# Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

Authors: Rudolf Schmid
Journal: Advances in Mathematical Physics
 (2010)

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## Abstract

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.


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## Body

## 1. Introduction

Lie groups play an important role in physical systems both as phase spaces and as symmetry groups. Infinite-dimensional Lie groups occur in the study of dynamical systems with an infinite number of degrees of freedom such as PDEs and in field theories. For such infinite-dimensional dynamical systems, diffeomorphism groups and various extensions and variations thereof, such as gauge groups, loop groups, and groups of Fourier integral operators, occur as symmetry groups and phase spaces. Symmetries are fundamental for Hamiltonian systems. They provide conservation laws (Noether currents) and reduce the number of degrees of freedom, that is, the dimension of the phase space.The topics selected for review aim to illustrate some of the ways infinite-dimensional geometry and global analysis can be used in mathematical problems of physical interest. The topics selected are the following.(1)
Infinite-Dimensional Lie Groups.(2)
Lie Groups as Symmetry Groups of Hamiltonian Systems.(3)
Applications.(4)
Gauge Theories, the Standard Model, and Gravity.(5)
SUSY (supersymmetry).


## 2. Infinite-Dimensional Lie Groups

### 2.1. Basic Definitions

A general theory of infinite-dimensional Lie groups is hardly developed. Even Bourbaki [1] only develops a theory of infinite-dimensional manifolds, but all of the important theorems about Lie groups are stated for finite-dimensional ones.An infinite-dimensionalLie group𝒢 is a group and an infinite-dimensional manifold with smooth group operations
(2.1)m:𝒢×𝒢→𝒢,m(g,h)=g·h,C∞,(2.2)i:𝒢→𝒢,i(g)=g-1,C∞.
Such a Lie group 𝒢 is locally diffeomorphic to an infinite-dimensional vector space. This can be a Banach space whose topology is given by a norm ∥·∥, a Hilbert space whose topology is given by an inner product 〈·,·〉, or a Frechet space whose topology is given by a metric but not by a norm. Depending on the choice of the topology on 𝒢, we talk about Banach, Hilbert, or Frechet Lie groups, respectively.TheLie algebra  𝔤 of a Lie group 𝒢 is defined as 𝔤={left invariant vector fields on 𝒢}≃Te𝒢 (tangent space at the identity e). The isomorphism is given (as in finite dimensions) by(2.3)ξ∈Te𝒢↦Xξ∈𝔤,Xξ(g):=TeLg(ξ),
and the Lie bracket on 𝔤 is induced by the Lie bracket of left invariant vector fields [ξ,η]=[Xξ,Xη](e),ξ,η∈Te𝒢.These definitions in infinite dimensions are identical with the definitions in finite dimensions. The big difference although is that infinite-dimensional manifolds, hence Lie groups, arenot locally compact. For Frechet Lie groups, we have the additional nontrivial difficulty of the question how to define differentiability of functions defined on a Frechet space; see the study by Keller in [2]. Hence the very definition of a Frechet manifold is not canonical. This problem does not arise for Banach- and Hilbert-Lie groups; the differential calculus extends in a straightforward manner from ℝn to Banach and Hilbert spaces, but not to Frechet spaces.


### 2.2. Finite- versus Infinite-Dimensional Lie Groups

Infinite-dimensional Lie groups areNOT locally compact. This causes some deficiencies of the Lie theory in infinite dimensions. We summarize some classical results in finite dimensions which are NOT true in general in infinite dimensions as follows.(1)
There is NO Implicit Function Theorem or Inverse Function Theorem in infinite dimensions! (except Nash-Moser-type theorems).(2)
IfG is a finite-dimensional Lie group, the exponential map  exp:𝔤→G is defined as follows. To each ξ∈𝔤, we assign the corresponding left invariant vector field Xξ defined by (2.3). We take the flow φξ(t) of Xξ and define exp(ξ)=φξ(1). The exponential map is a local diffeomorphism from a neighborhood of zero in 𝔤 onto a neighborhood of the identity in G; hence exp defines canonical coordinates on the Lie group G. This is not true in infinite dimensions.(3)
Iff1,f2:G1→G2 are smooth Lie group homomorphisms (i.e., fi(g·h)=fi(g)·fi(h),i=1,2) with Tef1=Tef2, then locally f1=f2. This is not true in infinite dimensions.(4)
Iff:G→H is a continuous group homomorphism between finite-dimensional Lie groups, then f is smooth. This is not true in infinite dimensions.(5)
If𝔤 is any finite-dimensional Lie algebra, then there exists a connected finite-dimensional Lie group G with 𝔤 as its Lie algebra; that is, 𝔤≃TeG. This is not true in infinite dimensions.(6)
IfG is a finite-dimensional Lie group and H⊂G is a closed subgroup, then H is a Lie subgroup (i.e., Lie group and submanifold). This is not true in infinite dimensions.(7)
IfG is a finite-dimensional Lie group with Lie algebra 𝔤 and 𝔥⊂𝔤 is a subalgebra, then there exists a unique connected Lie subgroup H⊂G with 𝔥 as its Lie algebra; that is, 𝔥≃TeH. This is not true in infinite dimensions.Some classical examples offinite-dimensional Lie groups are the matrix groups GL(n), SL(n), O(n), SO(n), U(n), SU(n), and Sp(n) with smooth group operations given by matrix multiplication and matrix inversion. The Lie algebra bracket is the commutator [A,B]=AB-BA with exponential map given by exp(A)=∑i=0∞(1/i!)Ai=eA.


### 2.3. Examples of Infinite-Dimensional Lie Groups

#### 2.3.1. The Vector Groups𝒢=(V,+)

LetV be a Banach space and take 𝒢=V with m(x,y)=x+y, i(x)=-x, and e=0, which makes 𝒢 into an Abelian Lie group; that is, m(x,y)=m(y,x). For the Lie algebra we have 𝔤≃TeV≃V. For u∈TeV the corresponding left invariant vector field Xu is given by Xu(v)=u, ∀v∈V; that is, Xu=const. Hence the Lie algebra 𝔤=V with the trivial Lie bracket [u,v]=0 is Abelian. For the exponential map we get exp:𝔤=V→𝒢=V,exp=idV.


#### 2.3.2. The General Linear Group𝒢=(GL(V),∘)

LetV be a Banach space and L(V,V) the space of bounded linear operators A:V→V. Then L(V,V) is a Banach space with the operator norm ∥A∥=sup∥x∥≤1∥A(x)∥, and the group 𝒢=GL(V) of all invertible elements is open in L(V,V). So GL(V) is a smooth Lie group with m(f,g)=f∘g,i(f)=f-1, and e=idV. Its Lie algebra is 𝔤=L(V,V) with the commutator bracket [A,B]=AB-BA and exponential map expA=eA.


#### 2.3.3. The Abelian Gauge Group𝒢=(C∞(M),+)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M) (smooth functions on M). With group operation being addition, that is, m(f,g)=f+g, i(f)=-f, and e=0. 𝒢 is an Abelian C∞ (addition is smooth) Frechet Lie group with Lie algebra 𝔤=TeC∞(M)≃C∞(M), with trivial bracket [ξ,η]=0, and exp=id. If we complete these spaces in the Ck-norm, k<∞ (denoted by 𝒢k), then 𝒢k is a Banach-Lie group, and if we complete in the Hs-Sobolev norm with s>(1/2)dimM then 𝒢s is a Hilbert-Lie group.


#### 2.3.4. The Abelian Gauge Group𝒢=(C∞(M,ℝ-{0}),·)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M,ℝ-{0}), with group operation being multiplication; that is, m(f,g)=f·g, i(f)=f-1, and e=1. For k<∞, Ck(M,ℝ-{0}) is open in C∞(M,ℝ), and if M is compact, then Ck(M,ℝ-{0}) is a Banach-Lie group. If s>(1/2)dimM, then Hs(M,ℝ-{0}) is closed under multiplication, and if M is compact, then Hs(M,ℝ-{0}) is a Hilbert-Lie group.


#### 2.3.5. Loop Group𝒢=(Ck(M,G),·)

We generalize the Abelian example (see Section2.3.4) by replacing ℝ-{0} with any finite-dimensional (non-Abelian) Lie group G. Let 𝒢=Ck(M,G) with pointwise group operations m(f,g)(x)=f(x)·g(x), x∈M, and i(f)(x)=(f(x))-1, where “·” and “(·)-1” are the operations in G. If k<∞ then Ck(M,G) is a Banach-Lie group. Let g denote the Lie algebra of G, then the Lie algebra of 𝒢=Ck(M,G) is 𝔤=Ck(M,g), with pointwise Lie bracket [ξ,η](x)=[ξ(x),η(x)], x∈M, the latter bracket being the Lie bracket in g. The exponential map exp:g→G defines the exponential map EXP:𝔤=Ck(M,g)→𝒢=Ck(M,G), EXP(ξ)=exp∘ξ, which is a local diffeomorphism. The same holds for Hs(M,G) if s>(1/2)dimM.Applications of these infinite-dimensional Lie groups are in gauge theories and quantum field theory, where they appear as groups of gauge transformations. We will discuss these in Section5.Special Case:𝒢=(Ck(S1,G),·)
As a special case of example mentioned in Section2.3.5 we take M=S1, the circle. Then 𝒢=Ck(S1,G)=ℒk(G) is called a loop group and 𝔤=Ck(S1,g)=𝔩k(g) is its loop algebra. They find applications in the theory of affine Lie algebras, Kac-Moody Lie algebras (central extensions), completely integrable systems, soliton equations (Toda, KdV, KP), and quantum field theory; see, for example, [3] and Section 5. Central extensions of loop algebras are examples of infinite-dimensional Lie algebras which need not have a corresponding Lie group.
Certain subgroups of loop groups play an important role in quantum field theory as groups of gauge transformations. We will discuss these in Section2.4.4.


### 2.4. Diffeomorphism Groups

Among the most important “classical” infinite-dimensional Lie groups are the diffeomorphism groups of manifolds. Their differential structure is not the one of a Banach Lie group as defined above. Nevertheless they have important applications.LetM be a compact manifold (the noncompact case is technically much more complicated but similar results are true; see the study by Eichhorn and Schmid in [4]) and let 𝒢=Diff∞(M) be the group of all smooth diffeomorphisms on M, with group operation being composition; that is, m(f,g)=f∘g, i(f)=f-1, and e=idM. For C∞ diffeomorphisms, Diff∞(M) is a Frechet manifold and there are nontrivial problems with the notion of smooth maps between Frechet spaces. There is no canonical extension of the differential calculus from Banach spaces (which is the same as for ℝn) to Frechet spaces; see the study by Keller in [2]. One possibility is to generalize the notion of differentiability. For example, if we use the so-called CΓ∞ differentiability, then 𝒢=Diff∞(M) becomes a CΓ∞ Lie group with CΓ∞ differentiable group operations. These notions of differentiability are difficult to apply to concrete examples. Another possibility is to complete Diff∞(M) in the Banach Ck-norm, 0≤k<∞, or in the Sobolev Hs-norm, s>(1/2)dimM. Then Diffk(M) and Diffs(M) become Banach and Hilbert manifolds, respectively. Then we consider the inverse limits of these Banach- and Hilbert-Lie groups, respectively:(2.4)Diff∞(M)=lim←Diffk(M)
becomes a so-called ILB- (Inverse Limit of Banach) Lie group, or with the Sobolev topologies(2.5)Diff∞(M)=lim←Diffs(M)
becomes a so-called ILH- (Inverse Limit of Hilbert) Lie group. See the study by Omori in [5] for details. Nevertheless, the group operations are not smooth, but have the following differentiability properties. If we equip the diffeomorphism group with the Sobolev Hs-topology, then Diffs(M) becomes a C∞ Hilbert manifold if s>(1/2)dimM and the group multiplication(2.6)m:Diffs+k(M)×Diffs(M)→Diffs(M)
is Ck differentiable; hence for k=0, m is only continuous on Diffs(M). The inversion(2.7)i:Diffs+k(M)→Diffs(M)
is Ck differentiable; hence for k=0, i is only continuous on Diffs(M). The same differentiability properties of m and i hold in the Ck topology.TheLie algebra of Diff∞(M) is given by 𝔤=TeDiff∞(M)≃Vec∞(M) being the space of smooth vector fields on M. Note that the space Vec(M) of all vector fields is a Lie algebra only for C∞ vector fields, but not for Ck or Hs vector fields if k<∞, s<∞, because one loses derivatives by taking brackets.The exponential map on the diffeomorphism group is given as follows. For any vector fieldX∈Vec∞(M), take its flow φt∈Diff∞(M), then define EXP:Vec∞(M)→Diff∞(M):X↦φ1, the flow at time t=1. The exponential map EXP is NOT a local diffeomorphism; it is not even locally surjective.We see that the diffeomorphism groups are not Lie groups in the classical sense, but what we callnested Lie groups. Nevertheless they have important applications as we will see.


#### 2.4.1. Subgroups ofDiff∞(M)

Several subgroups ofDiff∞(M) have important applications.


#### 2.4.2. Group of Volume-Preserving Diffeomorphisms

Letμ be a volume on M and (2.8)𝒢=Diffμ∞(M)={f∈Diff∞(M)∣f*μ=μ}
the group of volume-preserving diffeomorphisms. Diffμ∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.9)𝔤=Vecμ∞(M)={X∈Vec∞(M)∣divμX=0}
being the space of divergence-free vector fields on M. Vecμ∞(M) is a Lie subalgebra of Vec∞(M).Remark 2.1.
We cannot apply the finite-dimensional theorem that ifVecμ∞(M) is Lie algebra then there exists a Lie group whose Lie algebra it is; nor the one that if Diffμ∞(M)⊂Diff(M) is a closed subgroup then it is an Lie subgroup.
NeverthelessDiffμ∞(M) is an ILH-Lie group.


#### 2.4.3. Symplectomorphism Group

Letω be a symplectic 2-form on M and (2.10)𝒢=Diffω∞(M)={f∈xDiff∞(M)∣f*ω=ω}
the group of canonical transformations (or symplectomorphisms). Diffω∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.11)𝔤=Vecω∞(M)={X∈Vec∞(M)∣LXω=0}
being the space of locally Hamiltonian vector fields on M. Vecω∞(M) is a Lie subalgebra of Vec∞(M). Again Diffω∞(M) is an ILH-Lie group.


#### 2.4.4. Group of Gauge Transformations

The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups which are not only ILH-Lie groups but actually Hilbert-Lie groups.Letπ:P→M be a principal G bundle with G being a finite-dimensional Lie group (structure group) acting on P from the right p∈P, g∈G, and p·g∈P.TheGauge group𝒢 is the group of gauge transformations defined by(2.12)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,π(ϕ(p))=π(p)}.𝒢 is a group under composition, hence a subgroup of the diffeomorphism group Diff∞(P). Since a gauge transformation ϕ∈𝒢 preserves fibers, we can realize each such ϕ∈𝒢 via ϕ(p)=p·τ(p), where τ:P→G satisfies τ(p·g)=g-1τ(p)g, for p∈P, g∈G. Let(2.13)Gau(P)={τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}.Gau(P) is a group under pointwise multiplication, hence a subgroup of the loop group C∞(P,G) (see Section 2.4.3), which extends to a Hilbert-Lie group if equipped with the Hs-Sobolev topology. We give Gau(P) the induced topology and extend it to a Hilbert-Lie group denoted by Gaus(P). Another interpretation is that Gau(P) is isomorphic to C∞(AdP) the space of sections of the associated vector bundle Ad(P)=P×GG. Completed in the Hs Sobolev topology, we get Gaus(P)≃Hs(AdP).Letg denote the Lie algebra of G. Then the Lie algebra  𝔤𝔞𝔲(P) of Gau(P) is a subalgebra of the loop algebra Hs(P,g) under pointwise bracket in g, the finite-dimensional Lie algebra of G; that is, for any ξ,η∈Hs(P,g) the bracket is defined by [ξ,η]𝔤𝔞𝔲(P)(p)=[ξ(p),η(p)]g, p∈P. Then 𝔤𝔞𝔲s(P) is the subalgebra of Ad -invariant g-valued functions on P; that is,(2.14)𝔤𝔞𝔲(P)={ξ∈C∞(P,g);ξ(p·g)=Adg-1ξ(p)}.The Lie algebra𝔩𝔦𝔢𝒢 (running out of symbols) of the gauge group 𝒢 is the Lie subalgebra of 𝔛∞(P) consisting of all G-invariant vertical vector fields X on P; that is,(2.15)𝔩𝔦𝔢𝒢={X∈𝔛∞(P);Rg*X=X,X(p)∈g,g∈G,p∈P}
with commutator bracket [X1,X2]=X1X2-X2X1∈𝔩𝔦𝔢𝒢.On the other hand, the Lie algebra ofC∞(AdP) is C∞(ad(P)) being the space of sections of the associated vector bundle ad(P)≡(P×Gg)→M with pointwise bracket.We have three versions of gauge groups:𝒢,Gau(P), and C∞(AdP). They are all group isomorphic. There is a natural group isomorphism Gau(P)→𝒢:τ↦ϕ defined by ϕ(p)=p·τ(p), p∈P, which preserves the product τ1·τ2↦ϕ1∘ϕ2. Identifying 𝒢 with Gau(P), we can avoid the troubles with diffeomorphism groups and we can extend 𝒢 to a Hilbert-Lie group 𝒢s. So 𝒢s is actually a Hilbert-Lie group in the classical sense; that is, the group operations are C∞. Also the three Lie algebras 𝔩𝔦𝔢𝒢, 𝔤𝔞𝔲(P), and C∞(adP) are canonically isomorphic. Indeed, for s∈C∞(adP) define ξ∈𝔤𝔞𝔲(P)ξ:P→g by ξ(p·a):=Ada-1ξ(p); and for ξ∈𝔤𝔞𝔲(P) define s∈C∞(adP) by s(π(p)):=[p,ξ(p)].On the other hand, forξ∈𝔤𝔞𝔲(P) define Zξ∈𝔩𝔦𝔢𝒢 by (2.16)Zξ(p)=ddt∣t=0R(p,exptξ(p))(=ξ(p)*(p)),
that is, Zξ is the fundamental vector field on P, generated by ξ∈g. Zξ is invariant if and only if ξ(p·g)=Adg-1ξ(p).To topologize𝔩𝔦𝔢𝒢, we complete C∞(adP) in the Hs-Sobolev norm. If s>(1/2)dimM, then 𝔩𝔦𝔢𝒢s≃Hs(adP)≃𝔤𝔞𝔲s(P) are isomorphic Hilbert-Lie algebras.There is a natural exponential mapExp:𝔤𝔞𝔲(P)→Gau(P), which is a local diffeomorphism. Let exp:g→G  be the finite-dimensional exponential map. Then define(2.17)Exp:𝔤𝔞𝔲s(P)→Gaus(P):(Expξ)(p)=exp(ξ(p)),ξ∈𝔤𝔞𝔲s(P).
Or in terms of 𝒢, Exp:𝔩𝔦𝔢𝒢s→𝒢s:(Expξ)(p)=p·exp(ξp).We have the following theorem (Schmid [6]).Theorem 2.2.
Fors>(1/2)dimM,
(2.18)𝒢s≃Gaus(P)≃Hs(AdP)
is a smooth Hilbert-Lie group with Lie algebra
(2.19)𝔩𝔦𝔢𝒢s≃𝔤𝔞𝔲s(P)≃Hs(adP)
and smooth exponential map, which is a local diffeomorphism,
(2.20)EXP:𝔩𝔦𝔢𝒢s→𝒢s:(EXPξ)(p)=p·exp(ξ(p)).See [1–5, 7–19].


