Marathe K. Topics in Physical Mathematics

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198 6 Theory of Fields, I: Classical

E(g, φ,ω, ψ)=



L(g, φ,ω, ψ).

There are various groups associated with the geometric structures involved in

the construction of these ﬁelds, groups that have natural actions on them. We

shall require the Lagrangian to satisfy one or more of the following conditions:

i) naturality, ii) local regularity, iii) conformal invariance.

i) Naturality: Naturality with respect to the group of generalized gauge

transformations is deﬁned as follows. Let F ∈ Diﬀ

(P ) be a generalized

transformation covering f ∈ Diﬀ(M ). Then by naturality with respect to

Diff

(P )wemeanthat

L(f

∗

g, f

∗

φ, F

∗

ω, F

∗

ψ)=f

∗

L(g, φ,ω, ψ), (6.23)

where f

∗

is the induced action of f on W,andF

∗

is the action induced by

the generalized gauge transformation F on H. In the absence of the principal

bundle P this condition reduces to naturality with respect to Diﬀ(M)andis

Einstein’s condition of general covariance of physical laws derived from the

Lagrangian formalism. Further, in the absence of W(M ) this condition corre-

sponds to the covariance of gravitational ﬁeld equations when the Lagrangian

is taken to be the standard Einstein–Hilbert Lagrangian. If we require nat-

urality with respect to the group G, then condition (6.23) is precisely the

principle of gauge invariance introduced by Weyl. Since in this case f = id,

condition (6.23) becomes

L(g, φ, F

∗

ω, F

∗

ψ)=L(g, φ, ω, ψ).

The concept of natural tensors on a Riemannian manifold was introduced by

Epstein. It was extended to oriented Riemannian manifolds in [361], where a

functorial formulation of naturality is given and a complete classiﬁcation of

natural tensor ﬁelds is given under some regularity conditions.

ii) Local regularity: Given any coordinate chart on M and a local gauge

we can express the various potentials and ﬁelds with respect to induced bases.

We require that in this system the Lagrangian be expressible as a universal

polynomial in

(det g)

−1/2

, (det h)

−1/2

,∂

|α|

/∂x

,φ

|β|

,ω

|γ|

,ψ

|δ|

,...

where α, β, γ, δ,..., are suitable multi-indices (i.e., in the coeﬃcients and

derivatives of the potentials and ﬁelds in the induced bases). In physical

applications one often restricts the order of derivatives that can occur in this

local expression to at most two. For example in gravitation one considers

natural tensors satisfying the conditions that they contain derivatives up to

order 2 and depend linearly on the second order derivatives. Then it is well-

knownthatsuchtensorscanbeexpressedas

+ c

S + c

6.6 Matter Fields 199

where R

are the components of the Ricci tensor Ric and S is the scalar

curvature. Einstein’s equations with or without the cosmological constant

involve the above combination with suitable values of the constants c

Einstein’s ﬁeld equations and their generalization are discussed in the next

section.

Applying the classiﬁcation theorem of [361]toSO(4)-actions on the metric

and gauge ﬁelds, we get the following general form for the Lagrangian:

L(g, ω)=c

+ c

K

+ c

W



+ c

W

−



+ c

F

∧ F

 + c

F

∧ (∗F

), (6.24)

where S, K, W

, W

−

are the SO(4)-invariant components of the Rieman-

nian curvature and F

is the gauge ﬁeld of the gauge potential ω.Fora

suitable choice of constants in the above Lagrangian we obtain various topo-

logical invariants of M and P as well as the pure Yang–Mills action. For

example, the ﬁrst Pontryagin class of M is given by

(M)=

4π



(|W

−|W

−

The ﬁrst Pontryagin class of P is given by

(P )=

8π



(|F

−|F

−

which turns out to be the instanton number of P .

