Path integrals and anomalies in curved space

342 Pages·2006·1.734 MB·English

by  Fiorenzo Bastianelli, Peter van Nieuwenhuizen

Path Integrals and Anomalies in Curved Space
Fiorenzo Bastianelli 1
Dipartimento di Fisica, Universita(cid:181) di Bologna
and INFN, sezione di Bologna
via Irnerio 46, Bologna, Italy
and
Peter van Nieuwenhuizen 2
C.N. Yang Institute for Theoretical Physics
State University of New York at Stony Brook
Stony Brook, New York, 11794-3840, USA
1email: [email protected]
2email: [email protected]
Abstract
In this book we study quantum mechanical path integrals in curved
and (cid:176)at target space (nonlinear and linear sigma models), and use the
results to compute the anomalies of n-dimensional quantum fleld theories
coupled to external gravity and gauge flelds. Even though the quantum
fleld theories need not be supersymmetric, the corresponding quantum
mechanical models are often supersymmetric. Calculating anomalies us-
ing quantum mechanics is much simpler than using the full machinery of
quantum fleld theory.
In the flrst part of this book we give a complete derivation of the path
integrals for supersymmetric and non-supersymmetric nonlinear sigma
models describing bosonic and fermionic point particles (commuting co-
ordinates xi(t) and anticommuting variables ˆa(t) = ea(x(t))ˆi(t)) in
i
a curved target space with metric g (x) = ea(x)eb(x)– . All our cal-
ij i j ab
culations are performed in Euclidean space. We consider a flnite time
interval because this is what is needed for the applications to anomalies.
As these models contain double-derivative interactions, they are diver-
gent according to power counting, but ghost loops cancel the divergences.
Onlytheone-andtwo-loopgraphsarepowercountingdivergent, hencein
general the action may contain extra flnite local one- and two-loop coun-
terterms whose coe–cients should be flxed. They are flxed by imposing
suitablerenormalizationconditions. Toregularizeindividualdiagramswe
use three difierent regularization schemes:
(i) time slicing (TS), known from the work of Dirac and Feynman
(ii)moderegularization(MR),knownfrominstantonandsolitonphysics3
(iii) dimensional regularization on a flnite time interval (DR), discussed
in this book.
The renormalization conditions relate a given quantum Hamiltonian H^
to a corresponding quantum action S, which is the action which appears
in the exponent of the path integral. The particular flnite one- and two-
loop counterterms in S thus obtained are difierent for each regularization
scheme. In principle, any H^ with a deflnite ordering of the operators can
be taken as the starting point, and gives a corresponding path integral,
butforourphysicalapplicationsweshallflxtheseambiguitiesin H^ byre-
quiring that it maintains reparametrization and local Lorentz invariance
in target space (commutes with the quantum generators of these symme-
tries). Then there are no one-loop counterterms in the three schemes, but
only two-loop counterterms. Having deflned the regulated path integrals,
3Actually, the mode expansion was already used by Feynman and Hibbs to compute
the path integral for the harmonic oscillator.
i
thecontinuumlimitcanbetakenandrevealsthecorrect\Feynmanrules"
(the rules how to evaluate the integrals over products of distributions and
equal-time contractions) for each regularization scheme. All three regu-
larizationschemesgivethesameflnalanswerforthetransitionamplitude,
although the Feynman rules are difierent.
In the second part of this book we apply our methods to the evalua-
tionofanomaliesinn-dimensionalrelativisticquantumfleldtheorieswith
bosons and fermions in the loops (spin 0;1=2;1;3=2 and selfdual antisym-
metric tensor flelds) coupled to external gauge flelds and/or gravity. We
regulate the fleld-theoretical Jacobian for the symmetries whose anoma-
lies we want to compute with a factor exp( fl ), where is a covariant
¡ R R
regulator which is flxed by the symmetries of the quantum fleld theory,
and fl tends to zero only at the end of the calculation. Next we intro-
duceaquantum-mechanicalrepresentationoftheoperatorswhichenterin
the fleld-theoretical calculation. The regulator yields a corresponding
quantum mechanical Hamiltonian H^. We rewritRe the quantum mechani-
cal operator expression for the anomalies as a path integral on the flnite
timeinterval fl t 0foralinearornonlinearsigmamodelwithaction
¡ • •
S. For given spacetime dimension n, in the limit fl 0 only graphs with
!