## 2.1. Basic Definitions

A general theory of infinite-dimensional Lie groups is hardly developed. Even Bourbaki [1] only develops a theory of infinite-dimensional manifolds, but all of the important theorems about Lie groups are stated for finite-dimensional ones.An infinite-dimensionalLie group𝒢 is a group and an infinite-dimensional manifold with smooth group operations
(2.1)m:𝒢×𝒢→𝒢,m(g,h)=g·h,C∞,(2.2)i:𝒢→𝒢,i(g)=g-1,C∞.
Such a Lie group 𝒢 is locally diffeomorphic to an infinite-dimensional vector space. This can be a Banach space whose topology is given by a norm ∥·∥, a Hilbert space whose topology is given by an inner product 〈·,·〉, or a Frechet space whose topology is given by a metric but not by a norm. Depending on the choice of the topology on 𝒢, we talk about Banach, Hilbert, or Frechet Lie groups, respectively.TheLie algebra  𝔤 of a Lie group 𝒢 is defined as 𝔤={left invariant vector fields on 𝒢}≃Te𝒢 (tangent space at the identity e). The isomorphism is given (as in finite dimensions) by(2.3)ξ∈Te𝒢↦Xξ∈𝔤,Xξ(g):=TeLg(ξ),
and the Lie bracket on 𝔤 is induced by the Lie bracket of left invariant vector fields [ξ,η]=[Xξ,Xη](e),ξ,η∈Te𝒢.These definitions in infinite dimensions are identical with the definitions in finite dimensions. The big difference although is that infinite-dimensional manifolds, hence Lie groups, arenot locally compact. For Frechet Lie groups, we have the additional nontrivial difficulty of the question how to define differentiability of functions defined on a Frechet space; see the study by Keller in [2]. Hence the very definition of a Frechet manifold is not canonical. This problem does not arise for Banach- and Hilbert-Lie groups; the differential calculus extends in a straightforward manner from ℝn to Banach and Hilbert spaces, but not to Frechet spaces.


## 2.2. Finite- versus Infinite-Dimensional Lie Groups

Infinite-dimensional Lie groups areNOT locally compact. This causes some deficiencies of the Lie theory in infinite dimensions. We summarize some classical results in finite dimensions which are NOT true in general in infinite dimensions as follows.(1)
There is NO Implicit Function Theorem or Inverse Function Theorem in infinite dimensions! (except Nash-Moser-type theorems).(2)
IfG is a finite-dimensional Lie group, the exponential map  exp:𝔤→G is defined as follows. To each ξ∈𝔤, we assign the corresponding left invariant vector field Xξ defined by (2.3). We take the flow φξ(t) of Xξ and define exp(ξ)=φξ(1). The exponential map is a local diffeomorphism from a neighborhood of zero in 𝔤 onto a neighborhood of the identity in G; hence exp defines canonical coordinates on the Lie group G. This is not true in infinite dimensions.(3)
Iff1,f2:G1→G2 are smooth Lie group homomorphisms (i.e., fi(g·h)=fi(g)·fi(h),i=1,2) with Tef1=Tef2, then locally f1=f2. This is not true in infinite dimensions.(4)
Iff:G→H is a continuous group homomorphism between finite-dimensional Lie groups, then f is smooth. This is not true in infinite dimensions.(5)
If𝔤 is any finite-dimensional Lie algebra, then there exists a connected finite-dimensional Lie group G with 𝔤 as its Lie algebra; that is, 𝔤≃TeG. This is not true in infinite dimensions.(6)
IfG is a finite-dimensional Lie group and H⊂G is a closed subgroup, then H is a Lie subgroup (i.e., Lie group and submanifold). This is not true in infinite dimensions.(7)
IfG is a finite-dimensional Lie group with Lie algebra 𝔤 and 𝔥⊂𝔤 is a subalgebra, then there exists a unique connected Lie subgroup H⊂G with 𝔥 as its Lie algebra; that is, 𝔥≃TeH. This is not true in infinite dimensions.Some classical examples offinite-dimensional Lie groups are the matrix groups GL(n), SL(n), O(n), SO(n), U(n), SU(n), and Sp(n) with smooth group operations given by matrix multiplication and matrix inversion. The Lie algebra bracket is the commutator [A,B]=AB-BA with exponential map given by exp(A)=∑i=0∞(1/i!)Ai=eA.


## 2.3. Examples of Infinite-Dimensional Lie Groups

### 2.3.1. The Vector Groups𝒢=(V,+)

LetV be a Banach space and take 𝒢=V with m(x,y)=x+y, i(x)=-x, and e=0, which makes 𝒢 into an Abelian Lie group; that is, m(x,y)=m(y,x). For the Lie algebra we have 𝔤≃TeV≃V. For u∈TeV the corresponding left invariant vector field Xu is given by Xu(v)=u, ∀v∈V; that is, Xu=const. Hence the Lie algebra 𝔤=V with the trivial Lie bracket [u,v]=0 is Abelian. For the exponential map we get exp:𝔤=V→𝒢=V,exp=idV.


### 2.3.2. The General Linear Group𝒢=(GL(V),∘)

LetV be a Banach space and L(V,V) the space of bounded linear operators A:V→V. Then L(V,V) is a Banach space with the operator norm ∥A∥=sup∥x∥≤1∥A(x)∥, and the group 𝒢=GL(V) of all invertible elements is open in L(V,V). So GL(V) is a smooth Lie group with m(f,g)=f∘g,i(f)=f-1, and e=idV. Its Lie algebra is 𝔤=L(V,V) with the commutator bracket [A,B]=AB-BA and exponential map expA=eA.


### 2.3.3. The Abelian Gauge Group𝒢=(C∞(M),+)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M) (smooth functions on M). With group operation being addition, that is, m(f,g)=f+g, i(f)=-f, and e=0. 𝒢 is an Abelian C∞ (addition is smooth) Frechet Lie group with Lie algebra 𝔤=TeC∞(M)≃C∞(M), with trivial bracket [ξ,η]=0, and exp=id. If we complete these spaces in the Ck-norm, k<∞ (denoted by 𝒢k), then 𝒢k is a Banach-Lie group, and if we complete in the Hs-Sobolev norm with s>(1/2)dimM then 𝒢s is a Hilbert-Lie group.


### 2.3.4. The Abelian Gauge Group𝒢=(C∞(M,ℝ-{0}),·)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M,ℝ-{0}), with group operation being multiplication; that is, m(f,g)=f·g, i(f)=f-1, and e=1. For k<∞, Ck(M,ℝ-{0}) is open in C∞(M,ℝ), and if M is compact, then Ck(M,ℝ-{0}) is a Banach-Lie group. If s>(1/2)dimM, then Hs(M,ℝ-{0}) is closed under multiplication, and if M is compact, then Hs(M,ℝ-{0}) is a Hilbert-Lie group.


### 2.3.5. Loop Group𝒢=(Ck(M,G),·)

We generalize the Abelian example (see Section2.3.4) by replacing ℝ-{0} with any finite-dimensional (non-Abelian) Lie group G. Let 𝒢=Ck(M,G) with pointwise group operations m(f,g)(x)=f(x)·g(x), x∈M, and i(f)(x)=(f(x))-1, where “·” and “(·)-1” are the operations in G. If k<∞ then Ck(M,G) is a Banach-Lie group. Let g denote the Lie algebra of G, then the Lie algebra of 𝒢=Ck(M,G) is 𝔤=Ck(M,g), with pointwise Lie bracket [ξ,η](x)=[ξ(x),η(x)], x∈M, the latter bracket being the Lie bracket in g. The exponential map exp:g→G defines the exponential map EXP:𝔤=Ck(M,g)→𝒢=Ck(M,G), EXP(ξ)=exp∘ξ, which is a local diffeomorphism. The same holds for Hs(M,G) if s>(1/2)dimM.Applications of these infinite-dimensional Lie groups are in gauge theories and quantum field theory, where they appear as groups of gauge transformations. We will discuss these in Section5.Special Case:𝒢=(Ck(S1,G),·)
As a special case of example mentioned in Section2.3.5 we take M=S1, the circle. Then 𝒢=Ck(S1,G)=ℒk(G) is called a loop group and 𝔤=Ck(S1,g)=𝔩k(g) is its loop algebra. They find applications in the theory of affine Lie algebras, Kac-Moody Lie algebras (central extensions), completely integrable systems, soliton equations (Toda, KdV, KP), and quantum field theory; see, for example, [3] and Section 5. Central extensions of loop algebras are examples of infinite-dimensional Lie algebras which need not have a corresponding Lie group.
Certain subgroups of loop groups play an important role in quantum field theory as groups of gauge transformations. We will discuss these in Section2.4.4.


## 2.3.1. The Vector Groups𝒢=(V,+)

LetV be a Banach space and take 𝒢=V with m(x,y)=x+y, i(x)=-x, and e=0, which makes 𝒢 into an Abelian Lie group; that is, m(x,y)=m(y,x). For the Lie algebra we have 𝔤≃TeV≃V. For u∈TeV the corresponding left invariant vector field Xu is given by Xu(v)=u, ∀v∈V; that is, Xu=const. Hence the Lie algebra 𝔤=V with the trivial Lie bracket [u,v]=0 is Abelian. For the exponential map we get exp:𝔤=V→𝒢=V,exp=idV.


## 2.3.2. The General Linear Group𝒢=(GL(V),∘)

LetV be a Banach space and L(V,V) the space of bounded linear operators A:V→V. Then L(V,V) is a Banach space with the operator norm ∥A∥=sup∥x∥≤1∥A(x)∥, and the group 𝒢=GL(V) of all invertible elements is open in L(V,V). So GL(V) is a smooth Lie group with m(f,g)=f∘g,i(f)=f-1, and e=idV. Its Lie algebra is 𝔤=L(V,V) with the commutator bracket [A,B]=AB-BA and exponential map expA=eA.


## 2.3.3. The Abelian Gauge Group𝒢=(C∞(M),+)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M) (smooth functions on M). With group operation being addition, that is, m(f,g)=f+g, i(f)=-f, and e=0. 𝒢 is an Abelian C∞ (addition is smooth) Frechet Lie group with Lie algebra 𝔤=TeC∞(M)≃C∞(M), with trivial bracket [ξ,η]=0, and exp=id. If we complete these spaces in the Ck-norm, k<∞ (denoted by 𝒢k), then 𝒢k is a Banach-Lie group, and if we complete in the Hs-Sobolev norm with s>(1/2)dimM then 𝒢s is a Hilbert-Lie group.


## 2.3.4. The Abelian Gauge Group𝒢=(C∞(M,ℝ-{0}),·)

LetM be a finite-dimensional manifold and let 𝒢=C∞(M,ℝ-{0}), with group operation being multiplication; that is, m(f,g)=f·g, i(f)=f-1, and e=1. For k<∞, Ck(M,ℝ-{0}) is open in C∞(M,ℝ), and if M is compact, then Ck(M,ℝ-{0}) is a Banach-Lie group. If s>(1/2)dimM, then Hs(M,ℝ-{0}) is closed under multiplication, and if M is compact, then Hs(M,ℝ-{0}) is a Hilbert-Lie group.


## 2.3.5. Loop Group𝒢=(Ck(M,G),·)

We generalize the Abelian example (see Section2.3.4) by replacing ℝ-{0} with any finite-dimensional (non-Abelian) Lie group G. Let 𝒢=Ck(M,G) with pointwise group operations m(f,g)(x)=f(x)·g(x), x∈M, and i(f)(x)=(f(x))-1, where “·” and “(·)-1” are the operations in G. If k<∞ then Ck(M,G) is a Banach-Lie group. Let g denote the Lie algebra of G, then the Lie algebra of 𝒢=Ck(M,G) is 𝔤=Ck(M,g), with pointwise Lie bracket [ξ,η](x)=[ξ(x),η(x)], x∈M, the latter bracket being the Lie bracket in g. The exponential map exp:g→G defines the exponential map EXP:𝔤=Ck(M,g)→𝒢=Ck(M,G), EXP(ξ)=exp∘ξ, which is a local diffeomorphism. The same holds for Hs(M,G) if s>(1/2)dimM.Applications of these infinite-dimensional Lie groups are in gauge theories and quantum field theory, where they appear as groups of gauge transformations. We will discuss these in Section5.Special Case:𝒢=(Ck(S1,G),·)
As a special case of example mentioned in Section2.3.5 we take M=S1, the circle. Then 𝒢=Ck(S1,G)=ℒk(G) is called a loop group and 𝔤=Ck(S1,g)=𝔩k(g) is its loop algebra. They find applications in the theory of affine Lie algebras, Kac-Moody Lie algebras (central extensions), completely integrable systems, soliton equations (Toda, KdV, KP), and quantum field theory; see, for example, [3] and Section 5. Central extensions of loop algebras are examples of infinite-dimensional Lie algebras which need not have a corresponding Lie group.
Certain subgroups of loop groups play an important role in quantum field theory as groups of gauge transformations. We will discuss these in Section2.4.4.


## 2.4. Diffeomorphism Groups

Among the most important “classical” infinite-dimensional Lie groups are the diffeomorphism groups of manifolds. Their differential structure is not the one of a Banach Lie group as defined above. Nevertheless they have important applications.LetM be a compact manifold (the noncompact case is technically much more complicated but similar results are true; see the study by Eichhorn and Schmid in [4]) and let 𝒢=Diff∞(M) be the group of all smooth diffeomorphisms on M, with group operation being composition; that is, m(f,g)=f∘g, i(f)=f-1, and e=idM. For C∞ diffeomorphisms, Diff∞(M) is a Frechet manifold and there are nontrivial problems with the notion of smooth maps between Frechet spaces. There is no canonical extension of the differential calculus from Banach spaces (which is the same as for ℝn) to Frechet spaces; see the study by Keller in [2]. One possibility is to generalize the notion of differentiability. For example, if we use the so-called CΓ∞ differentiability, then 𝒢=Diff∞(M) becomes a CΓ∞ Lie group with CΓ∞ differentiable group operations. These notions of differentiability are difficult to apply to concrete examples. Another possibility is to complete Diff∞(M) in the Banach Ck-norm, 0≤k<∞, or in the Sobolev Hs-norm, s>(1/2)dimM. Then Diffk(M) and Diffs(M) become Banach and Hilbert manifolds, respectively. Then we consider the inverse limits of these Banach- and Hilbert-Lie groups, respectively:(2.4)Diff∞(M)=lim←Diffk(M)
becomes a so-called ILB- (Inverse Limit of Banach) Lie group, or with the Sobolev topologies(2.5)Diff∞(M)=lim←Diffs(M)
becomes a so-called ILH- (Inverse Limit of Hilbert) Lie group. See the study by Omori in [5] for details. Nevertheless, the group operations are not smooth, but have the following differentiability properties. If we equip the diffeomorphism group with the Sobolev Hs-topology, then Diffs(M) becomes a C∞ Hilbert manifold if s>(1/2)dimM and the group multiplication(2.6)m:Diffs+k(M)×Diffs(M)→Diffs(M)
is Ck differentiable; hence for k=0, m is only continuous on Diffs(M). The inversion(2.7)i:Diffs+k(M)→Diffs(M)
is Ck differentiable; hence for k=0, i is only continuous on Diffs(M). The same differentiability properties of m and i hold in the Ck topology.TheLie algebra of Diff∞(M) is given by 𝔤=TeDiff∞(M)≃Vec∞(M) being the space of smooth vector fields on M. Note that the space Vec(M) of all vector fields is a Lie algebra only for C∞ vector fields, but not for Ck or Hs vector fields if k<∞, s<∞, because one loses derivatives by taking brackets.The exponential map on the diffeomorphism group is given as follows. For any vector fieldX∈Vec∞(M), take its flow φt∈Diff∞(M), then define EXP:Vec∞(M)→Diff∞(M):X↦φ1, the flow at time t=1. The exponential map EXP is NOT a local diffeomorphism; it is not even locally surjective.We see that the diffeomorphism groups are not Lie groups in the classical sense, but what we callnested Lie groups. Nevertheless they have important applications as we will see.


### 2.4.1. Subgroups ofDiff∞(M)

Several subgroups ofDiff∞(M) have important applications.


### 2.4.2. Group of Volume-Preserving Diffeomorphisms

Letμ be a volume on M and (2.8)𝒢=Diffμ∞(M)={f∈Diff∞(M)∣f*μ=μ}
the group of volume-preserving diffeomorphisms. Diffμ∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.9)𝔤=Vecμ∞(M)={X∈Vec∞(M)∣divμX=0}
being the space of divergence-free vector fields on M. Vecμ∞(M) is a Lie subalgebra of Vec∞(M).Remark 2.1.
We cannot apply the finite-dimensional theorem that ifVecμ∞(M) is Lie algebra then there exists a Lie group whose Lie algebra it is; nor the one that if Diffμ∞(M)⊂Diff(M) is a closed subgroup then it is an Lie subgroup.
NeverthelessDiffμ∞(M) is an ILH-Lie group.


### 2.4.3. Symplectomorphism Group

Letω be a symplectic 2-form on M and (2.10)𝒢=Diffω∞(M)={f∈xDiff∞(M)∣f*ω=ω}
the group of canonical transformations (or symplectomorphisms). Diffω∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.11)𝔤=Vecω∞(M)={X∈Vec∞(M)∣LXω=0}
being the space of locally Hamiltonian vector fields on M. Vecω∞(M) is a Lie subalgebra of Vec∞(M). Again Diffω∞(M) is an ILH-Lie group.


