rλ δλR]eαδeµd4x+
t 8πGZ µ− 2 µ λ α
∂F(Q)
λ( √gR(x)d4x) ψ¯γβψ[e0eαδeµ+δe0]√gd3y
Z ∂Q Z β µ α β
i i
+Z d4x√geαµδeαµψ¯(2γβ D↔β −m)ψ+Z d4x√gδeβµψ¯(2γβ D↔µ)ψ
∂F(Q)
+λ ( √gδψ¯e0γαψd3y)( √gR(x)d4x)+ d4x√gδψ¯(iγαD m)ψ =0 (34)
∂Q Z α Z Z α−
So we have two equation, the first is:
∂F(Q)
(iγαDα−m+λ ∂Q eα0γαδ(z0−t0)(Z d4xR√g))ψ =0 (35)
which mean that again we have phase transformation ψ = ei∂F∂(QQ)λ(R√gR(x)d4x)θ(z0−t0)ψD that
eliminates the δ(z0 t ) term.
0
−
The second equation (after a modification of equation (24) ) is:
1 ∂F(Q)
[Rλ δλR]eα =8πGλ( √gR(x)d4x)ψ¯γβψδ(y0 t ) [e0eα +δαδ0]
− µ− 2 µ λ Z − 0 ∂Q β µ β µ
i i
+8πGeαµψ¯(2γα D↔α −m)ψ+8πG4ψ¯(γα D↔µ +γµ D↔α)ψ (36)
So, if we follow after the paragraph before and use the constraint of equation (28) we will
get:
1 i i
Rγµ gµγR= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − 4 4
∂F(Q)
8πλG( √gR(x)d4x)ψ¯γβeδg ψδ(y0 t ) [δ0δ0] (37)
− Z β δ0 − 0 ∂Q γ µ
if we express equation (37) in term of ψ , then equation (28) once again is obtained and
D
finally , once again, we will get the Einstein equation (32)
7 Mass coupling constant as a general function in curved
space
The action of Dirac is:
i
SD =Z d4x√gψ¯(2γα D↔α −m)ψ (38)
where we will defined the mass to be:
m=λF( √gψ¯e0γβψd3y)=λF(Q) (39)
Z β
6
and the action of the curved space is defined by equation (18), so the varation of the total
action is:
1 1
δS = √g[Rλ δλR]eαδeµd4x
t 8πGZ µ− 2 µ λ α −
∂F(Q)
λ( √gψ¯ψd4x) ψ¯γβψ[e0eαδeµ+δe0]√gd3y
Z ∂Q Z β µ α β
i i
+Z d4x√geαµδeαµψ¯(2γβ D↔β −m)ψ+Z d4x√gδeβµψ¯(2γβ D↔µ)ψ
∂F(Q)
−λ ∂Q (Z √gδψ¯eα0γαψd3y)(Z √gψ¯ψd4x)+Z d4x√gδψ¯(iγαDα−m)ψ =0 (40)
as we done at the last section, we can have two equation:
∂F(Q)
(iγαD m λ e0γαδ(z0 t )( d4xψ¯ψ√g))ψ =0 (41)
α− − ∂Q α − 0 Z
which mean that again we have a phase transformation ψ = e−i∂F∂(QQ)λ(Rψ¯ψd4x)θ(z0−t0)ψD that
eliminates the δ(z0 t ) term
0
−
and
1 i i
Rγµ gµγR= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − 4 4
∂F(Q)
+8πλG( √gψ¯ψd4x)ψ¯γβψδ(y0 t ) [eδg δ0] (42)
Z − 0 ∂Q β δγ µ
we can see that again we need to use the condition on the coordinate as in equation (28) so
finally with the condition we will have:
1 i i
Rγµ gµγR= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − 4 4
∂F(Q)
+8πλG(Z √gψ¯ψd4x)ψ¯γβeβδgδ0ψδ(y0−t0) ∂Q [δγ0δµ0] (43)
if we express equation (43) in term of ψ , then equation (41) once again is obtained and
D
finally , once again, we will get the Einstein equation (32)
8 Cosmological constant depending on the total charge in
the universe
We will show that we can make a cosmologicalconstantthat depends on the totalcharge of the
universe. we will start with the action:
i
SD =Z d4x√gψ¯(2γα D↔α −m)ψ (44)
where D = eµ∂ ieµη ωc σαb is the covariant derivative for fermion and eµ is the
α α µ − 4 α αc bµ α
vierbein and σαβ = i[γα,γβ] is the commutator of the Dirac gamma metric, and the spin
2
connection is ωc = ec∂ eν + eceσΓν where Γν is the Christoffel symbol and η is the
bµ ν µ b ν b σµ σµ αc
lorentzian metric. the second definition is the action of the gravity [4]
1
S = √g(R(x) ΛQ)d4x (45)
G −16πGZ −