a flnite number of loops on the worldline contribute. In this way the cal-
culation of the anomalies is transformed from a fleld-theoretical problem
to a problem in quantum mechanics. We give details of the derivation
of the chiral and gravitational anomalies as flrst given by Alvarez-Gaum¶e
and Witten, and discuss our own work on trace anomalies. For the for-
mer one only needs to evaluate one-loop graphs on the worldline, but for
the trace anomalies in 2 dimensions we need two-loop graphs, and for
the trace anomalies in 4 dimensions we compute three-loop graphs. We
obtain complete agreement with the results for these anomalies obtained
from other methods. We conclude with a detailed analysis of the gravita-
tional anomalies in 10 dimensional supergravities, both for classical and
for exceptional gauge groups.
ii
Preface
In 1983, L. Alvarez-Gaum¶e and E. Witten (AGW) wrote a fundamen-
tal article in which they calculated the one-loop gravitational anoma-
lies (anomalies in the local Lorentz symmetry of 4k + 2 dimensional
Minkowskian quantum fleld theories coupled to external gravity) of com-
plex chiral spin 1/2 and spin 3/2 flelds and real selfdual antisymmetric
tensor flelds1 [1]. They used two methods: a straightforward Feynman
graph calculation in 4k+2 dimensions with Pauli-Villars regularization,
and a quantum mechanical (QM) path integral method in which corre-
sponding nonlinear sigma models appeared. The former has been dis-
cussed in detail in an earlier book [3]. The latter method is the subject
of this book. AGW applied their formulas to N = 2B supergravity in
10 dimensions, which contains precisely one fleld of each kind, and found
that the sum of the gravitational anomalies cancels. Soon afterwards,
M.B. Green and J.H. Schwarz [4] calculated the gravitational anomalies
in one-loop string amplitudes, and concluded that these anomalies cancel
in string theory, and therefore should also cancel in N = 1 supergravity
with suitable gauge groups for the N = 1 matter couplings. Using the
formulas of AGW, one can indeed show that the sum of anomalies in
N = 1 supergravity coupled to super Yang-Mills theory with gauge group
SO(32) or E E , though nonvanishing, is in the technical sense exact:
8 8
£
it can be removed by adding a local counterterm to the action. These two
papers led to an explosion of interest in string theory.
We discussed these two papers in a series of internal seminars for ad-
vanced graduate students and faculty at Stony Brook (the \Friday semi-
nars"). Whereas the basic philosophy and methods of the paper by AGW
were clear, we stumbled on numerous technical problems and details.
Some of these became clearer upon closer reading, some became more
ba†ing. In a desire to clarify these issues we decided to embark on a
research project: the AGW program for trace anomalies. Since gravi-
tational and chiral anomalies only contribute at the one-worldline-loop
level in the QM method, one need not be careful with deflnitions of the
measureforthepathintegral, choiceofregulators, regularizationofdiver-
gent graphs etc. However, we soon noticed that for the trace anomalies
the opposite is true: if the fleld theory is deflned in n = 2k dimensions,
1Just as one can shift the axial anomaly from the axial-vector current to the vector
current, one can also shift the gravitational anomaly from the general coordinate
symmetrytothelocalLorentzsymmetry[2]. Conventionallyonechoosestopreserve
general coordinate invariance. However, AGW chose the symmetric vielbein gauge,
so that the symmetry whose anomalies they computed was a linear combination of
Einstein symmetry and a compensating local Lorentz symmetry.
iii
one needs (k +1)-loop graphs on the worldline in the QM method. As
a consequence, every detail in the calculation matters. Our program of
calculating trace anomalies turned into a program of studying path inte-
grals for nonlinear sigma models in phase space and conflguration space,
a notoriously di–cult and controversial subject. As already pointed out
by AGW, the QM nonlinear sigma models needed for spacetime fermions
(or selfdual antisymmetric tensor flelds in spacetime) have N = 1 (or
N = 2) worldline supersymmetry, even though the original fleld theories
were not spacetime supersymmetric. Thus we had also to wrestle with
the role of susy in the careful deflnitions and calculations of these QM
path integrals.