### 2.4.4. Group of Gauge Transformations

The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups which are not only ILH-Lie groups but actually Hilbert-Lie groups.Letπ:P→M be a principal G bundle with G being a finite-dimensional Lie group (structure group) acting on P from the right p∈P, g∈G, and p·g∈P.TheGauge group𝒢 is the group of gauge transformations defined by(2.12)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,π(ϕ(p))=π(p)}.𝒢 is a group under composition, hence a subgroup of the diffeomorphism group Diff∞(P). Since a gauge transformation ϕ∈𝒢 preserves fibers, we can realize each such ϕ∈𝒢 via ϕ(p)=p·τ(p), where τ:P→G satisfies τ(p·g)=g-1τ(p)g, for p∈P, g∈G. Let(2.13)Gau(P)={τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}.Gau(P) is a group under pointwise multiplication, hence a subgroup of the loop group C∞(P,G) (see Section 2.4.3), which extends to a Hilbert-Lie group if equipped with the Hs-Sobolev topology. We give Gau(P) the induced topology and extend it to a Hilbert-Lie group denoted by Gaus(P). Another interpretation is that Gau(P) is isomorphic to C∞(AdP) the space of sections of the associated vector bundle Ad(P)=P×GG. Completed in the Hs Sobolev topology, we get Gaus(P)≃Hs(AdP).Letg denote the Lie algebra of G. Then the Lie algebra  𝔤𝔞𝔲(P) of Gau(P) is a subalgebra of the loop algebra Hs(P,g) under pointwise bracket in g, the finite-dimensional Lie algebra of G; that is, for any ξ,η∈Hs(P,g) the bracket is defined by [ξ,η]𝔤𝔞𝔲(P)(p)=[ξ(p),η(p)]g, p∈P. Then 𝔤𝔞𝔲s(P) is the subalgebra of Ad -invariant g-valued functions on P; that is,(2.14)𝔤𝔞𝔲(P)={ξ∈C∞(P,g);ξ(p·g)=Adg-1ξ(p)}.The Lie algebra𝔩𝔦𝔢𝒢 (running out of symbols) of the gauge group 𝒢 is the Lie subalgebra of 𝔛∞(P) consisting of all G-invariant vertical vector fields X on P; that is,(2.15)𝔩𝔦𝔢𝒢={X∈𝔛∞(P);Rg*X=X,X(p)∈g,g∈G,p∈P}
with commutator bracket [X1,X2]=X1X2-X2X1∈𝔩𝔦𝔢𝒢.On the other hand, the Lie algebra ofC∞(AdP) is C∞(ad(P)) being the space of sections of the associated vector bundle ad(P)≡(P×Gg)→M with pointwise bracket.We have three versions of gauge groups:𝒢,Gau(P), and C∞(AdP). They are all group isomorphic. There is a natural group isomorphism Gau(P)→𝒢:τ↦ϕ defined by ϕ(p)=p·τ(p), p∈P, which preserves the product τ1·τ2↦ϕ1∘ϕ2. Identifying 𝒢 with Gau(P), we can avoid the troubles with diffeomorphism groups and we can extend 𝒢 to a Hilbert-Lie group 𝒢s. So 𝒢s is actually a Hilbert-Lie group in the classical sense; that is, the group operations are C∞. Also the three Lie algebras 𝔩𝔦𝔢𝒢, 𝔤𝔞𝔲(P), and C∞(adP) are canonically isomorphic. Indeed, for s∈C∞(adP) define ξ∈𝔤𝔞𝔲(P)ξ:P→g by ξ(p·a):=Ada-1ξ(p); and for ξ∈𝔤𝔞𝔲(P) define s∈C∞(adP) by s(π(p)):=[p,ξ(p)].On the other hand, forξ∈𝔤𝔞𝔲(P) define Zξ∈𝔩𝔦𝔢𝒢 by (2.16)Zξ(p)=ddt∣t=0R(p,exptξ(p))(=ξ(p)*(p)),
that is, Zξ is the fundamental vector field on P, generated by ξ∈g. Zξ is invariant if and only if ξ(p·g)=Adg-1ξ(p).To topologize𝔩𝔦𝔢𝒢, we complete C∞(adP) in the Hs-Sobolev norm. If s>(1/2)dimM, then 𝔩𝔦𝔢𝒢s≃Hs(adP)≃𝔤𝔞𝔲s(P) are isomorphic Hilbert-Lie algebras.There is a natural exponential mapExp:𝔤𝔞𝔲(P)→Gau(P), which is a local diffeomorphism. Let exp:g→G  be the finite-dimensional exponential map. Then define(2.17)Exp:𝔤𝔞𝔲s(P)→Gaus(P):(Expξ)(p)=exp(ξ(p)),ξ∈𝔤𝔞𝔲s(P).
Or in terms of 𝒢, Exp:𝔩𝔦𝔢𝒢s→𝒢s:(Expξ)(p)=p·exp(ξp).We have the following theorem (Schmid [6]).Theorem 2.2.
Fors>(1/2)dimM,
(2.18)𝒢s≃Gaus(P)≃Hs(AdP)
is a smooth Hilbert-Lie group with Lie algebra
(2.19)𝔩𝔦𝔢𝒢s≃𝔤𝔞𝔲s(P)≃Hs(adP)
and smooth exponential map, which is a local diffeomorphism,
(2.20)EXP:𝔩𝔦𝔢𝒢s→𝒢s:(EXPξ)(p)=p·exp(ξ(p)).See [1–5, 7–19].


## 2.4.1. Subgroups ofDiff∞(M)

Several subgroups ofDiff∞(M) have important applications.


## 2.4.2. Group of Volume-Preserving Diffeomorphisms

Letμ be a volume on M and (2.8)𝒢=Diffμ∞(M)={f∈Diff∞(M)∣f*μ=μ}
the group of volume-preserving diffeomorphisms. Diffμ∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.9)𝔤=Vecμ∞(M)={X∈Vec∞(M)∣divμX=0}
being the space of divergence-free vector fields on M. Vecμ∞(M) is a Lie subalgebra of Vec∞(M).Remark 2.1.
We cannot apply the finite-dimensional theorem that ifVecμ∞(M) is Lie algebra then there exists a Lie group whose Lie algebra it is; nor the one that if Diffμ∞(M)⊂Diff(M) is a closed subgroup then it is an Lie subgroup.
NeverthelessDiffμ∞(M) is an ILH-Lie group.


## 2.4.3. Symplectomorphism Group

Letω be a symplectic 2-form on M and (2.10)𝒢=Diffω∞(M)={f∈xDiff∞(M)∣f*ω=ω}
the group of canonical transformations (or symplectomorphisms). Diffω∞(M) is a closed subgroup of Diff∞(M) with Lie algebra (2.11)𝔤=Vecω∞(M)={X∈Vec∞(M)∣LXω=0}
being the space of locally Hamiltonian vector fields on M. Vecω∞(M) is a Lie subalgebra of Vec∞(M). Again Diffω∞(M) is an ILH-Lie group.


## 2.4.4. Group of Gauge Transformations

The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups which are not only ILH-Lie groups but actually Hilbert-Lie groups.Letπ:P→M be a principal G bundle with G being a finite-dimensional Lie group (structure group) acting on P from the right p∈P, g∈G, and p·g∈P.TheGauge group𝒢 is the group of gauge transformations defined by(2.12)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,π(ϕ(p))=π(p)}.𝒢 is a group under composition, hence a subgroup of the diffeomorphism group Diff∞(P). Since a gauge transformation ϕ∈𝒢 preserves fibers, we can realize each such ϕ∈𝒢 via ϕ(p)=p·τ(p), where τ:P→G satisfies τ(p·g)=g-1τ(p)g, for p∈P, g∈G. Let(2.13)Gau(P)={τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}.Gau(P) is a group under pointwise multiplication, hence a subgroup of the loop group C∞(P,G) (see Section 2.4.3), which extends to a Hilbert-Lie group if equipped with the Hs-Sobolev topology. We give Gau(P) the induced topology and extend it to a Hilbert-Lie group denoted by Gaus(P). Another interpretation is that Gau(P) is isomorphic to C∞(AdP) the space of sections of the associated vector bundle Ad(P)=P×GG. Completed in the Hs Sobolev topology, we get Gaus(P)≃Hs(AdP).Letg denote the Lie algebra of G. Then the Lie algebra  𝔤𝔞𝔲(P) of Gau(P) is a subalgebra of the loop algebra Hs(P,g) under pointwise bracket in g, the finite-dimensional Lie algebra of G; that is, for any ξ,η∈Hs(P,g) the bracket is defined by [ξ,η]𝔤𝔞𝔲(P)(p)=[ξ(p),η(p)]g, p∈P. Then 𝔤𝔞𝔲s(P) is the subalgebra of Ad -invariant g-valued functions on P; that is,(2.14)𝔤𝔞𝔲(P)={ξ∈C∞(P,g);ξ(p·g)=Adg-1ξ(p)}.The Lie algebra𝔩𝔦𝔢𝒢 (running out of symbols) of the gauge group 𝒢 is the Lie subalgebra of 𝔛∞(P) consisting of all G-invariant vertical vector fields X on P; that is,(2.15)𝔩𝔦𝔢𝒢={X∈𝔛∞(P);Rg*X=X,X(p)∈g,g∈G,p∈P}
with commutator bracket [X1,X2]=X1X2-X2X1∈𝔩𝔦𝔢𝒢.On the other hand, the Lie algebra ofC∞(AdP) is C∞(ad(P)) being the space of sections of the associated vector bundle ad(P)≡(P×Gg)→M with pointwise bracket.We have three versions of gauge groups:𝒢,Gau(P), and C∞(AdP). They are all group isomorphic. There is a natural group isomorphism Gau(P)→𝒢:τ↦ϕ defined by ϕ(p)=p·τ(p), p∈P, which preserves the product τ1·τ2↦ϕ1∘ϕ2. Identifying 𝒢 with Gau(P), we can avoid the troubles with diffeomorphism groups and we can extend 𝒢 to a Hilbert-Lie group 𝒢s. So 𝒢s is actually a Hilbert-Lie group in the classical sense; that is, the group operations are C∞. Also the three Lie algebras 𝔩𝔦𝔢𝒢, 𝔤𝔞𝔲(P), and C∞(adP) are canonically isomorphic. Indeed, for s∈C∞(adP) define ξ∈𝔤𝔞𝔲(P)ξ:P→g by ξ(p·a):=Ada-1ξ(p); and for ξ∈𝔤𝔞𝔲(P) define s∈C∞(adP) by s(π(p)):=[p,ξ(p)].On the other hand, forξ∈𝔤𝔞𝔲(P) define Zξ∈𝔩𝔦𝔢𝒢 by (2.16)Zξ(p)=ddt∣t=0R(p,exptξ(p))(=ξ(p)*(p)),
that is, Zξ is the fundamental vector field on P, generated by ξ∈g. Zξ is invariant if and only if ξ(p·g)=Adg-1ξ(p).To topologize𝔩𝔦𝔢𝒢, we complete C∞(adP) in the Hs-Sobolev norm. If s>(1/2)dimM, then 𝔩𝔦𝔢𝒢s≃Hs(adP)≃𝔤𝔞𝔲s(P) are isomorphic Hilbert-Lie algebras.There is a natural exponential mapExp:𝔤𝔞𝔲(P)→Gau(P), which is a local diffeomorphism. Let exp:g→G  be the finite-dimensional exponential map. Then define(2.17)Exp:𝔤𝔞𝔲s(P)→Gaus(P):(Expξ)(p)=exp(ξ(p)),ξ∈𝔤𝔞𝔲s(P).
Or in terms of 𝒢, Exp:𝔩𝔦𝔢𝒢s→𝒢s:(Expξ)(p)=p·exp(ξp).We have the following theorem (Schmid [6]).Theorem 2.2.
Fors>(1/2)dimM,
(2.18)𝒢s≃Gaus(P)≃Hs(AdP)
is a smooth Hilbert-Lie group with Lie algebra
(2.19)𝔩𝔦𝔢𝒢s≃𝔤𝔞𝔲s(P)≃Hs(adP)
and smooth exponential map, which is a local diffeomorphism,
(2.20)EXP:𝔩𝔦𝔢𝒢s→𝒢s:(EXPξ)(p)=p·exp(ξ(p)).See [1–5, 7–19].


## 3. Lie Groups as Symmetry Groups of Hamiltonian Systems

A short introduction and “crash course” to geometric mechanics can be found in the studies by Abraham and Marsden [20], Marsden [21], as well as Marsden and Ratiu [22]. For the general theory of infinite-dimensional manifolds and global analysis, see, for example, the studies by Bourbaki [9], Lang [14], as well as Palais [18].


### 3.1. Hamilton’s Equations on Poisson Manifolds

APoisson manifold is a manifold P (in general infinite-dimensional) equipped with a bilinear operation {·,·}, called Poisson bracket, on the space C∞(P) of smooth functions on P satisfying the following.(i)
(C∞(P),{·,·}) is a Lie algebra; that is, {·,·}:C∞(P)×C∞(P)→C∞(P) is bilinear, skew symmetric and satisfies the Jacobi identity {{F,G},H}+{{H,F},G}+{{G,H},F}=0 for all F,G,H∈C∞(P).(ii)
{·,·} satisfies the Leibniz rule; that is, {·,·} is a derivation in each factor: {F·G,H}=F·{G,H}+G·{F,H}, for all F,G,H∈C∞(P).The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The namePoisson manifold was coined by Lichnerowicz.For anyH∈C∞(P) we define the Hamiltonian vector field  XH by(3.1)XH(F)={F,H},F∈C∞(P).
It follows from (ii) that indeed XH defines a derivation on C∞(P), hence a vector field on P.  Hamilton’s equations of motion for a function F∈C∞(P) with Hamiltonian H∈C∞(P) (energy function) are then defined by the flow (integral curves) of the vector field XH; that is,(3.2)Ḟ=XH(F)={F,H},wherė=ddt.
We then call F a Hamiltonian system on P with energy (Hamiltonian function) H.


### 3.2. Examples of Poisson Manifolds and Hamilton’s Equations

Poisson manifolds are a generalization of symplectic manifolds on which Hamilton’s equations have a canonical formulated.


#### 3.2.1. Finite-Dimensional Classical Mechanics

For finite-dimensional classical mechanics we takeP=ℝ2n with coordinates (q1,…,qn,p1,…,pn) with the standard Poisson bracket for any two functions F(qi,pi), H(qi,pi) given by(3.3){F,H}=∑i=1n∂F∂pi∂H∂qi-∂H∂pi∂F∂qi.
Then the classical Hamilton’s equations are(3.4)q̇i={qi,H}=∂H∂pi,ṗi={pi,H}=-∂H∂qi,
where i=1,…,n. This finite-dimensional Hamiltonian system is a system of ordinary differential equations for which there are well-known existence and uniqueness theorems; that is, it has locally unique smooth solutions, depending smoothly on the initial conditions.Example 3.1 (Harmonic Oscillator).
As a concrete example we consider the harmonic oscillator. HereP=ℝ2 and the Hamiltonian (energy) is H(q,p)=(1/2)(q2+p2). Then Hamilton’s equations are
(3.5)q̇=p,ṗ=-q.


#### 3.2.2. Infinite-Dimensional Classical Field Theory

LetV be a Banach space and V* its dual space with respect to a pairing 〈·,·〉:V×V*→ℝ (i.e., 〈·,·〉 is a symmetric, bilinear, nondegenerate function). On P=V×V* we have the canonical Poisson bracket for F, H∈C∞(P), φ∈V, and π∈V*, given by(3.6){F,H}=〈δFδπ,δHδφ〉-〈δHδπ,δFδφ〉,
where the functional derivatives δF/δπ∈V, δF/δφ∈V* are the “duals” under the pairing 〈·,·〉 of the partial gradients D1F(π)∈V*,D2F(φ)∈V**≃V. The corresponding Hamilton’s equations are(3.7)φ̇={φ,H}=δHδπ,π̇={π,H}=-δHδφ.As a special case in finite dimensions, ifV≃ℝn, so that V*≃ℝn and P=V×V*≃ℝ2n, and the pairing is the standard inner product in ℝn, then the Poisson bracket (3.6) and Hamilton’s equations (3.7) are identical with (3.3) and (3.4), respectively.Example 3.2 (Wave Equations).
As a concrete example we consider the wave equations. LetV=C∞(ℝ3) and V*=Den(ℝ3) (densities) and the L2 pairing 〈φ,π〉=∫φ(x)π(x)dx. We take the Hamiltonian to be H(φ,π)=∫((1/2)π2+(1/2)|∇φ|2+F(φ))dx, where F is some function on V. Then Hamilton’s (3.7) become
(3.8)φ̇=π,π̇=∇2φ-F'(φ),where′=ddφ,
which imply the wave equation ∂2φ/∂t2=∇2φ-F'(φ). Different choices of F give different wave equations; for example, for F=0 we get the linear wave equation ∂2φ/∂t2=∇2φ. For F=(1/2)mφ we get the Klein-Gordon equation ∇2φ-∂2φ/∂t2=mφ. So these wave equations and the Klein-Gordon equation are infinite-dimensional Hamiltonian systems on P=C∞(ℝ3)×Den(ℝ3).


#### 3.2.3. Cotangent Bundles

The finite-dimensional examples of Poisson brackets (3.3) and Hamilton’s (3.4) and the infinite-dimensional examples (3.6) and (3.7) are the local versions of the general case where P=T*Q is the cotangent bundle (phase space) of a manifold Q (configuration space). If Q is an n-dimensional manifold, then T*Q is a 2n-Poisson manifold locally isomorphic to ℝ2n whose Poisson bracket is locally given by (3.3) and Hamilton’s equations are locally given by (3.4). If Q is an infinite-dimensional Banach manifold, then T*Q is a Poisson manifold locally isomorphic to V×V* whose Poisson bracket is given by (3.6) and Hamilton’s equations are locally given by (3.7).


#### 3.2.4. Symplectic Manifolds

All the examples above are special cases of symplectic manifolds(P,ω). That means that P is equipped with a symplectic structure ω which is a closed (dω=0), (weakly) nondegenerate 2-form on the manifold P. Then for any H∈C∞(P) the corresponding Hamiltonian vector field XH is defined by dH=ω(XH,·) and the canonical Poisson bracket is given by(3.9){F,H}=ω(XF,XH),F,H∈C∞(P).
For example, on ℝ2n the canonical symplectic structure ω is given by ω=∑i=1ndpi∧dqi=dθ, where θ=∑i=1npi∧dqi. The same formula for ω holds locally in T*Q for any finite-dimensional Q (Darboux’s Lemma). For the infinite-dimensional example P=V×V*, the symplectic form ω is given by ω((φ1,π1),(φ2,π2))=〈φ1,π2〉-〈φ2,π1〉. Again these two formulas for ω are identical if V=ℝn.Remark 3.3.
(A) IfP is a finite-dimensional symplectic manifold, then P is even dimensional.
(B) If the Poisson bracket{·,·} is nondegenerate, then {·,·} comes from a symplectic form ω; that is, {·,·} is given by (3.9).


#### 3.2.5. The Lie-Poisson Bracket

Not all Poisson brackets are of the form given in the above examples (3.3), (3.6), and (3.9); that is, not all Poisson manifolds are symplectic manifolds. An important class of Poisson bracket is the so-called Lie-Poisson bracket. It is defined on the dual of any Lie algebra. Let G be a Lie group with Lie algebra 𝔤=TeG≃{left invariant vector fields on G}, and let [·,·] denote the Lie bracket (commutator) on 𝔤. Let 𝔤* be the dual of a 𝔤 with respect to a pairing 〈·,·〉:𝔤*×𝔤→ℝ. Then for any F,H∈C∞(𝔤*) and μ∈𝔤*, the Lie-Poisson bracket is defined by(3.10){F,H}(μ)=±〈μ,[δFδμ,δHδμ]〉,
where δF/δμ,δH/δμ∈𝔤 are the "duals" of the gradients DF(μ),DH(μ)∈𝔤**≃𝔤 under the pairing 〈·,·〉. Note that the Lie-Poisson bracket is degenerate in general; for example, for G=SO(3) the vector space 𝔤* is 3 dimensional, so the Poisson bracket (3.10) cannot come from a symplectic structure. This Lie-Poisson bracket can also be obtained in a different way by taking the canonical Poisson bracket on T*G (locally given by (3.3) and (3.6)) and then restricting it to the fiber at the identity Te*G=𝔤*. In this sense the Lie-Poisson bracket (3.10) is induced from the canonical Poisson bracket on T*G. It is induced by the symmetry of left multiplication as we will discuss in Section 3.3.Example 3.4 (Rigid Body).
A concrete example of the Lie-Poisson bracket is given by the rigid body. HereG=SO(3) is the configuration space of a free rigid body. Identifying the Lie algebra (𝔰𝔬(3),[·,·]) with (ℝ3,×), where × is the vector product on ℝ3, and 𝔤*=𝔰𝔬(3)*≃ℝ3, the Lie-Poisson bracket translates into
(3.11){F,H}(m)=-m·(∇F×∇H).
For any F∈C∞(𝔰𝔬(3)*), we have (dF/dt)(m)=∇F·ṁ={F,H}(m)=-m·(∇F×∇H)=∇F·(m×∇H); hence ṁ=m×∇H. With the Hamiltonian H=(1/2)(m12/I12+m22/I22+m32/I32) we get Hamilton’s equation as
(3.12)ṁ1=I2-I3I2I3m2m3,ṁ2=I3-I1I3I1m3m1,ṁ3=I1-I2I1I2m1m2.
These are Euler’s equations for the free rigid body.


