Second Order Formalism for Fermions

13 Pages·0.11 MB·English

by  J. L. Cortés

DFTUZ 92/13
IPNO-TH 92/82
September 1992
SECOND ORDER FORMALISM
FOR FERMIONS
†
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9
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1

n
a
J
J. L. Cort´es1, J. Gamboa2 and L. Vel´azquez1
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1
1Departamento de F´ısica Teo´rica, Universidad de Zaragoza,

1
50009 Zaragoza, Spain.
v
1
7 2Divisio´n de Physique Theo´rique, Institut de Physique Nucleaire‡,
0
1 F-91406 Cedex, Orsay, France.
0
3
9
/
h
t Abstract. A new formulation of fermions based on a second order action is proposed.
-
p
e An analysis of the U(1) anomaly allows us to test the validity of the formalism at the
h
: quantum level. This formulation gives a new perspective to the introduction of parity non
v
i
X invariant interactions.
r
a
PACS 11.10.-z , 11.30.Rd
† Work partially supported by CICYT (Proyecto AEN 90-0030).
‡ Unite de Recherche des Universit´es Paris 11 et Paris 6 associe au CNRS
In order to avoid the difficulties of the Klein-Gordon equation which results from a
correspondence principle applied to the relativistic case, Dirac [1] was lead to introduce a
first order relativistic equation which describes spin-1 particles. The associated first order
2
lagrangian, in constrast to the second order bosonic lagrangian, is the root of many of the
characteristic properties of half integer spin particles. In particular the peculiarities of the
quantization of a fermionic field and the correspondence of physical degrees of freedom
with field components, the axial anomaly in even dimensions and the species doubling
which appears when a fermionic field is formulated on the lattice, are some examples of
features which are related to the use of a first order action.
SincetheoriginaldifficultiesintheKlein-Gordonequationarestillpresent intheDirac
action, a natural question is whether it is possible to find a path integral formulation of
fermions based on a second order action. It is not the first time that this idea has been
considered [2]. The study of the way how all the properties which seem to be a direct
consequence of a first order action can be understood in the second order formulation
together with a first look at the possible advantages of using a second order fermionic
action, is the main subject of this letter.
A formulation based on the use of two component spinors instead of Dirac spinors
and its non-trivial consequences on the introduction of parity violating interactions as well
as its possible application to the understanding of the weak decays, was considered long
ago [3]∗. For a recent attempt to use these ideas to identify a lattice formulation of chiral
gauge theories see [5].
At present the fashionable way to introduce parity violating interactions through a
generalization to the chiral case of the gauge principle which allows to understand the
strong and electromagnetic interactions, has lead to a formulation of chiral gauge theories
were many open problems remain to be solved. It is then probably interesting to keep an
open mind on this problem and to try new ways to introduce parity violating interactions.
The second order formalism of fermions presented in this work, which is based on
the identification of decoupled field components in the Dirac action, provides a framework
∗ Some related aspects of the canonical quantization of the Feynman and Gell-Mann
formulation and some variations of it were considered in [4].
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were the introduction of parity non invariant interactions presents new aspects which have
not an analog in the parity conserving case.