Although it only gradually dawned upon us, we have come to recognize
the problems with these susy and nonsusy QM path integrals as prob-
lems one should expect to encounter in any quantum fleld theory (QFT),
the only difierence being that these particular fleld theories have a one-
dimensional (flnite) spacetime, as a result of which inflnities in the sum of
Feynmangraphsforagivenprocesscancel. However,individualFeynman
graphs are power-counting divergent (because these models contain dou-
ble derivative interactions just like quantum gravity). This cancellation
of inflnities in the sum of graphs is perhaps the psychological reason why
thereisalmostnodiscussionofregularizationissuesintheearlyliterature
on the subject (in the 1950 and 1960’s). With the advent of the renor-
malization of gauge theories in the 1970’s also issues of regularization of
nonlinear sigma models were studied. It was found that most of the regu-
larizationschemesusedatthattime(thetime-slicingmethodofDiracand
Feynman,andthemoderegularizationmethodusedininstantonandsoli-
ton calculations of nonabelian gauge theories) broke general coordinate
invariance at intermediate stages, but that by adding noncovariant coun-
terterms, the flnal physical results were still general coordinate invariant
(we shall use the shorter term Einstein invariance for this symmetry in
thisbook). Thequestionthusarosehowtodeterminethosecounterterms,
and understand the relation between the counterterms of one regulariza-
tion scheme and those of other schemes. Once again, the answer to this
question could be found in the general literature on QFT: the imposition
of suitable renormalization conditions.
Aswetackledmoreandmoredi–cultproblems(4-loopgraphsfortrace
anomalies in six dimensions) it became clear to us that a scheme which
needed only covariant counterterms would be very welcome. Dimensional
regularization (DR) is such a scheme. It had been used by Kleinert and
Chervyakov [5] for the QM of a one dimensional target space on an infl-
nite worldline time interval (with a mass term added to regulate infrared
divergences). We have developed instead a version of dimensional regu-
iv
larization on a compact space; because the space is compact we do not
need to add by hand a mass term to regulate the infrared divergences
due to massless flelds. The counterterms needed in such an approach are
indeed covariant (both Einstein and locally Lorentz invariant).
The quantum mechanical path integral formalism can be used to com-
pute anomalies in quantum fleld theories. This application forms the
second part of this book. The anomalies are flrst written in the quantum
fleld theory as traces of a Jacobian with a regulator, TrJe fl , and then
¡ R
the limit fl 0 is taken. Chiral spin 1/2 and spin 3/2 flelds and selfdual
!
antisymmetric tensor (AT) flelds can produce anomalies in loop graphs
with external gravitons or external gauge (Yang-Mills) flelds. The treat-
mentofthespin3/2andATfleldsformedamajorobstacle. Inthearticle
by AGW the AT flelds are described by a bispinor ˆ , and the vector
fifl
index of the spin 3/2 fleld and the fl index of ˆ are treated difierently
fifl
from the spinor index of the spin 1/2 and spin 3/2 flelds and the fi index
of ˆ . In [1] one flnds the following transformation rule for the spin 3/2
fifl
fleld (in their notation)
– ˆ = ·iD ˆ +D · (Tab) ˆ (0.0.1)
· A i A a b AB B
¡
where ·i(x) yields an inflnitesimal coordinate transformation xi xi +
!
·i(x), and A = 1;2;:::n is the vector index of the spin 3/2 (gravitino)
fleld, while (Tab) = i(–a–b –b–a) is the generator of the Eu-
AB ¡ A B ¡ A B
clidean Lorentz group SO(n) in the vector representation. One would
expect that this transformation rule is a linear combination of an Ein-
steintransformation– ˆ = ·i@ ˆ (theindexAofˆ is(cid:176)at)andalocal
E A i A A
Lorentz rotation – ˆ = 1·i! ((cid:176)A(cid:176)B) flˆ +·i! Bˆ . However
lL Afi 4 iAB fi Afl iA Bfi
in(0.0.1)theterm·i! Bˆ islacking, andinsteadoneflndsthesecond
iA Bfi
term in (0.0.1) which describes a local Lorentz rotation with parameter
2(D · D · ) and this local Lorentz transformation only acts on the
a b b a
¡
vector index of the gravitino. We shall derive (0.0.1) from flrst principles,
and show that it is correct provided one uses a particular regulator .
R
The regulator for the spin 1/2 fleld ‚, for the gravitino ˆ , and for
A
the bispinor ˆ is in all cases the square of the fleld operator for ‚~,
fifl
ˆ~ and ˆ~ , where ‚~, ˆ~ and ˆ~ are obtained from ‚, ˆ and ˆ
A fifl A fifl A fifl
by multiplication by g1=4 = (dete m)1=2. These \twiddled flelds" were
„
used by Fujikawa, who pioneered the path integral approach to anomalies
[6]. An ordinary Einstein transformation of ‚~ is given by –‚~ = 1(»„@ +
2 „
@ »„)‚~, where the second derivative @ can also act on ‚~, and if one
„ „
evaluates the corresponding anomaly An = Tr1(»„@ +@ »„)e fl for
E 2 „ „ ¡ R
fl tending to zero by inserting a complete set of eigenfunctions ’~ of
n
R
v
with eigenvalues ‚ , one flnds
n
1
An = lim dx’~ (x) (»„@ +@ »„)e fl‚n’~ (x) : (0.0.2)
E fl 0 ⁄n 2 „ „ ¡ n
! Z
Thus the Einstein anomaly vanishes (partially integrate the second @ )
„
as long as the regulator is hermitian with respect to the inner product
(‚~ ;‚~ ) = dx‚~ (x)‚~ (x) (so that the ’~ form a complete set), and
1 2 ⁄1 2 n
as long as both ’~ (x) and ’~ (x) belong to the same complete set of
R n ⁄n
eigenstates (as in the case of plane waves g41eikx). One can always make
a unitary transformation from the ’~n to the set g14eikx, and this allows
explicitcalculationofanomaliesintheframeworkofquantumfleldtheory.