where Q= √gψ¯e0γαψd3y and √gΛQd4x is a charge dependent cosmologicalconstant like
α
term. The tRotal action is S =S R+S . So by variation we will get:
t D G
7
1 1
δS = √g[Rλ δλR]eαδeµd4x+
t 8πGZ µ− 2 µ λ α
Λ ΛQ
( √gd4x) ψ¯γβψ[e0eαδeµ+δe0]√gd3y+ √geαδeµd4x+
16πG Z Z β µ α β 16πGZ µ α
i i
Z d4x√geαµδeαµψ¯(2γβ D↔β −m)ψ+Z d4x√gδeβµψ¯(2γβ D↔µ)ψ
Λ
+ ( √gδψ¯e0γαψd3y)( √gd4x)+ d4x√gδψ¯(iγαD m)ψ =0 (46)
16πG Z α Z Z α−
because we have two different spaces, we can produce two equation. the first equation is:
Λ
( √gδψ¯e0γαψd3y)( √gd4x)+ d4x√gδψ¯(iγαD m)ψ =0 (47)
16πG Z α Z Z α−
we will use equation (4) and the fact that δψ¯a(x) = δ4(x z)δ and δψ(x) = 0 and after
δψ¯b(z) − ab δψ¯(z)
integration we get:
Λ
(iγαDα−m+ 16πGeα0γαδ(z0−t0)(Z d4x√g))ψ =0 (48)
which mean that again we have a gauge invariance transform which his solution is ψ =
ei16ΛπG(R√gd4x)θ(z0−t0)ψD.
The second equation that follows by the variation is:
1 ΛQ Λ
[Rλ δλR+ δλ]eα = ( √gd4x)ψ¯γβψδ(y0 t )[e0eα +δαδ0]
− µ− 2 µ 2 µ λ 2 Z − 0 β µ β µ
i i
+8πGeαµψ¯(2γα D↔α −m)ψ+8πG4ψ¯(γα D↔µ +γµ D↔α)ψ (49)
By contracting equation (49) with e = eδ η and use the fact that eαeδ η = g and
αγ γ αδ λ γ αδ λγ
the fact that ψ¯(2iγα D↔α −m)ψ = −16ΛπG( √gd4x)ψ¯eβ0γβψδ(y0−t0) we get the Einstein field
equation with a modification R
1 i i
Rγµ gµγ(R ΛQ)= 8πGψ¯( γγ D↔µ + γµ D↔γ)ψ
− 2 − − 4 4
Λ
( √gd4x)ψ¯γβψδ(y0 t )[eδg δ0] (50)
−2 Z − 0 β δγ µ
From here we follows the step as was done from equation (26) to equation (32) where now
ψ =ei16ΛπG(R√gd4x)θ(z0−t0)ψD and we have the Einstein equation with cosmologicalconstant:
1 i i
Rγµ gµγ(R ΛQ)= 8πGψ¯D( γγ D↔µ + γµ D↔γ)ψD (51)
− 2 − − 4 4
we got a cosmologicalconstant which depends on the charge of all the universe.
9 Using approximately conserved charges, the large num-
ber hypothesis
So far we have discussed the construction of the coupling constants as functions of conserved
quantities, like electric charge, but there are many known charges that although our standard
theories predict to be non conserved, there is still no direct evidence for it, for example baryon
number (the matter anti matter asymmetry may indicate to a violationof baryonnumber [5] in
the early universe, but does not prove it) . If we were to use such non conserved charge, then
8
we will not be able to restore Lorentz invariance, the time slice where the charge is defined will
be important, the theory constructed in this way could behave very different than the standard
one say in the early universe, but in more ”calm” periods, where there is no even a direct hint
of baryon number violation, the differences with the standard model would dissapear.
It is interesting also to note that A. Eddington thought of the possibility that Newton’s G
could be influence by the large number of particles [6]. In our case indeed we can formulate
a theory where G=G(N), where N is for example number of baryons, the theory will respect
Lorentz invariance if we work under the approximation that baryon number is conserved, not
exact in the standard model, but very close to exact indeed.