### 3.3. Reduction by Symmetries

The examples we have discussed so far are all canonical examples of Poisson brackets, defined either on a symplectic manifold(P,ω) or T*Q, or on the dual of a Lie algebra 𝔤*. Different, noncanonical Poisson brackets can arise from symmetries. Assume that a Lie group G is acting in a Hamiltonian way on the Poisson manifold (P,{·,·}). That means that we have a smooth map φ:G×P→P:φ(g,p)=g·p such that the induced maps φg=φ(g,·):P→P are canonical transformations, for each g∈G. In terms of Poisson manifolds, a canonical transformation is a smooth map that preserves the Poisson bracket. So the action of G on P is a Hamiltonian action if φg*{F,H}={φg*F,φg*H}, for all F,H∈C∞(P),g∈G. For any ξ∈𝔤 the canonical transformations φexp(tξ) generate a Hamiltonian vector field ξF on P and a momentum map J:P→𝔤* given by J(x)(ξ)=F(x), which is Ad* equivariant.If a Hamiltonian systemXH is invariant under a Lie group action, that is, H(φg(x))=H(x), then we obtain a reduced Hamiltonian system on a reduced phase space (reduced Poisson manifold). We recall the following Marsden-Weinstein reduction theorem [23].Theorem 3.5 (Reduction Theorem).
For a Hamiltonian action of a Lie groupG on a Poisson manifold (P,{·,·}), there is an equivariant momentum map J:P→𝔤* and for every regular μ∈𝔤* the reduced phase space Pμ≡J-1(μ)/Gμ carries an induced Poisson structure {·,·}μ (Gμ being the isotropy group). Any G-invariant Hamiltonian H on P defines a Hamiltonian Hμ on the reduced phase space Pμ, and the integral curves of the vector field XH project onto integral curves of the induced vector field X̂Hμ on the reduced space Pμ.Example 3.6 (Rigid Body).
The rigid body discussed above can be viewed as an example of this reduction theorem. IfP=T*G and G is acting on T*G by the cotangent lift of the left translation lg:G→G,lg(h)=gh, then the momentum map J:T*G→𝔤* is given by J(αg)=Te*Rg(αg) and the reduced phase space (T*G)μ=J-1(μ)/Gμ is isomorphic to the coadjoint orbit 𝒪μ through μ∈𝔤*. Each coadjoint orbit 𝒪μ carries a natural symplectic structure ωμ, and in this case, the reduced Lie-Poisson bracket {·,·}μ on the coadjoint orbit 𝒪μ is induced by the symplectic form ωμ on 𝒪μ as in (3.9). Furthermore T*G/G≃𝔤* and the induced Poisson bracket {·,·}μ on 𝒪μ are identical with the Lie-Poisson bracket restricted to the coadjoint orbit 𝒪μ⊂𝔤*. For the rigid body we apply this construction to G=SO(3).See [1, 8, 10, 17, 19–31].


## 3.1. Hamilton’s Equations on Poisson Manifolds

APoisson manifold is a manifold P (in general infinite-dimensional) equipped with a bilinear operation {·,·}, called Poisson bracket, on the space C∞(P) of smooth functions on P satisfying the following.(i)
(C∞(P),{·,·}) is a Lie algebra; that is, {·,·}:C∞(P)×C∞(P)→C∞(P) is bilinear, skew symmetric and satisfies the Jacobi identity {{F,G},H}+{{H,F},G}+{{G,H},F}=0 for all F,G,H∈C∞(P).(ii)
{·,·} satisfies the Leibniz rule; that is, {·,·} is a derivation in each factor: {F·G,H}=F·{G,H}+G·{F,H}, for all F,G,H∈C∞(P).The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The namePoisson manifold was coined by Lichnerowicz.For anyH∈C∞(P) we define the Hamiltonian vector field  XH by(3.1)XH(F)={F,H},F∈C∞(P).
It follows from (ii) that indeed XH defines a derivation on C∞(P), hence a vector field on P.  Hamilton’s equations of motion for a function F∈C∞(P) with Hamiltonian H∈C∞(P) (energy function) are then defined by the flow (integral curves) of the vector field XH; that is,(3.2)Ḟ=XH(F)={F,H},wherė=ddt.
We then call F a Hamiltonian system on P with energy (Hamiltonian function) H.


## 3.2. Examples of Poisson Manifolds and Hamilton’s Equations

Poisson manifolds are a generalization of symplectic manifolds on which Hamilton’s equations have a canonical formulated.


### 3.2.1. Finite-Dimensional Classical Mechanics

For finite-dimensional classical mechanics we takeP=ℝ2n with coordinates (q1,…,qn,p1,…,pn) with the standard Poisson bracket for any two functions F(qi,pi), H(qi,pi) given by(3.3){F,H}=∑i=1n∂F∂pi∂H∂qi-∂H∂pi∂F∂qi.
Then the classical Hamilton’s equations are(3.4)q̇i={qi,H}=∂H∂pi,ṗi={pi,H}=-∂H∂qi,
where i=1,…,n. This finite-dimensional Hamiltonian system is a system of ordinary differential equations for which there are well-known existence and uniqueness theorems; that is, it has locally unique smooth solutions, depending smoothly on the initial conditions.Example 3.1 (Harmonic Oscillator).
As a concrete example we consider the harmonic oscillator. HereP=ℝ2 and the Hamiltonian (energy) is H(q,p)=(1/2)(q2+p2). Then Hamilton’s equations are
(3.5)q̇=p,ṗ=-q.


### 3.2.2. Infinite-Dimensional Classical Field Theory

LetV be a Banach space and V* its dual space with respect to a pairing 〈·,·〉:V×V*→ℝ (i.e., 〈·,·〉 is a symmetric, bilinear, nondegenerate function). On P=V×V* we have the canonical Poisson bracket for F, H∈C∞(P), φ∈V, and π∈V*, given by(3.6){F,H}=〈δFδπ,δHδφ〉-〈δHδπ,δFδφ〉,
where the functional derivatives δF/δπ∈V, δF/δφ∈V* are the “duals” under the pairing 〈·,·〉 of the partial gradients D1F(π)∈V*,D2F(φ)∈V**≃V. The corresponding Hamilton’s equations are(3.7)φ̇={φ,H}=δHδπ,π̇={π,H}=-δHδφ.As a special case in finite dimensions, ifV≃ℝn, so that V*≃ℝn and P=V×V*≃ℝ2n, and the pairing is the standard inner product in ℝn, then the Poisson bracket (3.6) and Hamilton’s equations (3.7) are identical with (3.3) and (3.4), respectively.Example 3.2 (Wave Equations).
As a concrete example we consider the wave equations. LetV=C∞(ℝ3) and V*=Den(ℝ3) (densities) and the L2 pairing 〈φ,π〉=∫φ(x)π(x)dx. We take the Hamiltonian to be H(φ,π)=∫((1/2)π2+(1/2)|∇φ|2+F(φ))dx, where F is some function on V. Then Hamilton’s (3.7) become
(3.8)φ̇=π,π̇=∇2φ-F'(φ),where′=ddφ,
which imply the wave equation ∂2φ/∂t2=∇2φ-F'(φ). Different choices of F give different wave equations; for example, for F=0 we get the linear wave equation ∂2φ/∂t2=∇2φ. For F=(1/2)mφ we get the Klein-Gordon equation ∇2φ-∂2φ/∂t2=mφ. So these wave equations and the Klein-Gordon equation are infinite-dimensional Hamiltonian systems on P=C∞(ℝ3)×Den(ℝ3).


### 3.2.3. Cotangent Bundles

The finite-dimensional examples of Poisson brackets (3.3) and Hamilton’s (3.4) and the infinite-dimensional examples (3.6) and (3.7) are the local versions of the general case where P=T*Q is the cotangent bundle (phase space) of a manifold Q (configuration space). If Q is an n-dimensional manifold, then T*Q is a 2n-Poisson manifold locally isomorphic to ℝ2n whose Poisson bracket is locally given by (3.3) and Hamilton’s equations are locally given by (3.4). If Q is an infinite-dimensional Banach manifold, then T*Q is a Poisson manifold locally isomorphic to V×V* whose Poisson bracket is given by (3.6) and Hamilton’s equations are locally given by (3.7).


### 3.2.4. Symplectic Manifolds

All the examples above are special cases of symplectic manifolds(P,ω). That means that P is equipped with a symplectic structure ω which is a closed (dω=0), (weakly) nondegenerate 2-form on the manifold P. Then for any H∈C∞(P) the corresponding Hamiltonian vector field XH is defined by dH=ω(XH,·) and the canonical Poisson bracket is given by(3.9){F,H}=ω(XF,XH),F,H∈C∞(P).
For example, on ℝ2n the canonical symplectic structure ω is given by ω=∑i=1ndpi∧dqi=dθ, where θ=∑i=1npi∧dqi. The same formula for ω holds locally in T*Q for any finite-dimensional Q (Darboux’s Lemma). For the infinite-dimensional example P=V×V*, the symplectic form ω is given by ω((φ1,π1),(φ2,π2))=〈φ1,π2〉-〈φ2,π1〉. Again these two formulas for ω are identical if V=ℝn.Remark 3.3.
(A) IfP is a finite-dimensional symplectic manifold, then P is even dimensional.
(B) If the Poisson bracket{·,·} is nondegenerate, then {·,·} comes from a symplectic form ω; that is, {·,·} is given by (3.9).


### 3.2.5. The Lie-Poisson Bracket

Not all Poisson brackets are of the form given in the above examples (3.3), (3.6), and (3.9); that is, not all Poisson manifolds are symplectic manifolds. An important class of Poisson bracket is the so-called Lie-Poisson bracket. It is defined on the dual of any Lie algebra. Let G be a Lie group with Lie algebra 𝔤=TeG≃{left invariant vector fields on G}, and let [·,·] denote the Lie bracket (commutator) on 𝔤. Let 𝔤* be the dual of a 𝔤 with respect to a pairing 〈·,·〉:𝔤*×𝔤→ℝ. Then for any F,H∈C∞(𝔤*) and μ∈𝔤*, the Lie-Poisson bracket is defined by(3.10){F,H}(μ)=±〈μ,[δFδμ,δHδμ]〉,
where δF/δμ,δH/δμ∈𝔤 are the "duals" of the gradients DF(μ),DH(μ)∈𝔤**≃𝔤 under the pairing 〈·,·〉. Note that the Lie-Poisson bracket is degenerate in general; for example, for G=SO(3) the vector space 𝔤* is 3 dimensional, so the Poisson bracket (3.10) cannot come from a symplectic structure. This Lie-Poisson bracket can also be obtained in a different way by taking the canonical Poisson bracket on T*G (locally given by (3.3) and (3.6)) and then restricting it to the fiber at the identity Te*G=𝔤*. In this sense the Lie-Poisson bracket (3.10) is induced from the canonical Poisson bracket on T*G. It is induced by the symmetry of left multiplication as we will discuss in Section 3.3.Example 3.4 (Rigid Body).
A concrete example of the Lie-Poisson bracket is given by the rigid body. HereG=SO(3) is the configuration space of a free rigid body. Identifying the Lie algebra (𝔰𝔬(3),[·,·]) with (ℝ3,×), where × is the vector product on ℝ3, and 𝔤*=𝔰𝔬(3)*≃ℝ3, the Lie-Poisson bracket translates into
(3.11){F,H}(m)=-m·(∇F×∇H).
For any F∈C∞(𝔰𝔬(3)*), we have (dF/dt)(m)=∇F·ṁ={F,H}(m)=-m·(∇F×∇H)=∇F·(m×∇H); hence ṁ=m×∇H. With the Hamiltonian H=(1/2)(m12/I12+m22/I22+m32/I32) we get Hamilton’s equation as
(3.12)ṁ1=I2-I3I2I3m2m3,ṁ2=I3-I1I3I1m3m1,ṁ3=I1-I2I1I2m1m2.
These are Euler’s equations for the free rigid body.


## 3.2.1. Finite-Dimensional Classical Mechanics

For finite-dimensional classical mechanics we takeP=ℝ2n with coordinates (q1,…,qn,p1,…,pn) with the standard Poisson bracket for any two functions F(qi,pi), H(qi,pi) given by(3.3){F,H}=∑i=1n∂F∂pi∂H∂qi-∂H∂pi∂F∂qi.
Then the classical Hamilton’s equations are(3.4)q̇i={qi,H}=∂H∂pi,ṗi={pi,H}=-∂H∂qi,
where i=1,…,n. This finite-dimensional Hamiltonian system is a system of ordinary differential equations for which there are well-known existence and uniqueness theorems; that is, it has locally unique smooth solutions, depending smoothly on the initial conditions.Example 3.1 (Harmonic Oscillator).
As a concrete example we consider the harmonic oscillator. HereP=ℝ2 and the Hamiltonian (energy) is H(q,p)=(1/2)(q2+p2). Then Hamilton’s equations are
(3.5)q̇=p,ṗ=-q.


## 3.2.2. Infinite-Dimensional Classical Field Theory

LetV be a Banach space and V* its dual space with respect to a pairing 〈·,·〉:V×V*→ℝ (i.e., 〈·,·〉 is a symmetric, bilinear, nondegenerate function). On P=V×V* we have the canonical Poisson bracket for F, H∈C∞(P), φ∈V, and π∈V*, given by(3.6){F,H}=〈δFδπ,δHδφ〉-〈δHδπ,δFδφ〉,
where the functional derivatives δF/δπ∈V, δF/δφ∈V* are the “duals” under the pairing 〈·,·〉 of the partial gradients D1F(π)∈V*,D2F(φ)∈V**≃V. The corresponding Hamilton’s equations are(3.7)φ̇={φ,H}=δHδπ,π̇={π,H}=-δHδφ.As a special case in finite dimensions, ifV≃ℝn, so that V*≃ℝn and P=V×V*≃ℝ2n, and the pairing is the standard inner product in ℝn, then the Poisson bracket (3.6) and Hamilton’s equations (3.7) are identical with (3.3) and (3.4), respectively.Example 3.2 (Wave Equations).
As a concrete example we consider the wave equations. LetV=C∞(ℝ3) and V*=Den(ℝ3) (densities) and the L2 pairing 〈φ,π〉=∫φ(x)π(x)dx. We take the Hamiltonian to be H(φ,π)=∫((1/2)π2+(1/2)|∇φ|2+F(φ))dx, where F is some function on V. Then Hamilton’s (3.7) become
(3.8)φ̇=π,π̇=∇2φ-F'(φ),where′=ddφ,
which imply the wave equation ∂2φ/∂t2=∇2φ-F'(φ). Different choices of F give different wave equations; for example, for F=0 we get the linear wave equation ∂2φ/∂t2=∇2φ. For F=(1/2)mφ we get the Klein-Gordon equation ∇2φ-∂2φ/∂t2=mφ. So these wave equations and the Klein-Gordon equation are infinite-dimensional Hamiltonian systems on P=C∞(ℝ3)×Den(ℝ3).


## 3.2.3. Cotangent Bundles

The finite-dimensional examples of Poisson brackets (3.3) and Hamilton’s (3.4) and the infinite-dimensional examples (3.6) and (3.7) are the local versions of the general case where P=T*Q is the cotangent bundle (phase space) of a manifold Q (configuration space). If Q is an n-dimensional manifold, then T*Q is a 2n-Poisson manifold locally isomorphic to ℝ2n whose Poisson bracket is locally given by (3.3) and Hamilton’s equations are locally given by (3.4). If Q is an infinite-dimensional Banach manifold, then T*Q is a Poisson manifold locally isomorphic to V×V* whose Poisson bracket is given by (3.6) and Hamilton’s equations are locally given by (3.7).


## 3.2.4. Symplectic Manifolds

All the examples above are special cases of symplectic manifolds(P,ω). That means that P is equipped with a symplectic structure ω which is a closed (dω=0), (weakly) nondegenerate 2-form on the manifold P. Then for any H∈C∞(P) the corresponding Hamiltonian vector field XH is defined by dH=ω(XH,·) and the canonical Poisson bracket is given by(3.9){F,H}=ω(XF,XH),F,H∈C∞(P).
For example, on ℝ2n the canonical symplectic structure ω is given by ω=∑i=1ndpi∧dqi=dθ, where θ=∑i=1npi∧dqi. The same formula for ω holds locally in T*Q for any finite-dimensional Q (Darboux’s Lemma). For the infinite-dimensional example P=V×V*, the symplectic form ω is given by ω((φ1,π1),(φ2,π2))=〈φ1,π2〉-〈φ2,π1〉. Again these two formulas for ω are identical if V=ℝn.Remark 3.3.
(A) IfP is a finite-dimensional symplectic manifold, then P is even dimensional.
(B) If the Poisson bracket{·,·} is nondegenerate, then {·,·} comes from a symplectic form ω; that is, {·,·} is given by (3.9).


## 3.2.5. The Lie-Poisson Bracket

Not all Poisson brackets are of the form given in the above examples (3.3), (3.6), and (3.9); that is, not all Poisson manifolds are symplectic manifolds. An important class of Poisson bracket is the so-called Lie-Poisson bracket. It is defined on the dual of any Lie algebra. Let G be a Lie group with Lie algebra 𝔤=TeG≃{left invariant vector fields on G}, and let [·,·] denote the Lie bracket (commutator) on 𝔤. Let 𝔤* be the dual of a 𝔤 with respect to a pairing 〈·,·〉:𝔤*×𝔤→ℝ. Then for any F,H∈C∞(𝔤*) and μ∈𝔤*, the Lie-Poisson bracket is defined by(3.10){F,H}(μ)=±〈μ,[δFδμ,δHδμ]〉,
where δF/δμ,δH/δμ∈𝔤 are the "duals" of the gradients DF(μ),DH(μ)∈𝔤**≃𝔤 under the pairing 〈·,·〉. Note that the Lie-Poisson bracket is degenerate in general; for example, for G=SO(3) the vector space 𝔤* is 3 dimensional, so the Poisson bracket (3.10) cannot come from a symplectic structure. This Lie-Poisson bracket can also be obtained in a different way by taking the canonical Poisson bracket on T*G (locally given by (3.3) and (3.6)) and then restricting it to the fiber at the identity Te*G=𝔤*. In this sense the Lie-Poisson bracket (3.10) is induced from the canonical Poisson bracket on T*G. It is induced by the symmetry of left multiplication as we will discuss in Section 3.3.Example 3.4 (Rigid Body).
A concrete example of the Lie-Poisson bracket is given by the rigid body. HereG=SO(3) is the configuration space of a free rigid body. Identifying the Lie algebra (𝔰𝔬(3),[·,·]) with (ℝ3,×), where × is the vector product on ℝ3, and 𝔤*=𝔰𝔬(3)*≃ℝ3, the Lie-Poisson bracket translates into
(3.11){F,H}(m)=-m·(∇F×∇H).
For any F∈C∞(𝔰𝔬(3)*), we have (dF/dt)(m)=∇F·ṁ={F,H}(m)=-m·(∇F×∇H)=∇F·(m×∇H); hence ṁ=m×∇H. With the Hamiltonian H=(1/2)(m12/I12+m22/I22+m32/I32) we get Hamilton’s equation as
(3.12)ṁ1=I2-I3I2I3m2m3,ṁ2=I3-I1I3I1m3m1,ṁ3=I1-I2I1I2m1m2.
These are Euler’s equations for the free rigid body.