We shall use the regulator discussed above, and twiddled flelds, but
R
then cast the calculation of anomalies in terms of quantum mechanics.
Twenty year have passed since AGW wrote their renowned article. We
believe we have solved all major and minor problems we initially ran
into. The quantum mechanical approach to quantum fleld theory can be
applied to more problems than only anomalies. If future work on such
problems will proflt from the detailed account given in this book, our
scientiflc and geographical Odyssey has come to a good ending.
vi
Brief summary of the three regularization schemes
For experts who want a quick review of the main technical issues cov-
eredinthisbook, wegivehereabriefsummaryofthethreeregularization
schemes described in the main text, namely: time slicing (TS), mode reg-
ularization (MR), and dimensional regularization (DR). After this sum-
marywestartthisbookwithageneralintroductiontothesubjectofpath
integrals in curved space.
Time Slicing
WebeginwithbosonicsystemswithanarbitraryHamiltonianH^ quadratic
in momenta. Starting from the matrix element z exp( flH^) y (which
h j ¡h„ j i
we call the transition amplitude or transition element) with arbitrary but
a priori flxed operator ordering in H^, we insert complete sets of position
and momentum eigenstates, and obtain the discretized propagators and
vertices in closed form. These results tell us how to evaluate equal-time
contractions in the corresponding continuum Euclidean path integrals, as
well as products of distributions which are present in Feynman graphs,
such as
0 0
I = –((cid:190) ¿)(cid:181)((cid:190) ¿)(cid:181)(¿ (cid:190)) d(cid:190)d¿ :
¡ ¡ ¡
Z¡1Z¡1
It is found that –((cid:190) ¿) should be viewed as a Kronecker delta function,
¡
even in the continuum limit, and the step functions as functions with
(cid:181)(0) = 1 (yielding I = 1). Kronecker delta function here means that
2 4
–((cid:190) ¿)f((cid:190))d(cid:190) = f(¿), even when f((cid:190)) is a product of distributions.
¡
We show that the kernel x exp( †H^) x with † = fl=N may be
R h k+1j ¡h„ j ki
approximated by x (1 †H^) x . For linear sigma models this result
h k+1j ¡h„ j ki
is well-known and can be rigorously proven (\the Trotter formula"). For
nonlinear sigma models, the Hamiltonian H^ is rewritten in Weyl ordered
form(whichleadstoextratermsintheactionforthepathintegraloforder
h„ andh„2), andthemidpointrulefollowsautomatically(sonotbecausewe
require gauge invariance). The continuum path integrals thus obtained
arephase-spacepathintegrals. Byintegratingoutthemomentaweobtain
conflguration-space path integrals. We discuss the relation between both
ofthem(Matthews’theorem),bothforourquantummechanicalnonlinear
sigma models and also for 4-dimensional Yang-Mills theories.
The conflguration space path integrals contain new ghosts (anticom-
muting bi(¿), ci(¿) and commuting ai(¿)), obtained by exponentiating
the factors (detg (x(¿)))1=2 which result when one integrates out the
ij
momenta. At the one-loop level these ghosts merely remove the overall
–((cid:190) ¿) singularity in the x_x_ propagator, but at higher loops they are
¡
vii
as useful as in QCD and electroweak gauge theories. In QCD one can
choose a unitary gauge without ghosts, but calculations become horren-
dous. Similarly one could start without ghosts and try to renormalize
the theory in a consistent manner, but this is far more complicated than
working with ghosts. Since the ghosts arise when we integrate out the
momenta, it is natural to keep them. We stress that at any stage all
expressions are flnite and unambiguous once the operator H^ has been
specifled. As a result we do not have to flx normalization constants at the
end by physical arguments, but \the measure" is unambiguously derived
in explicit form. Several two-loop and three-loop examples are worked
out, and conflrm our path integral formalism in the sense that the results
agree with a direct evaluation using operator methods for the canonical
variables p^and x^.