The method we follow to obtain a second order formulation for fermions is to translate
at the level of the path integral formulation the derivation of a second order equation for
spinors. One way to do that is to rewrite the Dirac equation as a set of two first order
coupled equations. The starting point in the path integral formulation is the (euclidean)
first order gauge invariant action
1
S(1) = dDx mψ¯ψ + ψ¯(−→D/ ←D−/)ψ (1)
Z (cid:20) 2 − (cid:21)
where
−→Dµψ = (∂µψ ieAµψ),
−
ψ¯←D−µ = ∂µψ¯+ieAµψ¯ . (2)
(cid:0) (cid:1)
We assume an even space-time dimension D in order to decompose the Dirac field ψ
ψ 0 1+γ 1 γ
ψ = L + = D+1 ψ + − D+1 ψ,
(cid:18) 0 (cid:19) (cid:18)ψR(cid:19) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
1+γ 1 γ
ψ¯ = (ψ¯ 0)+(0 ψ¯ ) = ψ¯ D+1 +ψ¯ − D+1 , (3)
R L
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
where γ 2 = 1, γ ,γ = 0 and then
D+1 D+1 µ
{ }
ψ¯D/ψ = ψ¯ D/ ψ +ψ¯ D/ ψ . (4)
L L L R R R
If one introduces the variables
1 −→D/
χ = ψ , χ = √m(ψ + Lψ )
L L R R L
√m m
1 ←D−/
χ¯ = ψ¯ , χ¯ = √m(ψ¯ ψ¯ R), (5)
R R L L R
√m − m
then the first order action takes the form
1
S(1) = dDx m2χ¯ χ χ¯ (−→D/ −→D/ +←D−/ ←D−/ )χ +χ¯ χ
R L R R L R L L L R
Z (cid:20) − 2
1 1
+ χ¯ (−→D/ +←D−/ )χ χ¯ (←D−/ +−→D/ )χ , (6)
R R R R L L L L
2 − 2 (cid:21)
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where the last two terms which mix the two components reduce to a total derivative. The
variables χ ,χ¯ are auxiliary fields, they are decoupled from the gauge field and do not
R L
propagate. Then one can consider
S(2) = dDx m2χ¯ χ +χ¯ (←D−/ −→D/ )χ , (7)
R L R R L L
Z (cid:20) (cid:21)
asthe actionforthesecond order fermionicformulationwithallthe dynamicsconcentrated
on the anticonmutating fields χ¯ ,χ . One could anticipate from a naive counting of
R L
degrees of freedom, that a translation to a second order formulation must be acompanied
by a decoupling of half of the original fields.
One important point to remark is that a mass term is essential in order to go from
the first order action to the second order formulation. The massless case is special from
this point of view as well as the case of a chiral gauge theory where a mass term is not
allowed by gauge invariance.
The previous steps going from the Dirac fermionic action to the second order action
S(2) in (7) guarantee the equivalence of both formulations at the classical level. For
instance, it is straightforward to reconstruct the free Dirac propagator by combining the
propagators of the fermionic fields χ,χ¯ calculated with the action S(2) in the free case
(D ∂ )andtherelation(5)betweentheoriginalDiracfieldsψ andthefieldsχ,χ¯. What
µ µ
→
remains to be studied in order to establish the validity of the second order formulation is
to see that the quantum fluctuations do not destroy the equivalence with the Dirac first
order action.
In this sense one crucial point is to see whether the axial anomaly [6] is reproduced in
the present formulation. In order to study this question we will concentrate for simplicity
ontheabeliantwodimensionalcase. UsingtheWeylrepresentationforthetwodimensional
euclidean gamma matrices
γ = σ , γ = σ
1 2 2 1
−
γ = iγ γ = σ , (8)
5 1 2 3
−
and introducing for any vector the chiral components
a = a ia , (9)
± 1 2
±
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then one has in this case
D/ = iD
R −
−
D/ = iD , (10)
L +
and the second order lagrangian is given by
(2) = m2χ¯ χ χ¯ D D χ +“total derivatives”. (11)
R L R − + L
L { − }
In order to study how the anomaly is reproduced in the second order formulation let us
first rewrite the standard Fujikawa derivation of the anomaly in terms of the variables χ,χ¯
in (5). This will help us to understand the way the anomaly appears in the second order
formalism. In the Fujikawa method [7] the anomaly is understood as a non invariance of
the fermionic measure under the axial transformation. The fermionic measure is defined
in terms of the coefficients of the expansion of the fermionic field in eigenfuntions of the
Dirac operator
ψ ψ¯ = da da¯ , (12)
n n
D D
Yn
with
ψ = a ϕ
n n
Xn
ψ¯ = a¯ ϕ+, (13)
n n
Xn
and
D/ϕ = λ ϕ . (14)
n n n
If one makes use for the eigenfuntions ϕ of the spinorial decomposition in (3)
n
φ 0
ϕ = n + , (15)
n (cid:18) 0 (cid:19) (cid:18)η (cid:19)
n
and taking into account the spinorial structure of the two dimensional Dirac operator in
(10) one has
D D φ = λ2φ
− + n n n
D D η = λ2η . (16)
+ − n n n
5
Since
D/γ ϕ = λ γ ϕ
5 n n 5 n
−
we can consider λ > 0 (we do not consider the effect of zero modes in this discussion) and
n
there is a one to one correspondence between the measure of the Dirac field and a measure
of the components ψ ,ψ through the expansion in eigenfuntions of the operators D D
L R − +
and D D
+ −
ψ ψ = daLdaR, (17)
L R n n
D D
Yn
with
ψ = aLφ ,
L n n
Xn
ψ = aRη . (18)
R n n
Xn
If this expansion is plugged in (5) then it results into
χ = bLφ ,
L n n
Xn
χ = bRη . (19)
R n n
Xn
which leads to a fermionic measure
χ χ = dbLdbR, (20)
L R n n
D D
Yn
and a related factor for χ¯. Then the fermionic measure for the variables χ ,χ is defined
L R
in terms of the eigenfuntions of the operators D D and D D respectively.