## 3.3. Reduction by Symmetries

The examples we have discussed so far are all canonical examples of Poisson brackets, defined either on a symplectic manifold(P,ω) or T*Q, or on the dual of a Lie algebra 𝔤*. Different, noncanonical Poisson brackets can arise from symmetries. Assume that a Lie group G is acting in a Hamiltonian way on the Poisson manifold (P,{·,·}). That means that we have a smooth map φ:G×P→P:φ(g,p)=g·p such that the induced maps φg=φ(g,·):P→P are canonical transformations, for each g∈G. In terms of Poisson manifolds, a canonical transformation is a smooth map that preserves the Poisson bracket. So the action of G on P is a Hamiltonian action if φg*{F,H}={φg*F,φg*H}, for all F,H∈C∞(P),g∈G. For any ξ∈𝔤 the canonical transformations φexp(tξ) generate a Hamiltonian vector field ξF on P and a momentum map J:P→𝔤* given by J(x)(ξ)=F(x), which is Ad* equivariant.If a Hamiltonian systemXH is invariant under a Lie group action, that is, H(φg(x))=H(x), then we obtain a reduced Hamiltonian system on a reduced phase space (reduced Poisson manifold). We recall the following Marsden-Weinstein reduction theorem [23].Theorem 3.5 (Reduction Theorem).
For a Hamiltonian action of a Lie groupG on a Poisson manifold (P,{·,·}), there is an equivariant momentum map J:P→𝔤* and for every regular μ∈𝔤* the reduced phase space Pμ≡J-1(μ)/Gμ carries an induced Poisson structure {·,·}μ (Gμ being the isotropy group). Any G-invariant Hamiltonian H on P defines a Hamiltonian Hμ on the reduced phase space Pμ, and the integral curves of the vector field XH project onto integral curves of the induced vector field X̂Hμ on the reduced space Pμ.Example 3.6 (Rigid Body).
The rigid body discussed above can be viewed as an example of this reduction theorem. IfP=T*G and G is acting on T*G by the cotangent lift of the left translation lg:G→G,lg(h)=gh, then the momentum map J:T*G→𝔤* is given by J(αg)=Te*Rg(αg) and the reduced phase space (T*G)μ=J-1(μ)/Gμ is isomorphic to the coadjoint orbit 𝒪μ through μ∈𝔤*. Each coadjoint orbit 𝒪μ carries a natural symplectic structure ωμ, and in this case, the reduced Lie-Poisson bracket {·,·}μ on the coadjoint orbit 𝒪μ is induced by the symplectic form ωμ on 𝒪μ as in (3.9). Furthermore T*G/G≃𝔤* and the induced Poisson bracket {·,·}μ on 𝒪μ are identical with the Lie-Poisson bracket restricted to the coadjoint orbit 𝒪μ⊂𝔤*. For the rigid body we apply this construction to G=SO(3).See [1, 8, 10, 17, 19–31].


## 4. Applications

We now discuss some infinite-dimensional examples of reduced Hamiltonian systems.


### 4.1. Maxwell’s Equations

Maxwell’s equations of electromagnetism are a reduced Hamiltonian system with the Lie group𝒢=(C∞(M),+) discussed in Section 2.3.3 as symmetry group.LetE,B be the electric and magnetic fields on ℝ3, then Maxwell’s equations for a charge density ρ are(4.1)Ė=curlB,Ḃ=-curlE,(4.2)divB=0,divE=ρ.
Let A be the magnetic potential such that B=-curlA. As configuration space we take V=Vec(ℝ3), vector fields (potentials) on ℝ3, so A∈V, and as phase space we have P=T*V≃V×V*∋(A,E), with the standard L2 pairing 〈A,E〉=∫A(x)E(x)dx, and canonical Poisson bracket given by (3.6), which becomes(4.3){F,H}(A,E)=∫(δFδAδHδE-δHδAδFδE)dx.
As Hamiltonian we take the total electromagnetic energy (4.4)H(A,E)=12∫(|curlA|2+|E|2)dx.Then Hamilton’s equations in the canonical variablesA and E are Ȧ=δH/δE=E⇒Ḃ=-curlE and Ė=-δH/δA=-curl curlA=curlB. So the first two equations of Maxwell’s equations (4.1) are Hamilton’s equations; we get the third one automatically from the potential divB=-div curlA=0 and we obtain the 4th equation divE=ρ through the following symmetry (gauge invariance). The Lie group 𝒢=(C∞(ℝ3),+) acts on V by φ·A=A+∇φ, φ∈𝒢,A∈V. The lifted action to V×V* becomes φ·(A,E)=(A+∇φ,E), and has the momentum map J:V×V*→𝔤*≃{chargedensities}:(4.5)J(A,E)=divE.
With 𝔤=C∞(ℝ3) and 𝔤*=Den(ℝ3), we identify elements of 𝔤* with charge densities. The Hamiltonian H is 𝒢 invariant; that is, H(φ·(A,E))=H(A+∇φ,E)=H(A,E). Then the reduced phase space for ρ∈𝔤* is (V×V*)ρ=J-1(ρ)/G={(E,B)∣divE=ρ,divB=0} and the reduced Hamiltonian is(4.6)Hρ(E,B)=12∫(|E|2+|B|2)dx.
The reduced Poisson bracket becomes for any functions F,H on (V×V*)ρ(4.7){F,H}ρ(E,B)=∫(δFδE·curlδHδB-δHδE·curlδFδB)dx,
and a straightforward computation shows that(4.8)Ḟ={F,Hρ}ρ⇔{Ė=curlB,Ḃ=-curlE,divB=0,divE=ρ.
So Maxwell’s equations (4.1), (4.2) are an infinite-dimensional Hamiltonian system on this reduced phase space with respect to the reduced Poisson bracket.


### 4.2. Fluid Dynamics

Euler’s equations for an incompressible fluid(4.9)∂u∂t+u·∇u=-∇p,divu=0
are equivalent to the equations of geodesics on Diffμ∞(M). See the study by Marsden et al. in [15] for details.


### 4.3. Plasma Physics

The Maxwell-Vlasov’s equations are a reduced Hamiltonian system on a more complicated reduced space. See the study by Marsden et al. in [32] for details.Maxwell-Vlasov’s equations for a plasma densityf(x,v,t) generating the electric and magnetic fields E and B are the following set of equations:(4.10)∂f∂t+v·∂f∂x+(E+v×B)∂f∂v=0,∂B∂t=-curlE,∂E∂t=curlB-Jf,Jf=currentdensity,divE=ρf,ρf=chargedensity,divB=0.
This coupled nonlinear system of evolution equations is an infinite-dimensional Hamiltonian system of the form Ḟ={F,H}ρf on the reduced phase space(4.11)ℳ𝒱=(T*Diffω∞(ℝ6)×T*V)/C∞(ℝ6)
(V being the same space as in the example of Maxwell’s equations) with respect to the following reduced Poisson bracket, which is induced via gauge symmetry from the canonical Poisson bracket on T*Diffω∞(ℝ6)×T*V:(4.12){F,G}ρf(f,E,B)=∫f{δFδf,δGδf}dxdv+∫(δFδE·curlδGδB-δGδE·curlδFδB)dxdv+∫(δFδE·∂f∂vδGδf-δGδE·∂f∂vδFδf)dxdv+∫fB·(∂∂vδFδf×∂∂vδGδf)dxdv,
and with Hamiltonian(4.13)H(f,E,B)=12∫v2f(x,v,t)dv+12∫(|E|2+|B|2)dx.More complicated plasma models are formulated as Hamiltonian systems. For example, for the two-fluid model the phase space is a coadjoint orbit of the semidirect product (⋉) of the group 𝒢=Diff∞(ℝ6)⋉(C∞(ℝ6)×C∞(ℝ6)). For the MHD model, 𝒢=Diff∞(ℝ6)⋉(C∞(ℝ6)×Ω2(ℝ3)).


### 4.4. The KdV Equation and Fourier Integral Operators

There are many known examples of PDEs which are infinite-dimensional Hamiltonian systems, such as the Benjamin-Ono, Boussinesq, Harry Dym, KdV, KP equations, and others. In many cases the Poisson structures and Hamiltonians are given ad hoc on a formal level. We illustrate this with the KdV equation, where at least one of the three known Hamiltonian structures is well understood [33].The Korteweg-deVries (KdV) equation(4.14)ut+6uux+uxxx=0
is an infinite-dimensional Hamiltonian system with the Lie group of invertible Fourier integral operators as symmetry group. Gardner found that with the bracket(4.15){F,G}=∫02πδFδu∂∂xδGδudx
and Hamiltonian(4.16)H(u)=∫02π(u3+12ux3)dxu satisfies the KdV equation (4.14) if and only if (4.17)u̇={u,H}.The question is where this Poisson bracket (4.15) and Hamiltonian (4.16) come from? We showed [33–35] that this bracket is the Lie-Poisson bracket on a coadjoint orbit of Lie group 𝒢=FIO of invertible Fourier integral operators on the circle S1. We briefly summarize the following.A Fourier integral operators on a compact manifoldM is an operator (4.18)A:C∞(M)→C∞(M)
locally given by(4.19)A(u)(x)=(2π)-n∬eiφ(x,y,ξ)a(x,ξ)u(y)dydξ,
where φ(x,y,ξ) is a phase function with certain properties and the symbol a(x,ξ) belongs to a certain symbol class. A pseudodifferential operator is a special kind of Fourier integral operators, locally of the form(4.20)P(u)(x)=(2π)-n∫∫ei(x-y)·ξp(x,ξ)u(y)dydξ.
Denote by FIO and ΨDO the groups under composition (operator product) of invertible Fourier integral operators and invertible pseudodifferential operators on M, respectively. We have the following results.Both groupsΨDO and FIO are smooth infinite-dimensional ILH-Lie groups. The smoothness properties of the group operations (operator multiplication and inversion) are similar to the case of diffeomorphism groups (2.6), (2.7). The Lie algebras of both ILH-Lie groups ΨDO and FIO are the Lie algebras of all pseudodifferential operators under the commutator bracket. Moreover, FIO is a smooth infinite-dimensional principal fiber bundle over the diffeomorphism group of canonical transformations Diffω∞(T*M-{0}) with structure group (gauge group) ΨDO.For the KdV equation we take the special case whereM=S1. Then the Gardner bracket (4.15) is the Lie-Poisson bracket on the coadjoint orbit of FIO through the Schrodinger operator P∈ΨDO. Complete integrability of the KdV equation follows from the infinite system of conserved integral in involution given by Hk=Trace(Pk); in particular the Hamiltonian (4.16) equals H=H2.See the study by Adams et al. in [34, 35] for details.See [10, 15, 31–40].


## 4.1. Maxwell’s Equations

Maxwell’s equations of electromagnetism are a reduced Hamiltonian system with the Lie group𝒢=(C∞(M),+) discussed in Section 2.3.3 as symmetry group.LetE,B be the electric and magnetic fields on ℝ3, then Maxwell’s equations for a charge density ρ are(4.1)Ė=curlB,Ḃ=-curlE,(4.2)divB=0,divE=ρ.
Let A be the magnetic potential such that B=-curlA. As configuration space we take V=Vec(ℝ3), vector fields (potentials) on ℝ3, so A∈V, and as phase space we have P=T*V≃V×V*∋(A,E), with the standard L2 pairing 〈A,E〉=∫A(x)E(x)dx, and canonical Poisson bracket given by (3.6), which becomes(4.3){F,H}(A,E)=∫(δFδAδHδE-δHδAδFδE)dx.
As Hamiltonian we take the total electromagnetic energy (4.4)H(A,E)=12∫(|curlA|2+|E|2)dx.Then Hamilton’s equations in the canonical variablesA and E are Ȧ=δH/δE=E⇒Ḃ=-curlE and Ė=-δH/δA=-curl curlA=curlB. So the first two equations of Maxwell’s equations (4.1) are Hamilton’s equations; we get the third one automatically from the potential divB=-div curlA=0 and we obtain the 4th equation divE=ρ through the following symmetry (gauge invariance). The Lie group 𝒢=(C∞(ℝ3),+) acts on V by φ·A=A+∇φ, φ∈𝒢,A∈V. The lifted action to V×V* becomes φ·(A,E)=(A+∇φ,E), and has the momentum map J:V×V*→𝔤*≃{chargedensities}:(4.5)J(A,E)=divE.
With 𝔤=C∞(ℝ3) and 𝔤*=Den(ℝ3), we identify elements of 𝔤* with charge densities. The Hamiltonian H is 𝒢 invariant; that is, H(φ·(A,E))=H(A+∇φ,E)=H(A,E). Then the reduced phase space for ρ∈𝔤* is (V×V*)ρ=J-1(ρ)/G={(E,B)∣divE=ρ,divB=0} and the reduced Hamiltonian is(4.6)Hρ(E,B)=12∫(|E|2+|B|2)dx.
The reduced Poisson bracket becomes for any functions F,H on (V×V*)ρ(4.7){F,H}ρ(E,B)=∫(δFδE·curlδHδB-δHδE·curlδFδB)dx,
and a straightforward computation shows that(4.8)Ḟ={F,Hρ}ρ⇔{Ė=curlB,Ḃ=-curlE,divB=0,divE=ρ.
So Maxwell’s equations (4.1), (4.2) are an infinite-dimensional Hamiltonian system on this reduced phase space with respect to the reduced Poisson bracket.


## 4.2. Fluid Dynamics

Euler’s equations for an incompressible fluid(4.9)∂u∂t+u·∇u=-∇p,divu=0
are equivalent to the equations of geodesics on Diffμ∞(M). See the study by Marsden et al. in [15] for details.


## 4.3. Plasma Physics

The Maxwell-Vlasov’s equations are a reduced Hamiltonian system on a more complicated reduced space. See the study by Marsden et al. in [32] for details.Maxwell-Vlasov’s equations for a plasma densityf(x,v,t) generating the electric and magnetic fields E and B are the following set of equations:(4.10)∂f∂t+v·∂f∂x+(E+v×B)∂f∂v=0,∂B∂t=-curlE,∂E∂t=curlB-Jf,Jf=currentdensity,divE=ρf,ρf=chargedensity,divB=0.
This coupled nonlinear system of evolution equations is an infinite-dimensional Hamiltonian system of the form Ḟ={F,H}ρf on the reduced phase space(4.11)ℳ𝒱=(T*Diffω∞(ℝ6)×T*V)/C∞(ℝ6)
(V being the same space as in the example of Maxwell’s equations) with respect to the following reduced Poisson bracket, which is induced via gauge symmetry from the canonical Poisson bracket on T*Diffω∞(ℝ6)×T*V:(4.12){F,G}ρf(f,E,B)=∫f{δFδf,δGδf}dxdv+∫(δFδE·curlδGδB-δGδE·curlδFδB)dxdv+∫(δFδE·∂f∂vδGδf-δGδE·∂f∂vδFδf)dxdv+∫fB·(∂∂vδFδf×∂∂vδGδf)dxdv,
and with Hamiltonian(4.13)H(f,E,B)=12∫v2f(x,v,t)dv+12∫(|E|2+|B|2)dx.More complicated plasma models are formulated as Hamiltonian systems. For example, for the two-fluid model the phase space is a coadjoint orbit of the semidirect product (⋉) of the group 𝒢=Diff∞(ℝ6)⋉(C∞(ℝ6)×C∞(ℝ6)). For the MHD model, 𝒢=Diff∞(ℝ6)⋉(C∞(ℝ6)×Ω2(ℝ3)).


## 4.4. The KdV Equation and Fourier Integral Operators

There are many known examples of PDEs which are infinite-dimensional Hamiltonian systems, such as the Benjamin-Ono, Boussinesq, Harry Dym, KdV, KP equations, and others. In many cases the Poisson structures and Hamiltonians are given ad hoc on a formal level. We illustrate this with the KdV equation, where at least one of the three known Hamiltonian structures is well understood [33].The Korteweg-deVries (KdV) equation(4.14)ut+6uux+uxxx=0
is an infinite-dimensional Hamiltonian system with the Lie group of invertible Fourier integral operators as symmetry group. Gardner found that with the bracket(4.15){F,G}=∫02πδFδu∂∂xδGδudx
and Hamiltonian(4.16)H(u)=∫02π(u3+12ux3)dxu satisfies the KdV equation (4.14) if and only if (4.17)u̇={u,H}.The question is where this Poisson bracket (4.15) and Hamiltonian (4.16) come from? We showed [33–35] that this bracket is the Lie-Poisson bracket on a coadjoint orbit of Lie group 𝒢=FIO of invertible Fourier integral operators on the circle S1. We briefly summarize the following.A Fourier integral operators on a compact manifoldM is an operator (4.18)A:C∞(M)→C∞(M)
locally given by(4.19)A(u)(x)=(2π)-n∬eiφ(x,y,ξ)a(x,ξ)u(y)dydξ,
where φ(x,y,ξ) is a phase function with certain properties and the symbol a(x,ξ) belongs to a certain symbol class. A pseudodifferential operator is a special kind of Fourier integral operators, locally of the form(4.20)P(u)(x)=(2π)-n∫∫ei(x-y)·ξp(x,ξ)u(y)dydξ.
Denote by FIO and ΨDO the groups under composition (operator product) of invertible Fourier integral operators and invertible pseudodifferential operators on M, respectively. We have the following results.Both groupsΨDO and FIO are smooth infinite-dimensional ILH-Lie groups. The smoothness properties of the group operations (operator multiplication and inversion) are similar to the case of diffeomorphism groups (2.6), (2.7). The Lie algebras of both ILH-Lie groups ΨDO and FIO are the Lie algebras of all pseudodifferential operators under the commutator bracket. Moreover, FIO is a smooth infinite-dimensional principal fiber bundle over the diffeomorphism group of canonical transformations Diffω∞(T*M-{0}) with structure group (gauge group) ΨDO.For the KdV equation we take the special case whereM=S1. Then the Gardner bracket (4.15) is the Lie-Poisson bracket on the coadjoint orbit of FIO through the Schrodinger operator P∈ΨDO. Complete integrability of the KdV equation follows from the infinite system of conserved integral in involution given by Hk=Trace(Pk); in particular the Hamiltonian (4.16) equals H=H2.See the study by Adams et al. in [34, 35] for details.See [10, 15, 31–40].


## 5. Gauge Theories, the Standard Model, and Gravity

Here we will encounter various infinite-dimensional Lie groups and algebras such as diffeomorphism groups, loop groups, groups of gauge transformations, and their cohomologies.