10 Coupling constant depending on Pontryagin index
Until now we have defined the coupling constant to be a function of:
Q= ψ¯γ0ψd4x (52)
Z
Now we will mention the possibility to anther direction to another direction which the coupling
constantsare function ofthe Pontryaginindex orthe ”winding number”of some gaugefields:[7]
1 1 2
N = d4xTr(ǫµναβF F )= d4xTr(∂ (ǫµναβ(A F A A A ))) (53)
16π2 Z µν αβ 16π2 Z µ ν αβ − 3 α β ν
this number measures the number of time the hypersphere at infinity is wrapped around G =
µ
ǫµναβ(A F 2A A A ).
ν αβ − 3 α β ν
one can see that any local variation of A which leaves the boundaries unaffected douse not
µ
change N,therefore:
δN =0 (54)
so now by construction of the coupling constant to be function of N we will get the ordinary
DiracequationbecauseoftheδN =0situation. Noticehoweverthatthewindingnumberisonly
well defined for configurations of finite action in Euclidean space, like for example instantons.
11 Non local contribution motivated from ”infra red counter
terms”
In ref.[9] ,in order to cancel infra red divergence in 3 - dimension gauge theories , infra red
counter terms of the form:
c( jµd3y)( j d3x) (55)
Z Z µ
are introduced. This is shown in ref.[9] to be equivalent to the procedure developed in ref[11]
to cancel infra red divergence by introducing zero energy momentum photons. These infra red
divergences are related to the super renormalizabilty of the theory in 3- dimensions [10] We
generalize this idea and define the coupling constant to be proportionalto this term (and latter
tobeanarbitraryfunctionof(55)),welatterconnectthisprocedurewithourprevioustreatment.
We again begin by considering the action for the Dirac equation
i
S =Z d4xψ¯(2γµ ↔∂µ −eAµγµ−m)ψ (56)
but now we take the coupling constante to be proportionalto the appropriategeneralizationof
(55) in the context of a four dimensional theory.
λ λ
e= ( jµd4y)( j d4z)= ( ψ¯γµψd4y)( ψ¯γ ψd4z) (57)
2 Z Z µ 2 Z Z µ
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so the action will be
i λ
S =Z d4xψ¯(x)(2γµ ↔∂µ −m)ψ(x)−2(Z d4xψ¯(x)Aµγµψ(x))(Z ψ¯(y)γµψ(y)d4y)(Z ψ¯(z)γµψ(z)d4z)
(58)
ifweconsiderthefactthat δψ¯a(x) =δ4(x z)δ and δψ(x) =0wegettheequationofmotion
δψ¯b(z) − ab δψ¯(z)
δS λ
=(iγµ∂ m)ψ(z) ( ψ¯(z)γµψ(z)d4z)( ψ¯(z)γ ψ(z)d4z)A γµψ(z)
δψ¯(z) µ− − 2 Z Z µ µ
λ( ψ¯(x)A γνψ(x)d4x)( ψ¯(z)γµψ(z)d4z)γ ψ(z) (59)
− Z ν Z µ
whichcanbesimplifiedmorebytheuseofnewdefinitionbµ =λ( ψ¯(x)A γνψ(x)d4x)( ψ¯(z)γµψ(z)d4z)
e ν
which is a constant, and by the definition in equation (57) R R
δS
=[iγµ∂ m eA γµ bµγ ]ψ(z)=0 (60)
δψ¯(z) µ− − µ − e µ
The solution of this equation is
ψ =e−ibµexµψD (61)
where ψ is the solution of the equation
D
[iγµ∂ m eA γµ]ψ =0 (62)
µ µ D
− −
So we have just gauge transformation. We can connect this kind of contractions to what we did
before in the page by generalizing the way we introduce the charge in the action from
e=λ d4xδ(t t )ψ¯γ0ψ (63)
Z − 0
to
e= d4x λ δ(t t )ψ¯γ0ψ λ =λ (64)
Z i − i i
X X
i i
and can use also instead of λ δ(t t ) an arbitrary function of time f(t)
i i − i
P
e= f(t)ψ¯(t,x)γ0ψ(t,x)d4x, f(t)dt=λ (65)
Z Z
from the special case f(t)=const, we have to deal with
ψ¯γ0ψd4x=n ψ¯γµψd4x (66)
Z µZ
where n =(1,0,0,0). Notice that in principle n can be an arbitrary constant vector, then all
µ µ
the procedure goes thought, i.e. the non covariant terms are possible to eliminate by a gauge
transformation. The desired construction is obtained when choosing for n also ψ¯γ ψd4x
µ µ
leading them to R
1
( ψ¯γµψd4x)( ψ¯γ ψd4x) (67)
2 Z Z µ
the 1 factorisincludednow,becausen hasbecomedynamicalandits variationalsocontribute
2 µ
to the equations of motion.
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