− + + −
Once the measure has been identified then one can repeat the standard Fujikawa
evaluation of the anomaly. First one sees the effect over the measure,through the change
of the coefficients in the expansion of a chiral transformation ψ′(x) = eiα(x)γ5ψ(x)
db′Ldb′R = 1 i d2xα(x)A(x) dbLdbR, (21)
n n n n
(cid:18) − Z (cid:19)
Yn Yn
where A(x) = lim AL(x) AR(x) and
M→∞
(cid:20) − (cid:21)
D D
AL(x) = φ+(x)exp[ − +]φ (x), (22)
n − M2 n
Xn
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D D
AR(x) = η+(x)exp[ − +]η (x), (23)
n − M2 n
Xn
AL,AR being the constributions to the anomaly from the two components χ ,χ . The
L R
origin of the anomaly from this point of view is the difference (D D = D D ) between
+ − − +
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the operators involved in the two factors of the fermionic measure. The last step is to use
the basis independence of the trace involved in AL,AR to replace the eigenfuntions φ ,η
n n
by plane waves which allow a straightforward generalization of the Fujikawa’s calculation.
Making use of the identities
D D = D D +iF , (24)
− + µ µ 12
D D = D D iF , (25)
+ − µ µ 12
−
one obtains
1
A(x) = F , (26)
12
−2π
which can be combined with a similar contribution from χ¯ to reproduce the standard form
of the axial 2-dimensional anomaly. One remark to be made from this derivation of the
anomaly is that the variable χ plays an important role in this evaluation; even though the
R
χ component is completely decoupled at the level of the action it still envolves a gauge
R
field dependence in the definition of the measure which shows up in the axial anomaly.
Note that the axial U(1) global transformation of the original Dirac field
ψ′ = eiαψ , ψ′ = e−iαψ ,
L L R R
translates into
χ′ = eiαχ ,
L L
χ′ = e−iαχ +(eiα e−iα)D/ χ , (27)
R R L L
−
which involves the decoupled component χ in a non-trivial way.
R
Now, one can ask if the second order formulation, where χ is absent, will be able to
R
reproduce completely the anomaly. We will answer this question by a direct calculation of
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theanomalyinthesecond orderformulation. Our derivationisbased ontheeffectivegauge
field action generated by the fermion field fluctuations, which formally can be written as
(D/ D/ m2)
R L
Γ = lndet − . (28)
− (cid:20) (∂ ∂ m2) (cid:21)
− +
−
In order to study the anomaly one has to introduce an external field A5 which acts as a
µ
source for the axial current. This is done by replacing the Dirac operatorD/ by
/ =D/ iγ γ A5, (29)
µ 5 µ
D −
and the two dimensional anomaly can be identified from the term linear in the fields A
µ
and A5 in the expansion of the effective action Γ(A,A5)
µ
d2k
Γ (A,A5) = e A5( k)Π5 (k)A (k). (30)
2 Z (2π)2 µ − µν ν
When wecalculatetheeffectiveactionforthefirst order formulationthestandard result for
the U(1) anomaly follows directly from the rotational and gauge invariance of the Dirac
action (1). Since the term χ¯ χ neglected when going to the second order formalism
L R
respects these symmetries one can expect that the same result for the U(1) anomaly will
be obtained when the second order action (7) is used. Let us see explicitly the derivation of
this result from a direct calculation of the axial polarizationtensor Π5 . When the effective
µν
action in (28),with D replaced by ,is expanded in powers of A and A5 then one obtains
µ µ
D
d2l k k k k
Π5 = 4 P(l) (l+ ) (l+ ) P(l+ )P(l ) , (31a)
+− Z (2π)2(cid:20) − 2 − 2 + 2 − 2 (cid:21)
d2l k k k k
Π5 = 4 (l+ ) (l ) P(l+ )P(l ), (31b)
++ − Z (2π)2 2 + − 2 + 2 − 2
d2l k k k k
Π5 = 4 (l+ ) (l ) P(l+ )P(l ), (31c)
−− Z (2π)2 2 − − 2 − 2 − 2
d2l k k k k
Π5 = 4 P(l) (l ) (l ) P(l+ )P(l ) , (31d)
−+ − Z (2π)2(cid:20) − − 2 + − 2 − 2 − 2 (cid:21)
where
P(q) = [q2 +m2]−1, (32)
is the bosonic free propagator.