To satisfy the conditions of naturality and local regularity for ﬁelds coupled

to gauge ﬁelds physicists often start with ordinary derivatives of associated

ﬁelds and the coupling is achieved by replacing these by gauge covariant

derivatives in the Lagrangian. This is called the principle of minimal cou-

pling (or minimal interaction). These two requirements can also be for-

mulated by taking the Lagrangian to be deﬁned on sections of suitable jet

bundles on the space of connections. Using this approach a generalization of

the classical theorem of Utiyama has been obtained [153].

iii) Conformal invariance:Aconformal transformation of a manifold

M is a diﬀeomorphism f : M → M such that

∗

g = e

2σ

The condition of conformal invariance of the Lagrangian may be expressed

as follows

L(e

2σ

g, φ, ω, ψ)=L(g,φ, ω, ψ), ∀σ ∈F(M).

In general, Lagrangians satisfying the conditions of naturality and regularity

need not satisfy the condition of conformal invariance. This condition is often

used to select parameters such as the dimension of the base space and the rank

200 6 Theory of Fields, I: Classical

of the representation. A particular case of (6.24) is the Yang–Mills Lagrangian

with action

E =

8π



∧∗F

It is an example of a Lagrangian that is conformally invariant only if

dim M =4.

A large number of Lagrangians satisfying the naturality and regularity

requirements are used in the physics literature. They are broadly classiﬁed

into Lagrangians where gauge and other force ﬁelds are coupled to ﬁelds of

bosonic matter, which has integral spin, and to ﬁelds of fermionic matter,

which has half-integral spin. For example, a bosonic Lagrangian is given by

boson

(g,ω, ψ)=F



+ ∇ψ

Sψ

− V (ψ),

where V is the potential function taken to be a gauge-invariant polynomial

of degree ≤ 4 on the ﬁbers of E.IfM is a spin manifold and if Σ is a bundle

associated to the spin bundle, then we deﬁne an E-valued spinor to be a

section of Σ ⊗ E. An example of a fermion Lagrangian is given by

fermion

(g,ω, ξ)=F



+ D(ξ),ξ,

where ξ ∈ Γ (Σ ⊗E)andD is the Dirac operator on E-valued spinors. Several

important properties of coupled ﬁeld equations are studied in [314,58].

6.7 Gravitational Field Equations

There are several ways of deriving Einstein’s gravitational ﬁeld equations.

For example, as we observed in the previous section, we can consider natural

tensors satisfying the conditions that they contain derivatives of the fun-

damental (pseudo-metric) tensor up to order 2 and depend linearly on the

second order derivatives. Then we obtain the tensor

+ c

S + c

where R

are the components of the Ricci tensor Ric and S is the scalar

curvature. Requiring this tensor to be divergenceless and using the Bianchi

identities leads to the relation c

+2c

= 0 between the constants c

Choosing c

=1andc

= 0 we obtain Einstein’s equations (without the

cosmological constant), which may be expressed as

E = −T, (6.25)

where E := Ric−

Sg is the Einstein tensor and T is an energy-momentum

tensor on the space-time manifold which acts as the source term. Now the

6.7 Gravitational Field Equations 201

Bianchi identities satisﬁed by the curvature tensor imply that

div E := δ(g



E)=0.

Hence, if Einstein’s equations (6.25) are satisﬁed, then for consistency we

must have

div T =0. (6.26)

Equation (6.26) is called the diﬀerential (or local) law of conservation of

energy and momentum. However, integral (or global) conservation laws can

be obtained by integrating equation (6.26) only if the space-time manifold

admits Killing vectors. Thus, equation (6.26) has no clear physical meaning,

except in special cases. An interesting discussion of this point is given by

Sachs and Wu [331]. Einstein was aware of the tentative nature of the right

hand side of equation (6.25), but he believed strongly in the expression on

the left hand side. By taking the trace of both sides of equation (6.25)weare

led to the condition

S = t (6.27)

where t denotes the trace of the energy-momentum tensor. The physical

meaning of this condition seems even more obscure than that of condi-

tion (6.26). If we modify equation (6.25) by adding the cosmological term

cg (c is called the cosmological constant) to the left hand side of equa-

tion (6.25), we obtain Einstein’s equation with cosmological constant

E + cg = −T. (6.28)