We then extend our results to fermionic systems. We deflne and use
coherent states, deflne Weyl ordering and derive a fermionic midpoint
rule, and obtain also the fermionic discretized propagators and vertices in
closed form, with similar conclusions as for the continuum path integral
for the bosonic case.
Particular attention is paid to the operator treatment of Majorana
fermions. It is shown that \fermion-doubling" (by adding a full set of
noninteracting Majorana fermions) and \fermion halving" (by combining
pairs of Majorana fermions into Dirac fermions) yield difierent propaga-
tors and vertices but the same physical results such as anomalies.
Mode Regularization
As quantum mechanics can be viewed as a one-dimensional quantum
fleld theory (QFT), we can follow the same approach in quantum me-
chanics as familiar from four-dimensional quantum fleld theories. One
way to formulate quantum fleld theory is to expand flelds into a complete
set of functions, and integrate in the path integral over the coe–cients of
these functions. One could try to derive this approach from flrst princi-
ples, starting for example from canonical methods for operators, but we
shall follow a difierent approach for mode regularization. Namely we flrst
writedownformalrulesforthepathintegralinmoderegularizationwith-
out derivation, and a posteriori flx all ambiguities and free coe–cients by
consistency conditions.
We start from the formal sum over paths weighted by the phase fac-
tor containing the classical action (which is like the Boltzmann factor of
statistical mechanics in our Euclidean treatment), and next we suitably
deflne the space of paths. We parametrize all paths as a background
trajectory, which takes into account the boundary conditions, and quan-
tum (cid:176)uctuations, which vanish at the time boundaries. Quantum (cid:176)uc-
tuations are expanded into a complete set of functions (the sines) and
path integration is generated by integration over all Fourier coe–cients
viii
appearing in the mode expansion of the quantum flelds. General covari-
ance demands a nontrivial measure x = detg (x(t))dnx(t). This
D t ij
measure is formally a scalar, but it is not trqanslationally invariant un-
Q
der xi(t) xi(t) + †i(t). To derive propagators it is more convenient
!
to exponentiate the nontrivial part of the measure by using ghost flelds
detg (x(t)) DaDbDc exp( dt 1 g (x)(aiaj +bicj)). At this
t ij » ¡ 2 ij
stagqe the construction is still formal, and one regulates it by integrating
Q R R
over only a flnite number of modes, i.e. by cutting ofi the Fourier sums
at a large mode number M. This makes all expressions well-deflned and
flnite. For example in a perturbative expansion all Feynman diagrams
are unambiguous and give flnite results. This regularization is in spirit
equivalent to a standard momentum cut-ofi in QFT. The continuum limit
is achieved by sending M to inflnity. Thanks to the presence of the ghost
flelds (i.e. of the nontrivial measure) there is no need to cancel inflnities
(i.e. to perform inflnite renormalization). This procedure deflnes a con-
sistent way of doing path integration, but it cannot determine the overall
normalizationofthepathintegral(inQFTitisgenericallyinflnite). More
generally one would like to know how MR is related to other regulariza-
tion schemes. As is well-known, in QFT difierent regularization schemes
are related to each other by local counterterms. Deflning the necessary
renormalization conditions introduces a speciflc set of counterterms of or-
der h„ and h„2, and flxes all of these ambiguities. We do this last step
by requiring that the transition amplitude computed in the MR scheme
satisfles the Schro˜dinger equation with an a priori flxed Hamiltonian H^
(the same as for time slicing). The fact that one-dimensional nonlinear
sigma models are super-renormalizable guarantees that the counterterms
needed to match MR with other regularization schemes (and also needed
to recover general coordinate invariance, which is broken by the TS and
MR regularizations) are not generated beyond two loops.
Dimensional Regularization
Thedimensionallyregulatedpathintegralcanbedeflnedfollowingsteps
similar to those used in the deflnition of the MR scheme, but the regular-
ization of the ambiguous Feynman diagrams is achieved difierently. One
extends the one dimensional compact time coordinate fl t 0 by
¡ • •
adding D extra non-compact (cid:176)at dimensions. The propagators on the
worldline are now a combined sum-integral, where the integral is a mo-
mentum integral as usual in dimensional regularization. At this stage
these momentum space integrals deflne expressions where the variable D
can be analytically continued into the complex plane. We are not able
to perform explicitly these momentum integrals, but we assume that for
arbitrary D all expressions are regulated and deflne analytic functions,
possiblywithpolesonlyatintegerdimensions,asinusualdimensionalreg-
ix