### 5.1. Gauge Theories: Yang-Mills, QED, and QCD

Consider a principalG-bundle π:P→M, with M being a compact, orientable Riemannian manifold (e.g., M=S4,T4) and G a compact non-Abelian gauge group with Lie algebra g. Let 𝒜 be the infinite-dimensional affine space of connection 1-forms on P. So each A∈𝒜 is a g-valued, equivariant 1-form on P (also called vector potential) and defines the covariant derivative of any field φ by DAφ=dφ+(1/2)[A,φ]. The curvature 2-form FA (or field strength) is a g-valued 2-form and is defined as FA=DAA=dA+(1/2)[A,A]. They are locally given by A=Aμdxμ and F=(1/2)Fμνdxμ∧dxν, where Fμν=∂μAν-∂νAμ+[Aμ,Aν].In pure Yang-Mills theory the action functional is given by(5.1)S(A)=12∥FA∥2=12∫MTr(FμνFμν),
and the Yang-Mills equations become globally(5.2)d*FA=0.
With added fermionic field ψ interaction, the action becomes(5.3)S(A,ψ)=12∥FA∥2+〈∂Aψ,ψ〉,
where ψ is a section of the spin bundle 𝒮pin±(M) and, ∂A:𝒮pin±(M)→𝒮pin∓(M) is the induced Dirac operator.


#### 5.1.1. Gauge Invariance

In gauge theories the symmetry group is the group of gauge transformations. The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups, as discussed in Section2.4.4.The group𝒢 of gauge transformations of the principal G-bundle π:P→M is given by (5.4)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,πϕ(p)=π(p)}≅{τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}=Gau(P)
which is a smooth Hilbert-Lie group with smooth group operations [6].We only sketch here what role this infinite-dimensional gauge group𝒢 plays in these quantum field theories. A good reference for this topic is the study by Deligne et al. in [41, 42].The gauge group𝒢 acts on 𝒜 via pullback ϕ∈𝒢, A∈𝒜, ϕ·A=(ϕ-1)*A∈𝒜, or under the isomorphism (see Section 2.4.4) 𝒢≅Gau(P), ϕ⇔τ we have Gau(P) acting on 𝒜 by τ·A=τAτ-1+τdτ-1. Hence the covariant derivative transforms as Dτ·A=τDAτ-1, and the action on the field is τ·FA:=Fτ·A=τFAτ-1.The action functional (the Yang-Mills functional) isS(A)=∥FA∥2, locally given by ∥FA∥2=(1/2)∫MTr(FμνFμν). This action is gauge invariant S(ϕ·A)=S(A), ϕ∈𝒢, so the Yang-Mills functional is defined on the orbit space ℳ=𝒜/𝒢. The space ℳ is in general not a manifold since the action of 𝒢 on 𝒜 is not free. If we restrict to irreducible connections, then ℳ is a smooth infinite-dimensional manifold and 𝒜→ℳ is an infinite-dimensional principal fiber bundle with structure group 𝒢.For self-dual connectionsFA=*FA (instantons) on a compact 4-manifold, the moduli space ℳ={A∈𝒜;Aself-dual}/𝒢 is a smooth finite-dimensional manifold. Self-dual connections absolutely minimize the Yang-Mills action integral (5.5)YM(A)=∫Ω∥FA∥2,Ω⊂Mcompact.
The Feynman path integral quantizes the action and we get the probability amplitude (5.6)W(f)=∫𝒜/𝒢e-S(A)f(A)𝒟(A)
for any gauge-invariant functional f(A).Let𝒢 be the group of gauge transformations. So ϕ∈𝒢⇔ϕ:P→P is a diffeomorphism over idM; that is, ϕ(p·g)=ϕ(p)·g, p∈P, g∈G. Then 𝒢 acts on 𝒜 and 𝒮pin±(M) by ϕ·A=(ϕ-1)*A and ϕ·ψ=(ϕ-1)*ψ. The action functionals S are gauge invariant:(5.7)Yang-Mills:S(ϕ·A)=S(A),A∈𝒜,ϕ∈𝒢,(5.8)QED:S(ϕ·A,ϕ·ψ)=S(A,ψ),A∈𝒜,ψ∈𝒮pin±(M),ϕ∈𝒢.


#### 5.1.2. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)

In classical field theory, one considers a Lagrangianℒ(ϕi,∂μϕi) of the fields φi:ℝn→ℝ, i=1,…,k, and ∂μ=∂/∂xμ and the corresponding action functional S=∫ℒ(ϕi,∂μϕi)dnx. The variational principle δS=0 then leads to the Euler-Lagrange equations of motion(5.9)∂ℒ∂ϕi-∂μ∂ℒ∂(∂μϕi)=0.In QED and QCD the Lagrangian is more complicated of the form(5.10)ℒ(A,ψ,φ)=-14g2TrFμνFμν-iψ¯[γμ(∂μ+ieAμ)+m]ψ+(DAμφ)†(DAμφ)-m2φ†φ,
where Aμ(x) is a potential 1-form (boson), and the field strength F is given by Fμν=∂μAν-∂νAμ+[Aμ,Aν]. In QED the gauge group of the principal bundle is G=U(1), and in QCD we have G=SU(2). The Dirac γ-matrices are γi=(0-σiσi0), where σi are the Pauli matrices (canonical basis of 𝔰𝔲(2)) and ψ¯=ψ†γo is the Pauli adjoint with γo=(0110), m is the electron mass, e is the electron charge, and g is a coupling constant.


#### 5.1.3. The Equations of Motion

The variational principle of the Lagrangian (5.10) with respect to the fields A,ψ, and φ gives the corresponding Euler-Lagrange equations of motion. They describe, for instance, the motion of an electron ψ(x) (fermion, spinor) in an electromagnetic field F, interacting with a bosonic field φ. We get, from the variational principle, δS/δAμ=0⇒∂μFμν=eψ¯γνψ, which are Maxwell’s equations for G=U(1).In the free case, that is, whenψ=0, we get ∂μFμν=0, the vacuum Maxwell equations.ForG=SU(2) these equations become DμFμν=0, the Yang-Mills equations. Moreover, δS/δψ=0⇒i(∂A-m)ψ=0, which are Dirac’s equations, where ∂A=γμ(∂μ+ieAμ)=γμDAμ. In the free case, that is, when A=0, we get i(∂A-m)ψ=0, the classical Dirac equation.


#### 5.1.4. Chiral Symmetry

The chiral symmetry is the symmetry that leads to anomalies and the BRST invariance. In QCD the chiral symmetry of the Fermi fieldψ is given by ψ↦eiβγ5ψ, where β is a constant and γ5=iγoγ1γ2γ3. The classical Noether current of this symmetry is given by Jμ=ψ¯γμγ5ψ which is conserved; that is, ∂μJμ=0.This conservation law breaks down after quantization; one gets(5.11)∂μJμ=2imψ¯γ5ψ-g28π2TrFμνFμν≡ω≠0.
This value ω is called the chiral anomaly.


### 5.2. Quantization

The quantization is given by the Feynman path integral:(5.12)∫𝒜/𝒢∫×𝒮pineiS(A,ψ)ℱ(A,ψ)𝒟A𝒟ψ=〈ℱ(A,ψ)〉
which computes the expectation value 〈ℱ(A,ψ)〉 of the function ℱ(A,ψ). This is an integral over two infinite-dimensional spaces: the gauge orbit space 𝒜/𝒢 and the fermionic Berezin integral over the spin space 𝒮pin±(M). These integrals are mathematically not defined but physicists compute them by gauge fixing; that is, fixing a section σ:𝒜/𝒢→𝒜, (e.g., σ(A)=∂μAμ=0, the Lorentz gauge) and then integrating over the section σ. Such a section does not exist globally, but only locally (Gribov ambiguity!). The effect of such a gauge fixing is that one gets extra terms in the Lagrangian (gauge-fixing terms) and one has to introduce new fields, so-called ghost fields η via the Faddeev-Popov procedure. The such obtained effective Lagrangian is no longer gauge invariant. This effective Lagrangian has the form in QCD:(5.13)ℒeff(A,ψ,η)=12Tr(FμνFμν)kineticenergy+12αTr(∂μAμ)2gauge-fixingterm-g∂μη¯DAμηghostterm+⋯interactionterms.We can write this globally as(5.14)ℒeff=12∥FA∥2+12∥σ(A)∥2+η¯ℳη+⋯,
where ℳ=(δ/δϕ)(σ(ϕ·A)) is the Faddeev-Popov determinant, acting like the Jacobian of the global gauge variation δ/δϕ over the section σ. Writing this term in the exponent of the action functional like a “fermionic Gaussian integral” leads to the Faddeev-Popov ghost fields η,η¯ in the form detℳ=∫e-η¯ℳηdη¯dη.The effective Lagrangianℒeff is NOT gauge invariant but has a new symmetry, called BRST symmetry.


### 5.3. BRST Symmetry

Named after Becchi et al. [43] and Tyutin who discovered this invariance in 1975-76, the BRST operator 𝔰 is given as follows:(5.15)𝔰A=dη+[A,η]𝔰η=-12[η,η]ℒeffisBRSTinvariant.
Note that the BRST operator 𝔰 mixes bosons (A) and fermions (η). This is an example of supersymmetry which we will discuss in Section 6. Also, the BRST operator 𝔰 is nilpotent; that is, 𝔰2=0. The question arises whether this operator 𝔰 is the coboundary operator of some kind of cohomology. The affirmative answer is given by the following theorem (Schmid [6, 44]).Theorem 5.1.
Let𝒞q,p(𝔩𝔦𝔢𝒢,Ωloc) be the Chevalley-Eilenberg complex of the Lie algebra 𝔩𝔦𝔢𝒢 of infinitesimal gauge transformations, with respect to the induced adjoint representation on local forms Ωloc, with boundary operator
(5.16)δloc:𝒞q,p(𝔩𝔦𝔢𝒢,Ωloc)→𝒞q+1,p(𝔩𝔦𝔢𝒢,Ωloc),δloc2=0.
Then with 𝔰:=(-1)p+1/(q+1)δloc, one has 𝔰2=0 and the following.(1)
Forq=0,p=1,A∈𝒜⊂𝒞0,1, then 𝔰A=dη+[A,η].(2)
Forq=1,p=0,η∈𝒞1,0, then 𝔰η=-(1/2)[η,η], the Maurer-Cartan form.(3)
The chiral anomalyω (given by (5.11)) is represented as cohomology class of this complex [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).


#### 5.3.1. The Chevalley-Eilenberg Cohomology

We are now going to explain the previous theorem, in particular the general definition of the Chevalley-Eilenberg [45] complex and the corresponding cohomology.LetG be a Lie group with Lie algebra 𝔤 and let σ be a representation of 𝔤 on the vector space W. Denote by 𝒞q(𝔤,W) the space of W-valued q-cochains on 𝔤 and define the coboundary operator δ:𝒞q(𝔤,W)→𝒞q+1(𝔤,W) by(5.17)δΦ(ξo,…,ξq)=∑i=0q(-1)iσ(ξi)Φ(ξo,…,ξ̂i,…,ξq)+∑i<j(-1)i+jΦ(σ(ξi)ξj,…,ξ̂i,…,ξ̂j,…,ξq).
We have δ2=0, and define the Lie algebra cohomology of 𝔤 with respect to (σ,W) as ℋ*(𝔤,W)=kerδ/imδ. This is called the Chevalley-Eilenberg cohomology [45] of the Lie algebra 𝔤 with respect to the representation σ.


#### 5.3.2. Anomalies

The Noether current induced by the chiral symmetry (after quantization) for the free case (ψ=0), that is, for pure Yang-Mills becomes(5.18)∂μJμ=-g28π2ɛμνρτTrFμνFρτ=-14π2ɛμνρτTr∂μ(Aν∂ρAτ+23AμAρAτ)=ω≠0anomaly.
See (5.11).Note the similarity with the Chern-Simon Lagrangian(5.19)ℒ(A)=∫MTr(AdA+23A3).We are going to derive a representation of the chiral anomalyω in the BRST cohomology that is [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).The question is “if𝔰ω=0, does there exist a local functional F(A), such that ω=𝔰(F(A))? That is, is ω BRST 𝔰-exact? The answer in general is NO; that is, ω represents a nontrivial cohomology class. This class is given by the Chern-Weil homotopy.LetÃ=A+η∈𝒞0,1×𝒞1,0 and F̃≡sÃ+Ã2=FÃ. For t∈[0,1], let F̃t=tF̃+(t2-t)Ã2 and define the Chern-Simons form(5.20)ω2q-1≡q∫01Tr(ÃF̃tq-1)dt,
we get(5.21)𝔰ω2q-1=TrF̃q.
We write ω2q-1 as sum of homogeneous terms in ghost number (upper index) and degree (lower index) ω2q-1=ω2q-10+ω2q-21+ω2q-32+⋯+ω02q-1. Let ω(X,A)=∫Mω2q-21(X).Theorem 5.2 (see Schmid [46]).
The formω(X,A)=∫M∫01ÃF̃tq-1(X)dt satisfies the Wess-Zumino consistency condition (𝔰ω)(X0,X1,A)=0 and represents the chiral anomaly [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).We have an explicit form of the anomaly in(2q-2) dimensions:
(5.22)ω2q-21=q(q-1)∫01(1-t)Tr(ηδloc(ÃF̃tq-2))dt.
So for q=2 the non-Abelian anomaly in 2 dimensions becomes ω21=Tr(ηδlocÃ), and for q=3 the non-Abelian anomaly in 4 dimensions becomes(5.23)ω41=Tr(ηδloc(ÃδlocÃ+12Ã3)).


### 5.4. The Standard Model

The standard model is a Yang-Mills gauge theory. Recall that the freeYang-Mills equations are DA*F=0, where A is a connection 1-form (vector potential), and F is the associated curvature 2-form (field) on the principal bundle P. The connection A defines the covariant derivative DA and the curvature F given by F=DAA=dA+(1/2)[A,A], or locally Fμν=∂μAν-∂νAμ+[Aμ,Aν], and DμFμν=0. Again the connection A is the fundamental object.For different choices of the gauge Lie groupG, we obtain the 3 theories that make up the standard model. For G=U(1) on a trivial bundle (i.e., global symmetry, which gives charge conservation) the curvature 2-form F is simply the electromagnetic field, and the Yang-Mills equations DA*F=0 are Maxwell’s equations  dF=0, locally ∂μFμν=0. For G=U(1) as local gauge group we get the quantum mechanical symmetry and the equations of motion are Dirac’s equations. Combing the two, we get QED as a U(1) gauge theory. For G=SU(N) we get the full non-Abelian Yang-Mills equations DA*F=0. For weak interactions with G=SU(2) and combining the two (spontaneous symmetry breaking, Higgs), we get the Glashow-Weinberg-Salam model as SU(2)×U(1) Yang-Mills theory of electroweak interactions. For G=SU(3) we obtain the Yang-Mills equations DA*F=0 for strong interactions and the equations of motion for QCD. Finally that standard model is a SU(3)×SU(2)×U(1) gauge theory governed by the corresponding Yang-Mills equations DA*F=0. Recall that F is the curvature in the corresponding principal bundle determined by the connection A.For interactions, all the relevant fields involved can be considered as sections of corresponding associated vector bundles induced by representations of the gauge groups, for example, the Dirac operator on the associated spin bundle (induced by the spin representation ofSU(2)) acting on spinors (sections of this bundle). The vector potentials are the corresponding connection 1-forms and the Yang-Mills fields are the corresponding curvature 2-forms on these bundles over spacetime.Again we do not need the metric and the curvature is determined by the potential, so the potential is the fundamental object.


### 5.5. Gravity

#### 5.5.1. Stop Looking for Gravitons

Stop looking for the graviton, not because it had been found but because it does not exist. The graviton is supposed to be the particle that communicates the gravitational force. But the gravitational force is not a fundamental force.Gravity is geometry. One might as well search for the Corioliston for the coriolis force or the Centrifugiton for the centrifugal force.Since Einstein in the 1920s, physicists have tried to unify what are considered the four fundamental forces, namely, electromagnetism, weak and strong nuclear forces, and the gravitational force. In the 1970s, the three nongravitational forces were unified in the standard model. At high enough energy (about1015 GeV) they become the same force.Since then, with all the string theory, SUSY, branes, and extra dimensions, the gravitational force could not be incorporated into GUT that includes all 4 forces and no graviton has been found experimentally. The reason is simple: not many people, including Einstein himself, take/took the general theory of relativity seriously enough, according to which we know that the gravitational force does not exist as fundamental force but as geometry! We do not feel it. What we feel is the resistance of the solid ground on which we stand. In general relativity, free-falling objects follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics because of the mechanical resistance of matter. Newton’s apple falls downward because the spacetime in which we exist iscurved. The “gravitational force” is not a force but it is the geometry of spacetime as Einstein observed in [47, page 137]:“Die Koeffizienten(gμν)dieser Mertik beschreiben in Bezug auf das gewählte Koordinatensystem zugleich das Gravitationsfeld.”(“The coefficients(gμν) of this metric with respect to the chosen coordinate system describe at the same time the gravitational field”) [47, page 146]:“Aus pysikalischen Gründen bestand die Überzeugung, dass das metrische Feld zugleich das Gravitationsfeld sei.”(“For physical reasons there was the conviction that the metric field was at the same time the gravitational field”).Therefore GUT, the grand unified theory had been completed since the 1970s with the standard model. Since the gravitational force does not exist as a fundamental force, there is nothing more to unify as forces. If we want to unify all four theories, then it has to be done in a geometric way. The equations governing gravity as well as the standard model are allcurvature equations, Einstein’s equation, and the Yang-Mills equations.


#### 5.5.2. Einstein’s Vacuum Field Equations

Let(M,g) be spacetime with Lorentzian metric g. Then Einstein’s vacuum field equations are (5.24)Ric=0,
where Ric is the Ricci curvature of the Lorentz metric g. These are the Euler-Lagrange equations for the Lagrangian ℒ(g)=∫R(g)μ(g), where μ(g)=-detgd4x and R(g) is the scalar curvature of g.Or in general, locally, in terms of the stress-energy tensorTμν, Einstein’s equations are Gμν=κTμν with the Einstein tensor Gμν=Rμν-(1/2)gμνR. The stress-energy tensor Tμν is the conserved Noether current corresponding to spacetime translation invariance.The Levi-Civita connectionΓ of the Riemannian metric g is given by (5.25)Γμνλ=12gλσ(∂gσμ∂xν+∂gσν∂xμ-∂gμν∂xσ).The curvature tensorR and the Ricci curvature Ric in Einstein’s field equations are completely determined by the connection Γ.First the curvature tensorR is locally given by (5.26)Rμνκλ=(∂Γμνλ∂xκ-∂Γμκλ∂xν)+(ΓμνηΓκηλ-ΓμκηΓνηλ).
Taking its trace, we get the Rici tensor Ric as (Ric)μν=Rμλνλ.So we can express Einstein’s equations completely in terms of the connection (potential)Γ; we do not need the metric g; also the curvature R is determined by the potential Γ. So the potential Γ is the fundamental object.The free motion in spacetime is along geodesic curvesγ(t) which again are expressed in terms of the connection by (5.27)γ̈α+Γβναγ̇βγ̇ν=0.


#### 5.5.3. Symmetry

In general relativity the diffeomorphism group plays the role of a symmetry group of coordinate transformations. Then the vacuum Einstein’s field equationsRic(g)=0 are invariant under coordinate transformations, that is, under the action of Diff∞(M). Denote by ℳ the space of all metrics g on M. Then, Einstein’s field equations Ric(g)=0 are a Hamiltonian system on the reduced space P=M/Diff∞(M); see the study by Marsden et al. in [15] for details.