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The integrals in (31) are logarithmically divergent and an ultraviolet regularization is
required in order to evaluate the effective action. It is convenient to make a decomposition
Π5 = Π¯5 +Π˜5 . (33)
µν µν µν
The first term is given by the axial polarization tensor with a substraction of the integrand
at zero external momentum in order to have a convergent integral. All the regularization
dependence will be concentrated on the momentum independent contribution Π˜5 .
µν
The regularization independent contribution can be written as
Π¯ = 4I , Π¯ = 4I ,
++ ++ −− −−
−
Π¯ = Π¯ = 4m2B, (34)
+− −+
−
where
d2l k k k k
I = (l+ ) (l ) P(l+ )P(l ) l l P2(l) , (35)
µν Z (2π)2(cid:26) 2 µ − 2 ν 2 − 2 − µ ν (cid:27)
d2l k k
B = P(l+ )P(l ) P2(l) , (36)
Z (2π)2(cid:26) 2 − 2 − (cid:27)
and by covariance arguments
k k
µ ν
I = C +Dg , (37)
µν µν
k2
where m2B, C, and D will be functions of k2 .
m2
If one uses a gauge invariant regularization of the effective action then
k Π5 = k Π¯5 +k Π˜5 = 0, (38)
ν µν ν µν ν µν
and this is enough to fix the regularization dependent component Π˜
Π˜ = Π˜ = 0,
++ −−
Π˜ = Π˜ = 4(m2B C). (39)
+− −+
− − −
When this result is combined with Π¯ in (34) one finds
k k
Π5 = 4iCǫ µ ρ, (40)
µν ρν k2
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and the anomaly can be read from
k Π5 = 4iCk ǫ . (41)
µ µν ρ ρν
In fact, since the axial polarization tensor is related to the effective action by (30),
then one has in the limit m 0
→
∂ δΓ(A,A5) ie
= 2ieC(m = 0)ǫ F = ǫ F , (42)
ρν ρν ρν ρν
∂x δA5(x) (cid:12) −2π
µ µ (cid:12)A5=0
(cid:12)
(cid:12)
which is the standard result for the anomaly of the axial current as obtained from the
first order Dirac action. Then one can rederive the anomaly from the effective action of
the second order formulation, which makes manifest the decoupling at the quantum level
of the auxiliary field χ . This is in contrast with the derivation of the anomaly from
R
the fermionic measure where χ was an essential ingredient. In fact the current one is
R
considering in the second order formulation is given by
δS(2)
J(2)(x) = = i χ¯ ←D−/ (γ ) χ +χ¯ (γ ) −→D/ χ , (43)
µ5 δA5(x) − (cid:20) R R µ L L R µ R L L(cid:21)
µ
which is the Noether current associated to the transformation
χ′ = eiαχ , χ¯′ = χ¯ eiα. (44)
L L R R
Then the perturbative calculation based on the efective action leads to
δΓ
∂ < J(2) >= ∂ =< δS(2) > +2AL (45)
µ µ5 µδA5(x)
µ
where the first term on the right side takes into in account the variationof the second order
actionunder the global transformation (44)and the second term 2AL isthe contribution to
the Ward identity from the measure which in the second order formalism involves χ ,χ¯
L R
exclusively.
The contribution to the anomaly from the decoupled variables is included in the
second order formulation in the variation of the action under a chiral transformation of
the dynamical variables. This is the way the anomaly is reproduced in this context.
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