This equation also leads to the consistency condition (6.26), but condi-

tion (6.27) is changed to

S = t +4c. (6.29)

Using (6.29), equation (6.28) can be rewritten in the following form

K = −(T −

tg), (6.30)

where

K = −(Ric −

Sg) (6.31)

is the trace-free part of the Ricci tensor of g. We call equation (6.30) gen-

eralized ﬁeld equations of gravitation. We now show that these equations

arise naturally in a geometric formulation of Einstein’s equations. We begin

by deﬁning a tensor of curvature type.

Deﬁnition 6.2 Let C be a tensor of type (4, 0) on M. We can regard C as a

quadrilinear mapping (pointwise) so that for each x ∈ M, C

can be identiﬁed

with a multilinear map

: T

∗

(M) × T

∗

(M) × T

∗

(M) × T

∗

(M) → R.

202 6 Theory of Fields, I: Classical

We say that the tensor C is of curvature type if C

satisﬁes the following

conditions for each x ∈ M and for all α, β, γ, δ ∈ T

∗

(M);

1. C

(α, β, γ, δ)=−C

(β,α,γ, δ),

2. C

(α, β, γ, δ)=−C

(α, β, δ, γ),

3. C

(α, β, γ, δ)+C

(α, γ, δ, β)+C

(α, δ, γ, β)=0.

From the above deﬁnition it follows that a tensor C of curvature type also

satisﬁes the following condition:

(α, β, γ, δ)=C

(γ,δ,α,β), ∀x ∈ M.

Example 6.3 The Riemann–Christoﬀel curvature tensor is of curvature

type. Indeed, the deﬁnition of tensors of curvature type is modeled after this

fundamental example. Another important example of a tensor of curvature

type is the tensor G deﬁned by

(α, β, γ, δ)=g

(α, γ)g

(β,δ) − g

(α, δ)g

(β,γ), ∀x ∈ M,

where g is the fundamental or metric tensor of M.

We now deﬁne the curvature product of two symmetric tensors of type

(2, 0) on M . The curvature product was introduced in [264] and used in [273]

to obtain a geometric formulation of Einstein’s equations.

Deﬁnition 6.3 Let g and T be two symmetric tensors of type (2, 0) on M.

The curvature product of g and T , denoted by g ×

T , is a tensor of type

(4, 0) deﬁned by

(g ×

T )

(α, β, γ, δ):=



g(α, γ)T (β, δ)+g(β,δ)T (α, γ)

−g(α, δ)T (β,γ) −g(β,γ)T (α, δ)



for all x ∈ M and α, β, γ, δ ∈ T

∗

(M).

In the following proposition we collect some important properties of the

curvature product and tensors of curvature type.

Proposition 6.9 Let g and T be two symmetric tensors of type (2, 0) on M

and let C be a tensor of curvature type on M . Then we have the following:

1. g ×

T = T ×

2. g ×

T is a tensor of curvature type:

3. g ×

g = G, where G is the tensor deﬁned in Example 6.3;

4. G

induces a pseudo-inner product on Λ

(M), ∀x ∈ M;

5. C

induces a symmetric, linear transformation of Λ

(M), ∀x ∈ M.

We denote the Hodge star operator on Λ

(M)byJ

.Thefactthat

M is a Lorentz 4-manifold implies that J

deﬁnes a complex structure on

(M), ∀x ∈ M. Using this complex structure we can give a natural structure

of a complex vector space to Λ

(M). Then we have the following proposition.