### 5.6. Conclusions

The relation between the connectionΓ occurring in Einstein’s equations and the connection A in Yang-Mills equations is as follows. A is a Lie algebra-valued 1-form on a principal G-bundle (P,π,M) over spacetime (M,g) (or any associated vector bundle given by representations of G). The Levi-Civita connection Γ is a connection 1-form in this sense on the tangent bundle TM (frame bundle) with G=GL(n). So in this sense  general relativity and the standard model are Yang-Mills gauge theories.Therefore all four theories, electromagnetism, weak interaction, strong interaction, and gravity, are unified as  curvature equations in vector bundles over spacetime. Different interactions require different bundles.There is no hierarchy problem because there is no fundamental gravitational force. The question why gravitational interaction is so much weaker than electroweak and strong interactions is meaningless, comparing apples with oranges. Why are so many physicists still talking about gravitational force? It is like as if we are still talking about “sun rise” and “sun set”, 500 years after Copernicus! only worse; these are serious scientists trying to unify all four “forces” to a TOE.I am not saying that there are no open problems in physics. Of course there is still the problem of unifying quantum mechanics and general relativity on a geometric level (not as forces). The question is “how does spacetime look at the Planck scale? Do we have to modify spacetime to incorporate quantum mechanics or quantum mechanics to accommodate spacetime, or both? We need a theory of quantum gravity. There are several theories in the developing stage that promise to accomplish this.(i)
Superstring theory by E. Witten et al.(ii)
Discrete spacetime at Planck length by R. Loll in "Causal dynamical triangulation" and by J. Ambjorn, J. Jurkiewicz, and R. Loll in "The Universe from Scratch" [arXiv: hep-th/0509010].(iii)
Spacetime quantization: loop quantum gravity by L. Smolin in “Three Roads to Quantum Gravity” (London: Weidenfeld and Nicholson, 2000) and by S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin in “Quantum Gravity and the Standard Model”, preprint 2006.(iv)
Geometric formulation of quantum mechanics by A. Ashtekar and T. A. Schilling [arXiv: gr-qc/9706069].(v)
Deterministic quantum mechanics at Planck scale by G. t’Hooft in “Quantum Gravity as a Dissipative Deterministic System.”(vi)
Branes and new dimensions: parallel universes by L. Randell et al., D. Deutsch, PS. In the brane world, gravity is again singled out as the only force not confined to one brane.(vii)
Noncommutative Geometry. A. Connes describes the standard model form general relativity.(viii)
The most recent new development is by Verlinde [48]. He agrees that gravity is not a fundamental force, but explains it as an emergent force (entropic force) caused by a change in the amount of information (entropy) associated with the positions of bodies of matter.See [6, 11, 27, 36, 41–59].


## 5.1. Gauge Theories: Yang-Mills, QED, and QCD

Consider a principalG-bundle π:P→M, with M being a compact, orientable Riemannian manifold (e.g., M=S4,T4) and G a compact non-Abelian gauge group with Lie algebra g. Let 𝒜 be the infinite-dimensional affine space of connection 1-forms on P. So each A∈𝒜 is a g-valued, equivariant 1-form on P (also called vector potential) and defines the covariant derivative of any field φ by DAφ=dφ+(1/2)[A,φ]. The curvature 2-form FA (or field strength) is a g-valued 2-form and is defined as FA=DAA=dA+(1/2)[A,A]. They are locally given by A=Aμdxμ and F=(1/2)Fμνdxμ∧dxν, where Fμν=∂μAν-∂νAμ+[Aμ,Aν].In pure Yang-Mills theory the action functional is given by(5.1)S(A)=12∥FA∥2=12∫MTr(FμνFμν),
and the Yang-Mills equations become globally(5.2)d*FA=0.
With added fermionic field ψ interaction, the action becomes(5.3)S(A,ψ)=12∥FA∥2+〈∂Aψ,ψ〉,
where ψ is a section of the spin bundle 𝒮pin±(M) and, ∂A:𝒮pin±(M)→𝒮pin∓(M) is the induced Dirac operator.


### 5.1.1. Gauge Invariance

In gauge theories the symmetry group is the group of gauge transformations. The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups, as discussed in Section2.4.4.The group𝒢 of gauge transformations of the principal G-bundle π:P→M is given by (5.4)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,πϕ(p)=π(p)}≅{τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}=Gau(P)
which is a smooth Hilbert-Lie group with smooth group operations [6].We only sketch here what role this infinite-dimensional gauge group𝒢 plays in these quantum field theories. A good reference for this topic is the study by Deligne et al. in [41, 42].The gauge group𝒢 acts on 𝒜 via pullback ϕ∈𝒢, A∈𝒜, ϕ·A=(ϕ-1)*A∈𝒜, or under the isomorphism (see Section 2.4.4) 𝒢≅Gau(P), ϕ⇔τ we have Gau(P) acting on 𝒜 by τ·A=τAτ-1+τdτ-1. Hence the covariant derivative transforms as Dτ·A=τDAτ-1, and the action on the field is τ·FA:=Fτ·A=τFAτ-1.The action functional (the Yang-Mills functional) isS(A)=∥FA∥2, locally given by ∥FA∥2=(1/2)∫MTr(FμνFμν). This action is gauge invariant S(ϕ·A)=S(A), ϕ∈𝒢, so the Yang-Mills functional is defined on the orbit space ℳ=𝒜/𝒢. The space ℳ is in general not a manifold since the action of 𝒢 on 𝒜 is not free. If we restrict to irreducible connections, then ℳ is a smooth infinite-dimensional manifold and 𝒜→ℳ is an infinite-dimensional principal fiber bundle with structure group 𝒢.For self-dual connectionsFA=*FA (instantons) on a compact 4-manifold, the moduli space ℳ={A∈𝒜;Aself-dual}/𝒢 is a smooth finite-dimensional manifold. Self-dual connections absolutely minimize the Yang-Mills action integral (5.5)YM(A)=∫Ω∥FA∥2,Ω⊂Mcompact.
The Feynman path integral quantizes the action and we get the probability amplitude (5.6)W(f)=∫𝒜/𝒢e-S(A)f(A)𝒟(A)
for any gauge-invariant functional f(A).Let𝒢 be the group of gauge transformations. So ϕ∈𝒢⇔ϕ:P→P is a diffeomorphism over idM; that is, ϕ(p·g)=ϕ(p)·g, p∈P, g∈G. Then 𝒢 acts on 𝒜 and 𝒮pin±(M) by ϕ·A=(ϕ-1)*A and ϕ·ψ=(ϕ-1)*ψ. The action functionals S are gauge invariant:(5.7)Yang-Mills:S(ϕ·A)=S(A),A∈𝒜,ϕ∈𝒢,(5.8)QED:S(ϕ·A,ϕ·ψ)=S(A,ψ),A∈𝒜,ψ∈𝒮pin±(M),ϕ∈𝒢.


### 5.1.2. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)

In classical field theory, one considers a Lagrangianℒ(ϕi,∂μϕi) of the fields φi:ℝn→ℝ, i=1,…,k, and ∂μ=∂/∂xμ and the corresponding action functional S=∫ℒ(ϕi,∂μϕi)dnx. The variational principle δS=0 then leads to the Euler-Lagrange equations of motion(5.9)∂ℒ∂ϕi-∂μ∂ℒ∂(∂μϕi)=0.In QED and QCD the Lagrangian is more complicated of the form(5.10)ℒ(A,ψ,φ)=-14g2TrFμνFμν-iψ¯[γμ(∂μ+ieAμ)+m]ψ+(DAμφ)†(DAμφ)-m2φ†φ,
where Aμ(x) is a potential 1-form (boson), and the field strength F is given by Fμν=∂μAν-∂νAμ+[Aμ,Aν]. In QED the gauge group of the principal bundle is G=U(1), and in QCD we have G=SU(2). The Dirac γ-matrices are γi=(0-σiσi0), where σi are the Pauli matrices (canonical basis of 𝔰𝔲(2)) and ψ¯=ψ†γo is the Pauli adjoint with γo=(0110), m is the electron mass, e is the electron charge, and g is a coupling constant.


### 5.1.3. The Equations of Motion

The variational principle of the Lagrangian (5.10) with respect to the fields A,ψ, and φ gives the corresponding Euler-Lagrange equations of motion. They describe, for instance, the motion of an electron ψ(x) (fermion, spinor) in an electromagnetic field F, interacting with a bosonic field φ. We get, from the variational principle, δS/δAμ=0⇒∂μFμν=eψ¯γνψ, which are Maxwell’s equations for G=U(1).In the free case, that is, whenψ=0, we get ∂μFμν=0, the vacuum Maxwell equations.ForG=SU(2) these equations become DμFμν=0, the Yang-Mills equations. Moreover, δS/δψ=0⇒i(∂A-m)ψ=0, which are Dirac’s equations, where ∂A=γμ(∂μ+ieAμ)=γμDAμ. In the free case, that is, when A=0, we get i(∂A-m)ψ=0, the classical Dirac equation.


### 5.1.4. Chiral Symmetry

The chiral symmetry is the symmetry that leads to anomalies and the BRST invariance. In QCD the chiral symmetry of the Fermi fieldψ is given by ψ↦eiβγ5ψ, where β is a constant and γ5=iγoγ1γ2γ3. The classical Noether current of this symmetry is given by Jμ=ψ¯γμγ5ψ which is conserved; that is, ∂μJμ=0.This conservation law breaks down after quantization; one gets(5.11)∂μJμ=2imψ¯γ5ψ-g28π2TrFμνFμν≡ω≠0.
This value ω is called the chiral anomaly.


## 5.1.1. Gauge Invariance

In gauge theories the symmetry group is the group of gauge transformations. The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups, as discussed in Section2.4.4.The group𝒢 of gauge transformations of the principal G-bundle π:P→M is given by (5.4)𝒢={ϕ∈Diff∞(P);ϕ(p·g)=ϕ(p)·g,πϕ(p)=π(p)}≅{τ∈C∞(P,G);τ(p·g)=g-1τ(p)g}=Gau(P)
which is a smooth Hilbert-Lie group with smooth group operations [6].We only sketch here what role this infinite-dimensional gauge group𝒢 plays in these quantum field theories. A good reference for this topic is the study by Deligne et al. in [41, 42].The gauge group𝒢 acts on 𝒜 via pullback ϕ∈𝒢, A∈𝒜, ϕ·A=(ϕ-1)*A∈𝒜, or under the isomorphism (see Section 2.4.4) 𝒢≅Gau(P), ϕ⇔τ we have Gau(P) acting on 𝒜 by τ·A=τAτ-1+τdτ-1. Hence the covariant derivative transforms as Dτ·A=τDAτ-1, and the action on the field is τ·FA:=Fτ·A=τFAτ-1.The action functional (the Yang-Mills functional) isS(A)=∥FA∥2, locally given by ∥FA∥2=(1/2)∫MTr(FμνFμν). This action is gauge invariant S(ϕ·A)=S(A), ϕ∈𝒢, so the Yang-Mills functional is defined on the orbit space ℳ=𝒜/𝒢. The space ℳ is in general not a manifold since the action of 𝒢 on 𝒜 is not free. If we restrict to irreducible connections, then ℳ is a smooth infinite-dimensional manifold and 𝒜→ℳ is an infinite-dimensional principal fiber bundle with structure group 𝒢.For self-dual connectionsFA=*FA (instantons) on a compact 4-manifold, the moduli space ℳ={A∈𝒜;Aself-dual}/𝒢 is a smooth finite-dimensional manifold. Self-dual connections absolutely minimize the Yang-Mills action integral (5.5)YM(A)=∫Ω∥FA∥2,Ω⊂Mcompact.
The Feynman path integral quantizes the action and we get the probability amplitude (5.6)W(f)=∫𝒜/𝒢e-S(A)f(A)𝒟(A)
for any gauge-invariant functional f(A).Let𝒢 be the group of gauge transformations. So ϕ∈𝒢⇔ϕ:P→P is a diffeomorphism over idM; that is, ϕ(p·g)=ϕ(p)·g, p∈P, g∈G. Then 𝒢 acts on 𝒜 and 𝒮pin±(M) by ϕ·A=(ϕ-1)*A and ϕ·ψ=(ϕ-1)*ψ. The action functionals S are gauge invariant:(5.7)Yang-Mills:S(ϕ·A)=S(A),A∈𝒜,ϕ∈𝒢,(5.8)QED:S(ϕ·A,ϕ·ψ)=S(A,ψ),A∈𝒜,ψ∈𝒮pin±(M),ϕ∈𝒢.


## 5.1.2. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)

In classical field theory, one considers a Lagrangianℒ(ϕi,∂μϕi) of the fields φi:ℝn→ℝ, i=1,…,k, and ∂μ=∂/∂xμ and the corresponding action functional S=∫ℒ(ϕi,∂μϕi)dnx. The variational principle δS=0 then leads to the Euler-Lagrange equations of motion(5.9)∂ℒ∂ϕi-∂μ∂ℒ∂(∂μϕi)=0.In QED and QCD the Lagrangian is more complicated of the form(5.10)ℒ(A,ψ,φ)=-14g2TrFμνFμν-iψ¯[γμ(∂μ+ieAμ)+m]ψ+(DAμφ)†(DAμφ)-m2φ†φ,
where Aμ(x) is a potential 1-form (boson), and the field strength F is given by Fμν=∂μAν-∂νAμ+[Aμ,Aν]. In QED the gauge group of the principal bundle is G=U(1), and in QCD we have G=SU(2). The Dirac γ-matrices are γi=(0-σiσi0), where σi are the Pauli matrices (canonical basis of 𝔰𝔲(2)) and ψ¯=ψ†γo is the Pauli adjoint with γo=(0110), m is the electron mass, e is the electron charge, and g is a coupling constant.


## 5.1.3. The Equations of Motion

The variational principle of the Lagrangian (5.10) with respect to the fields A,ψ, and φ gives the corresponding Euler-Lagrange equations of motion. They describe, for instance, the motion of an electron ψ(x) (fermion, spinor) in an electromagnetic field F, interacting with a bosonic field φ. We get, from the variational principle, δS/δAμ=0⇒∂μFμν=eψ¯γνψ, which are Maxwell’s equations for G=U(1).In the free case, that is, whenψ=0, we get ∂μFμν=0, the vacuum Maxwell equations.ForG=SU(2) these equations become DμFμν=0, the Yang-Mills equations. Moreover, δS/δψ=0⇒i(∂A-m)ψ=0, which are Dirac’s equations, where ∂A=γμ(∂μ+ieAμ)=γμDAμ. In the free case, that is, when A=0, we get i(∂A-m)ψ=0, the classical Dirac equation.


## 5.1.4. Chiral Symmetry

The chiral symmetry is the symmetry that leads to anomalies and the BRST invariance. In QCD the chiral symmetry of the Fermi fieldψ is given by ψ↦eiβγ5ψ, where β is a constant and γ5=iγoγ1γ2γ3. The classical Noether current of this symmetry is given by Jμ=ψ¯γμγ5ψ which is conserved; that is, ∂μJμ=0.This conservation law breaks down after quantization; one gets(5.11)∂μJμ=2imψ¯γ5ψ-g28π2TrFμνFμν≡ω≠0.
This value ω is called the chiral anomaly.


## 5.2. Quantization

The quantization is given by the Feynman path integral:(5.12)∫𝒜/𝒢∫×𝒮pineiS(A,ψ)ℱ(A,ψ)𝒟A𝒟ψ=〈ℱ(A,ψ)〉
which computes the expectation value 〈ℱ(A,ψ)〉 of the function ℱ(A,ψ). This is an integral over two infinite-dimensional spaces: the gauge orbit space 𝒜/𝒢 and the fermionic Berezin integral over the spin space 𝒮pin±(M). These integrals are mathematically not defined but physicists compute them by gauge fixing; that is, fixing a section σ:𝒜/𝒢→𝒜, (e.g., σ(A)=∂μAμ=0, the Lorentz gauge) and then integrating over the section σ. Such a section does not exist globally, but only locally (Gribov ambiguity!). The effect of such a gauge fixing is that one gets extra terms in the Lagrangian (gauge-fixing terms) and one has to introduce new fields, so-called ghost fields η via the Faddeev-Popov procedure. The such obtained effective Lagrangian is no longer gauge invariant. This effective Lagrangian has the form in QCD:(5.13)ℒeff(A,ψ,η)=12Tr(FμνFμν)kineticenergy+12αTr(∂μAμ)2gauge-fixingterm-g∂μη¯DAμηghostterm+⋯interactionterms.We can write this globally as(5.14)ℒeff=12∥FA∥2+12∥σ(A)∥2+η¯ℳη+⋯,
where ℳ=(δ/δϕ)(σ(ϕ·A)) is the Faddeev-Popov determinant, acting like the Jacobian of the global gauge variation δ/δϕ over the section σ. Writing this term in the exponent of the action functional like a “fermionic Gaussian integral” leads to the Faddeev-Popov ghost fields η,η¯ in the form detℳ=∫e-η¯ℳηdη¯dη.The effective Lagrangianℒeff is NOT gauge invariant but has a new symmetry, called BRST symmetry.


## 5.3. BRST Symmetry

Named after Becchi et al. [43] and Tyutin who discovered this invariance in 1975-76, the BRST operator 𝔰 is given as follows:(5.15)𝔰A=dη+[A,η]𝔰η=-12[η,η]ℒeffisBRSTinvariant.
Note that the BRST operator 𝔰 mixes bosons (A) and fermions (η). This is an example of supersymmetry which we will discuss in Section 6. Also, the BRST operator 𝔰 is nilpotent; that is, 𝔰2=0. The question arises whether this operator 𝔰 is the coboundary operator of some kind of cohomology. The affirmative answer is given by the following theorem (Schmid [6, 44]).Theorem 5.1.
Let𝒞q,p(𝔩𝔦𝔢𝒢,Ωloc) be the Chevalley-Eilenberg complex of the Lie algebra 𝔩𝔦𝔢𝒢 of infinitesimal gauge transformations, with respect to the induced adjoint representation on local forms Ωloc, with boundary operator
(5.16)δloc:𝒞q,p(𝔩𝔦𝔢𝒢,Ωloc)→𝒞q+1,p(𝔩𝔦𝔢𝒢,Ωloc),δloc2=0.
Then with 𝔰:=(-1)p+1/(q+1)δloc, one has 𝔰2=0 and the following.(1)
Forq=0,p=1,A∈𝒜⊂𝒞0,1, then 𝔰A=dη+[A,η].(2)
Forq=1,p=0,η∈𝒞1,0, then 𝔰η=-(1/2)[η,η], the Maurer-Cartan form.(3)
The chiral anomalyω (given by (5.11)) is represented as cohomology class of this complex [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).


### 5.3.1. The Chevalley-Eilenberg Cohomology

We are now going to explain the previous theorem, in particular the general definition of the Chevalley-Eilenberg [45] complex and the corresponding cohomology.LetG be a Lie group with Lie algebra 𝔤 and let σ be a representation of 𝔤 on the vector space W. Denote by 𝒞q(𝔤,W) the space of W-valued q-cochains on 𝔤 and define the coboundary operator δ:𝒞q(𝔤,W)→𝒞q+1(𝔤,W) by(5.17)δΦ(ξo,…,ξq)=∑i=0q(-1)iσ(ξi)Φ(ξo,…,ξ̂i,…,ξq)+∑i<j(-1)i+jΦ(σ(ξi)ξj,…,ξ̂i,…,ξ̂j,…,ξq).
We have δ2=0, and define the Lie algebra cohomology of 𝔤 with respect to (σ,W) as ℋ*(𝔤,W)=kerδ/imδ. This is called the Chevalley-Eilenberg cohomology [45] of the Lie algebra 𝔤 with respect to the representation σ.