6.7 Gravitational Field Equations 203

Proposition 6.10 Let L : Λ

(M) → Λ

(M) be a real, linear transforma-

tion. Then the following are equivalent:

1. L commutes with J

;

2. L is a complex linear transformation of the complex vector space Λ

(M);

3. The matrix of L with respect to a G

-orthonormal basis of Λ

(M) is of

the form



−BA



(6.32)

where A, B are real 3 × 3 matrices.

We now deﬁne the gravitational tensor W

,ofcurvaturetype,whichin-

cludes the source term.

Deﬁnition 6.4 Let M be a space-time manifold with fundamental tensor g

and let T be a symmetric tensor of type (2, 0) on M. Then the gravitational

tensor W

is deﬁned by

:= R + g ×

T, (6.33)

where R is the Riemann–Christoﬀel curvature tensor of type (4, 0).

We are now in a position to give a geometric formulation of the generalized

ﬁeld equations of gravitation.

Theorem 6.11 Let W

denote the gravitational tensor deﬁned by (6.33)with

source tensor T .WealsodenotebyW

the linear transformation of Λ

(M)

induced by W

. Then the following are equivalent:

1. g satisﬁes the generalized ﬁeld equations of gravitation (6.30);

2. W

commutes with J

;

3. W

is a complex linear transformation of the complex vector space Λ

(M).

We shall call the triple (M,g,T)ageneralized gravitational ﬁeld if any

one of the conditions of Theorem 6.11 is satisﬁed. Generalized gravitational

ﬁeld equations were introduced by the author in [264]. Their mathemati-

cal properties have been studied in [275, 266, 287]. Solutions of Marathe’s

generalized gravitational ﬁeld equations that are not solutions of Einstein’s

equations are discussed in [69]. We note that the above theorem and the last

condition in Proposition 6.8 can be used to discuss the Petrov classiﬁcation

of gravitational ﬁelds (see Petrov [316]). The tensor W

canbeusedinplace

of R in the usual deﬁnition of sectional curvature to deﬁne the gravitational

sectional curvature on the Grassmann manifold of non-degenerate 2-planes

over M and to give a further geometric characterization of gravitational ﬁeld

equations. We observe that the generalized ﬁeld equations of gravitation con-

tain Einstein’s equations with or without the cosmological constant as special

cases. Solutions of the source-free generalized ﬁeld equations are called gravi-

tational instantons. If the base manifold is Riemannian, then gravitational

204 6 Theory of Fields, I: Classical

instantons correspond to Einstein spaces. A detailed discussion of the struc-

ture of Einstein spaces and their moduli spaces may be found in [39].

We note that, over a compact, 4-dimensional, Riemannian manifold (M,g),

the gravitational instantons that are not solutions of the vacuum Einstein

equations are critical points of the quadratic, Riemannian functional or action

(g) deﬁned by

(g)=



In fact, using any polynomial in S of degree > 1 in the above action leads,

generically to the gravitational instanton equations. Furthermore, the stan-

dard Hilbert–Einstein action

(g)=



Sdv

also leads to the generalized ﬁeld equations when the variation of the action

is restricted to metrics of volume 1.

There are several diﬀerences between the Riemannian functionals used in

theories of gravitation and the Yang–Mills functional used to study gauge ﬁeld

theories. The most important diﬀerence is that the Riemannian functionals

are dependent on the bundle of frames of M or its reductions, while the Yang–

Mills functional can be deﬁned on any principal bundle over M. However, we

have the following interesting theorem [20].

Theorem 6.12 Let (M,g) be a compact, 4-dimensional, Riemannian man-

ifold. Let Λ

(M) denote the bundle of self-dual 2-forms on M with induced

metric G

. Then the Levi-Civita connection λ

on M satisﬁes the gravita-

tional instanton equations if and only if the Levi-Civita connection λ

(M) satisﬁes the Yang–Mills instanton equations.

6.8 Geometrization Conjecture and Gravity

The classiﬁcation problem for low-dimensional manifolds is a natural ques-

tion after the success of the case of surfaces by the uniformization theorem. In

1905, Poincar´e formulated his famous conjecture, which states, in the smooth

case: A closed, simply connected 3-manifold is diﬀeomorphic to S

,thestan-

dard sphere. A great deal of work in 3-dimensional topology the century that

followed was motivated by this. In the 1980s Thurston studied hyperbolic

manifolds. This led him to his “geometrization conjecture” about the exis-

tence of homogeneous metrics on all 3-manifolds. It includes the Poincar´e

conjecture as a special case. We already discussed this in Chapter 2. In the

case of 4-manifolds there is at present no analogue of the geometrization

conjecture. We now discuss brieﬂy the main idea behind Perelman’s proof of

6.8 Geometrization Conjecture and Gravity 205

the geometrization conjecture and the relation of the perturbed Ricci ﬂow

equations to Einstein’s equations in Euclidean gravity.

The Ricci ﬂow equations

∂g

∂t

= −2R

for a Riemannian metric g were introduced by Hamilton in [183]. They form

a system of nonlinear second order partial diﬀerential equations. Hamilton

proved that this equation has a unique solution for a short time for any

smooth metric on a closed manifold. The evolution equation for the metric

leads to the evolution equations for the curvature and Ricci tensors and for

the scalar curvature. By developing a maximum principle for tensors Hamil-

ton proved that the Ricci ﬂow preserves the positivity of the Ricci tensor in

dimension 3 and that of the curvature operator in dimension 4 [184]. In each

of these cases he proved that the evolving metrics converge to metrics of con-

stant positive curvature (modulo scaling). These and a series of further papers

led him to conjecture that the Ricci ﬂow with surgeries could be used to prove

the Thurston geometrization conjecture. In a series of e-prints Perelman de-

veloped the essential framework for implementing the Hamilton program. We

would like to add that the full Einstein equations with dilaton ﬁeld as source

play a fundamental role in Perelman’s work (see, arXiv.math.DG/0211159,

0303109, 0307245 for details) on the geometrization conjecture. A corollary

of this work is the proof of the long standing Poincar´e conjecture. Perelman

was awarded the Fields medal at the ICM 2006 in Madrid for his proof of the

Poincar´e and the geometrization conjectures. His ideas and methods have al-

ready found many applications in analysis and geometry. On March 18, 2010

Perelman was awarded the Clay Mathematics Institute’s ﬁrst millenium prize

of one million dollars for his resolution of the Poincar´e conjecture. A complete

proof of the geometrization conjecture through application of the Hamilton–

Perelman theory of the Ricci ﬂow has now appeared in [70] in a special issue

dedicated to the memory of S.-S. Chern,

one of the greatest mathematicians

of the twentieth century.

The Ricci ﬂow is perturbed by a scalar ﬁeld, which corresponds in string

theory to the dilaton. It is supposed to determine the overall strength of all

interactions. The low energy eﬀective action of the dilaton ﬁeld coupled to

gravity is given by the action functional

F(g, f)=



(R + |∇f|

−f

dv.

I ﬁrst met Prof. Chern and his then newly arrived student S.-T. Yau in 1973 at the

AMS summer workshop on diﬀerential geometry held at Stanford University. Chern was

a gourmet and his conference dinners were always memorable. I attended the ﬁrst one

in 1973 and the last one in 2002 on the occassion of the ICM satellite conference at his

institute in Tianjin. In spite of his advanced age and poor health he participated in the

entire program and then continued with his duties as President of the ICM in Beijing.

206 6 Theory of Fields, I: Classical

Note that when f is the constant function the action reduces to the classical

Hilbert–Einstein action. The ﬁrst variation can be written as

δF(g, f)=



[−δg

+∇

f∇

f)+(

δg

−δf)(2Δf −|∇f|

+R)]dm,

where dm = e

−f

dv. If m =



−f

dv is kept ﬁxed, then the second term in

the variation is zero and then the symmetric tensor −(R

+ ∇

f∇

f)isthe

gradient ﬂow of the action functional F



(R+|∇f|

)dm . The choice

of m is similar to the choice of a gauge. All choices of m lead to the same

ﬂow, up to diﬀeomorphism, if the ﬂow exists. We remark that in the quan-

tum ﬁeld theory of the 2-dimensional nonlinear σ-model, the Ricci ﬂow can

be considered as an approximation to the renormalization group ﬂow. This

suggests gradient-ﬂow-like behavior for the Ricci ﬂow, from the physical point

of view. Perelman’s calculations conﬁrm this result. The functional F

has

also a geometric interpretation in terms of the classical Bochner–Lichnerowicz

formulas with the metric measure replaced by the dilaton twisted measure

dm.

The corresponding variational equations are

−

= −(∇

∇

f −

(Δf)g

These are the usual Einstein equations with the energy-momentum tensor of

the dilaton ﬁeld as source. They lead to the decoupled evolution equations

)

= −2(R

+ ∇

∇

f),f

= −R − Δf.

After applying a suitable diﬀeomorphism these equations lead to the gra-

dient ﬂow equations. This modiﬁed Ricci ﬂow can be pushed through the

singularities by surgery and rescaling. A detailed case by case analysis is

then used to prove Thurston’s geometrization conjecture [70]. This includes

as a special case the classical Poincar´e conjecture. A complete proof of the

Poincar´e conjecture without appealing to the Thurston geometrization con-

jecture may be found in the book [289] by Morgan and Tian. In fact, they

prove a more general result which implies a closely related stronger conjec-

ture called the 3-dimensional spherical space-form conjecture. This conjecture

states that a closed 3-manifold with ﬁnite fundamental group is diﬀeomor-

phic to a 3-dimensional spherical space-form, i.e., the quotient of S

by free,

linear action of a ﬁnite subgroup of the orthogonal group O(4).

Chapter 7

Theory of Fields, II: Quantum and

Topological

7.1 Introduction

Quantization of classical ﬁelds is an area of fundamental importance in mod-

ern mathematical physics. Although there is no satisfactory mathematical

theory of quantization of classical dynamical systems or ﬁelds, physicists

have developed several methods of quantization that can be applied to spe-

ciﬁc problems. Most successful among these is QED (quantum electro-

dynamics), the theory of quantization of electromagnetic ﬁelds. The physi-

cal signiﬁcance of electromagnetic ﬁelds is thus well understood at both the

classical and the quantum level. Electromagnetic theory is the prototype of

classical gauge theories. It is therefore natural to try to extend the methods

of QED to the quantization of other gauge ﬁeld theories. The methods of

quantization may be broadly classiﬁed as non-perturbative and perturbative.

The literature pertaining to each of these areas is vast. See for example, the

two volumes [95,96] edited by Deligne, et al. which contain the lectures given

at the Institute for Advanced Study, Princeton, during a special year devoted

to quantum ﬁelds and strings; the book by Nash [298], and [41, 354,89]. For

a collection of lectures covering various aspects of quantum ﬁeld theory, see,

for example, [134,133, 376].

Our aim in this chapter is to illustrate each of these methods by discussing

some speciﬁc examples where a reasonably clear mathematical formulation

is possible. A brief account of the non-perturbative methods in the quan-

tization of gauge ﬁelds is given in Section 7.2. A widely used perturbative

method in gauge theories is that of semiclassical approximation. We devote

Section 7.3 to a mathematical formulation of semiclassical approximation in

Euclidean Yang–Mills theory. This requires a detailed knowledge of the geom-

etry of the moduli spaces of instantons and methods of regularization for the

inﬁnite-dimensional quantities. Two such methods are indicated in this sec-

tion. Both classical and quantum ﬁeld theories lead to topological invariants

of base manifolds on which they are deﬁned. We call these topological ﬁeld

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 7, 207

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