### 5.3.2. Anomalies

The Noether current induced by the chiral symmetry (after quantization) for the free case (ψ=0), that is, for pure Yang-Mills becomes(5.18)∂μJμ=-g28π2ɛμνρτTrFμνFρτ=-14π2ɛμνρτTr∂μ(Aν∂ρAτ+23AμAρAτ)=ω≠0anomaly.
See (5.11).Note the similarity with the Chern-Simon Lagrangian(5.19)ℒ(A)=∫MTr(AdA+23A3).We are going to derive a representation of the chiral anomalyω in the BRST cohomology that is [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).The question is “if𝔰ω=0, does there exist a local functional F(A), such that ω=𝔰(F(A))? That is, is ω BRST 𝔰-exact? The answer in general is NO; that is, ω represents a nontrivial cohomology class. This class is given by the Chern-Weil homotopy.LetÃ=A+η∈𝒞0,1×𝒞1,0 and F̃≡sÃ+Ã2=FÃ. For t∈[0,1], let F̃t=tF̃+(t2-t)Ã2 and define the Chern-Simons form(5.20)ω2q-1≡q∫01Tr(ÃF̃tq-1)dt,
we get(5.21)𝔰ω2q-1=TrF̃q.
We write ω2q-1 as sum of homogeneous terms in ghost number (upper index) and degree (lower index) ω2q-1=ω2q-10+ω2q-21+ω2q-32+⋯+ω02q-1. Let ω(X,A)=∫Mω2q-21(X).Theorem 5.2 (see Schmid [46]).
The formω(X,A)=∫M∫01ÃF̃tq-1(X)dt satisfies the Wess-Zumino consistency condition (𝔰ω)(X0,X1,A)=0 and represents the chiral anomaly [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).We have an explicit form of the anomaly in(2q-2) dimensions:
(5.22)ω2q-21=q(q-1)∫01(1-t)Tr(ηδloc(ÃF̃tq-2))dt.
So for q=2 the non-Abelian anomaly in 2 dimensions becomes ω21=Tr(ηδlocÃ), and for q=3 the non-Abelian anomaly in 4 dimensions becomes(5.23)ω41=Tr(ηδloc(ÃδlocÃ+12Ã3)).


## 5.3.1. The Chevalley-Eilenberg Cohomology

We are now going to explain the previous theorem, in particular the general definition of the Chevalley-Eilenberg [45] complex and the corresponding cohomology.LetG be a Lie group with Lie algebra 𝔤 and let σ be a representation of 𝔤 on the vector space W. Denote by 𝒞q(𝔤,W) the space of W-valued q-cochains on 𝔤 and define the coboundary operator δ:𝒞q(𝔤,W)→𝒞q+1(𝔤,W) by(5.17)δΦ(ξo,…,ξq)=∑i=0q(-1)iσ(ξi)Φ(ξo,…,ξ̂i,…,ξq)+∑i<j(-1)i+jΦ(σ(ξi)ξj,…,ξ̂i,…,ξ̂j,…,ξq).
We have δ2=0, and define the Lie algebra cohomology of 𝔤 with respect to (σ,W) as ℋ*(𝔤,W)=kerδ/imδ. This is called the Chevalley-Eilenberg cohomology [45] of the Lie algebra 𝔤 with respect to the representation σ.


## 5.3.2. Anomalies

The Noether current induced by the chiral symmetry (after quantization) for the free case (ψ=0), that is, for pure Yang-Mills becomes(5.18)∂μJμ=-g28π2ɛμνρτTrFμνFρτ=-14π2ɛμνρτTr∂μ(Aν∂ρAτ+23AμAρAτ)=ω≠0anomaly.
See (5.11).Note the similarity with the Chern-Simon Lagrangian(5.19)ℒ(A)=∫MTr(AdA+23A3).We are going to derive a representation of the chiral anomalyω in the BRST cohomology that is [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).The question is “if𝔰ω=0, does there exist a local functional F(A), such that ω=𝔰(F(A))? That is, is ω BRST 𝔰-exact? The answer in general is NO; that is, ω represents a nontrivial cohomology class. This class is given by the Chern-Weil homotopy.LetÃ=A+η∈𝒞0,1×𝒞1,0 and F̃≡sÃ+Ã2=FÃ. For t∈[0,1], let F̃t=tF̃+(t2-t)Ã2 and define the Chern-Simons form(5.20)ω2q-1≡q∫01Tr(ÃF̃tq-1)dt,
we get(5.21)𝔰ω2q-1=TrF̃q.
We write ω2q-1 as sum of homogeneous terms in ghost number (upper index) and degree (lower index) ω2q-1=ω2q-10+ω2q-21+ω2q-32+⋯+ω02q-1. Let ω(X,A)=∫Mω2q-21(X).Theorem 5.2 (see Schmid [46]).
The formω(X,A)=∫M∫01ÃF̃tq-1(X)dt satisfies the Wess-Zumino consistency condition (𝔰ω)(X0,X1,A)=0 and represents the chiral anomaly [ω]∈ℋBRST1,0(𝔩𝔦𝔢𝒢,Ωloc).We have an explicit form of the anomaly in(2q-2) dimensions:
(5.22)ω2q-21=q(q-1)∫01(1-t)Tr(ηδloc(ÃF̃tq-2))dt.
So for q=2 the non-Abelian anomaly in 2 dimensions becomes ω21=Tr(ηδlocÃ), and for q=3 the non-Abelian anomaly in 4 dimensions becomes(5.23)ω41=Tr(ηδloc(ÃδlocÃ+12Ã3)).


## 5.4. The Standard Model

The standard model is a Yang-Mills gauge theory. Recall that the freeYang-Mills equations are DA*F=0, where A is a connection 1-form (vector potential), and F is the associated curvature 2-form (field) on the principal bundle P. The connection A defines the covariant derivative DA and the curvature F given by F=DAA=dA+(1/2)[A,A], or locally Fμν=∂μAν-∂νAμ+[Aμ,Aν], and DμFμν=0. Again the connection A is the fundamental object.For different choices of the gauge Lie groupG, we obtain the 3 theories that make up the standard model. For G=U(1) on a trivial bundle (i.e., global symmetry, which gives charge conservation) the curvature 2-form F is simply the electromagnetic field, and the Yang-Mills equations DA*F=0 are Maxwell’s equations  dF=0, locally ∂μFμν=0. For G=U(1) as local gauge group we get the quantum mechanical symmetry and the equations of motion are Dirac’s equations. Combing the two, we get QED as a U(1) gauge theory. For G=SU(N) we get the full non-Abelian Yang-Mills equations DA*F=0. For weak interactions with G=SU(2) and combining the two (spontaneous symmetry breaking, Higgs), we get the Glashow-Weinberg-Salam model as SU(2)×U(1) Yang-Mills theory of electroweak interactions. For G=SU(3) we obtain the Yang-Mills equations DA*F=0 for strong interactions and the equations of motion for QCD. Finally that standard model is a SU(3)×SU(2)×U(1) gauge theory governed by the corresponding Yang-Mills equations DA*F=0. Recall that F is the curvature in the corresponding principal bundle determined by the connection A.For interactions, all the relevant fields involved can be considered as sections of corresponding associated vector bundles induced by representations of the gauge groups, for example, the Dirac operator on the associated spin bundle (induced by the spin representation ofSU(2)) acting on spinors (sections of this bundle). The vector potentials are the corresponding connection 1-forms and the Yang-Mills fields are the corresponding curvature 2-forms on these bundles over spacetime.Again we do not need the metric and the curvature is determined by the potential, so the potential is the fundamental object.


## 5.5. Gravity

### 5.5.1. Stop Looking for Gravitons

Stop looking for the graviton, not because it had been found but because it does not exist. The graviton is supposed to be the particle that communicates the gravitational force. But the gravitational force is not a fundamental force.Gravity is geometry. One might as well search for the Corioliston for the coriolis force or the Centrifugiton for the centrifugal force.Since Einstein in the 1920s, physicists have tried to unify what are considered the four fundamental forces, namely, electromagnetism, weak and strong nuclear forces, and the gravitational force. In the 1970s, the three nongravitational forces were unified in the standard model. At high enough energy (about1015 GeV) they become the same force.Since then, with all the string theory, SUSY, branes, and extra dimensions, the gravitational force could not be incorporated into GUT that includes all 4 forces and no graviton has been found experimentally. The reason is simple: not many people, including Einstein himself, take/took the general theory of relativity seriously enough, according to which we know that the gravitational force does not exist as fundamental force but as geometry! We do not feel it. What we feel is the resistance of the solid ground on which we stand. In general relativity, free-falling objects follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics because of the mechanical resistance of matter. Newton’s apple falls downward because the spacetime in which we exist iscurved. The “gravitational force” is not a force but it is the geometry of spacetime as Einstein observed in [47, page 137]:“Die Koeffizienten(gμν)dieser Mertik beschreiben in Bezug auf das gewählte Koordinatensystem zugleich das Gravitationsfeld.”(“The coefficients(gμν) of this metric with respect to the chosen coordinate system describe at the same time the gravitational field”) [47, page 146]:“Aus pysikalischen Gründen bestand die Überzeugung, dass das metrische Feld zugleich das Gravitationsfeld sei.”(“For physical reasons there was the conviction that the metric field was at the same time the gravitational field”).Therefore GUT, the grand unified theory had been completed since the 1970s with the standard model. Since the gravitational force does not exist as a fundamental force, there is nothing more to unify as forces. If we want to unify all four theories, then it has to be done in a geometric way. The equations governing gravity as well as the standard model are allcurvature equations, Einstein’s equation, and the Yang-Mills equations.


### 5.5.2. Einstein’s Vacuum Field Equations

Let(M,g) be spacetime with Lorentzian metric g. Then Einstein’s vacuum field equations are (5.24)Ric=0,
where Ric is the Ricci curvature of the Lorentz metric g. These are the Euler-Lagrange equations for the Lagrangian ℒ(g)=∫R(g)μ(g), where μ(g)=-detgd4x and R(g) is the scalar curvature of g.Or in general, locally, in terms of the stress-energy tensorTμν, Einstein’s equations are Gμν=κTμν with the Einstein tensor Gμν=Rμν-(1/2)gμνR. The stress-energy tensor Tμν is the conserved Noether current corresponding to spacetime translation invariance.The Levi-Civita connectionΓ of the Riemannian metric g is given by (5.25)Γμνλ=12gλσ(∂gσμ∂xν+∂gσν∂xμ-∂gμν∂xσ).The curvature tensorR and the Ricci curvature Ric in Einstein’s field equations are completely determined by the connection Γ.First the curvature tensorR is locally given by (5.26)Rμνκλ=(∂Γμνλ∂xκ-∂Γμκλ∂xν)+(ΓμνηΓκηλ-ΓμκηΓνηλ).
Taking its trace, we get the Rici tensor Ric as (Ric)μν=Rμλνλ.So we can express Einstein’s equations completely in terms of the connection (potential)Γ; we do not need the metric g; also the curvature R is determined by the potential Γ. So the potential Γ is the fundamental object.The free motion in spacetime is along geodesic curvesγ(t) which again are expressed in terms of the connection by (5.27)γ̈α+Γβναγ̇βγ̇ν=0.


### 5.5.3. Symmetry

In general relativity the diffeomorphism group plays the role of a symmetry group of coordinate transformations. Then the vacuum Einstein’s field equationsRic(g)=0 are invariant under coordinate transformations, that is, under the action of Diff∞(M). Denote by ℳ the space of all metrics g on M. Then, Einstein’s field equations Ric(g)=0 are a Hamiltonian system on the reduced space P=M/Diff∞(M); see the study by Marsden et al. in [15] for details.


## 5.5.1. Stop Looking for Gravitons

Stop looking for the graviton, not because it had been found but because it does not exist. The graviton is supposed to be the particle that communicates the gravitational force. But the gravitational force is not a fundamental force.Gravity is geometry. One might as well search for the Corioliston for the coriolis force or the Centrifugiton for the centrifugal force.Since Einstein in the 1920s, physicists have tried to unify what are considered the four fundamental forces, namely, electromagnetism, weak and strong nuclear forces, and the gravitational force. In the 1970s, the three nongravitational forces were unified in the standard model. At high enough energy (about1015 GeV) they become the same force.Since then, with all the string theory, SUSY, branes, and extra dimensions, the gravitational force could not be incorporated into GUT that includes all 4 forces and no graviton has been found experimentally. The reason is simple: not many people, including Einstein himself, take/took the general theory of relativity seriously enough, according to which we know that the gravitational force does not exist as fundamental force but as geometry! We do not feel it. What we feel is the resistance of the solid ground on which we stand. In general relativity, free-falling objects follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics because of the mechanical resistance of matter. Newton’s apple falls downward because the spacetime in which we exist iscurved. The “gravitational force” is not a force but it is the geometry of spacetime as Einstein observed in [47, page 137]:“Die Koeffizienten(gμν)dieser Mertik beschreiben in Bezug auf das gewählte Koordinatensystem zugleich das Gravitationsfeld.”(“The coefficients(gμν) of this metric with respect to the chosen coordinate system describe at the same time the gravitational field”) [47, page 146]:“Aus pysikalischen Gründen bestand die Überzeugung, dass das metrische Feld zugleich das Gravitationsfeld sei.”(“For physical reasons there was the conviction that the metric field was at the same time the gravitational field”).Therefore GUT, the grand unified theory had been completed since the 1970s with the standard model. Since the gravitational force does not exist as a fundamental force, there is nothing more to unify as forces. If we want to unify all four theories, then it has to be done in a geometric way. The equations governing gravity as well as the standard model are allcurvature equations, Einstein’s equation, and the Yang-Mills equations.


## 5.5.2. Einstein’s Vacuum Field Equations

Let(M,g) be spacetime with Lorentzian metric g. Then Einstein’s vacuum field equations are (5.24)Ric=0,
where Ric is the Ricci curvature of the Lorentz metric g. These are the Euler-Lagrange equations for the Lagrangian ℒ(g)=∫R(g)μ(g), where μ(g)=-detgd4x and R(g) is the scalar curvature of g.Or in general, locally, in terms of the stress-energy tensorTμν, Einstein’s equations are Gμν=κTμν with the Einstein tensor Gμν=Rμν-(1/2)gμνR. The stress-energy tensor Tμν is the conserved Noether current corresponding to spacetime translation invariance.The Levi-Civita connectionΓ of the Riemannian metric g is given by (5.25)Γμνλ=12gλσ(∂gσμ∂xν+∂gσν∂xμ-∂gμν∂xσ).The curvature tensorR and the Ricci curvature Ric in Einstein’s field equations are completely determined by the connection Γ.First the curvature tensorR is locally given by (5.26)Rμνκλ=(∂Γμνλ∂xκ-∂Γμκλ∂xν)+(ΓμνηΓκηλ-ΓμκηΓνηλ).
Taking its trace, we get the Rici tensor Ric as (Ric)μν=Rμλνλ.So we can express Einstein’s equations completely in terms of the connection (potential)Γ; we do not need the metric g; also the curvature R is determined by the potential Γ. So the potential Γ is the fundamental object.The free motion in spacetime is along geodesic curvesγ(t) which again are expressed in terms of the connection by (5.27)γ̈α+Γβναγ̇βγ̇ν=0.


## 5.5.3. Symmetry

In general relativity the diffeomorphism group plays the role of a symmetry group of coordinate transformations. Then the vacuum Einstein’s field equationsRic(g)=0 are invariant under coordinate transformations, that is, under the action of Diff∞(M). Denote by ℳ the space of all metrics g on M. Then, Einstein’s field equations Ric(g)=0 are a Hamiltonian system on the reduced space P=M/Diff∞(M); see the study by Marsden et al. in [15] for details.


## 5.6. Conclusions

The relation between the connectionΓ occurring in Einstein’s equations and the connection A in Yang-Mills equations is as follows. A is a Lie algebra-valued 1-form on a principal G-bundle (P,π,M) over spacetime (M,g) (or any associated vector bundle given by representations of G). The Levi-Civita connection Γ is a connection 1-form in this sense on the tangent bundle TM (frame bundle) with G=GL(n). So in this sense  general relativity and the standard model are Yang-Mills gauge theories.Therefore all four theories, electromagnetism, weak interaction, strong interaction, and gravity, are unified as  curvature equations in vector bundles over spacetime. Different interactions require different bundles.There is no hierarchy problem because there is no fundamental gravitational force. The question why gravitational interaction is so much weaker than electroweak and strong interactions is meaningless, comparing apples with oranges. Why are so many physicists still talking about gravitational force? It is like as if we are still talking about “sun rise” and “sun set”, 500 years after Copernicus! only worse; these are serious scientists trying to unify all four “forces” to a TOE.I am not saying that there are no open problems in physics. Of course there is still the problem of unifying quantum mechanics and general relativity on a geometric level (not as forces). The question is “how does spacetime look at the Planck scale? Do we have to modify spacetime to incorporate quantum mechanics or quantum mechanics to accommodate spacetime, or both? We need a theory of quantum gravity. There are several theories in the developing stage that promise to accomplish this.(i)
Superstring theory by E. Witten et al.(ii)
Discrete spacetime at Planck length by R. Loll in "Causal dynamical triangulation" and by J. Ambjorn, J. Jurkiewicz, and R. Loll in "The Universe from Scratch" [arXiv: hep-th/0509010].(iii)
Spacetime quantization: loop quantum gravity by L. Smolin in “Three Roads to Quantum Gravity” (London: Weidenfeld and Nicholson, 2000) and by S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin in “Quantum Gravity and the Standard Model”, preprint 2006.(iv)
Geometric formulation of quantum mechanics by A. Ashtekar and T. A. Schilling [arXiv: gr-qc/9706069].(v)
Deterministic quantum mechanics at Planck scale by G. t’Hooft in “Quantum Gravity as a Dissipative Deterministic System.”(vi)
Branes and new dimensions: parallel universes by L. Randell et al., D. Deutsch, PS. In the brane world, gravity is again singled out as the only force not confined to one brane.(vii)
Noncommutative Geometry. A. Connes describes the standard model form general relativity.(viii)
The most recent new development is by Verlinde [48]. He agrees that gravity is not a fundamental force, but explains it as an emergent force (entropic force) caused by a change in the amount of information (entropy) associated with the positions of bodies of matter.See [6, 11, 27, 36, 41–59].


## 6. SUSY (Supersymmetry)

Supersymmetry (SUSY) is an important idea in quantum filed theory and string theories. The BRST symmetry we described in Section5.3 is an example of SUSY. Now we give a summary of a mathematical description of super Hamiltonian systems on supersymplectic supermanifolds, state a generalization of the Marsden-Weinstein reduction theorem in this context, and illustrate the method with examples. This is a very technical topic, so we only give a brief sketch; for details see the studies Glimm in [60] and Tuynman in [61].The classical Marsden-Weinstein reduction theorem is a geometrical result stating that if a Lie groupG acts on a symplectic manifold P by symplectomorphisms and admits an equivariant momentum map J:P→𝔤*, then, for any regular value μ∈𝔤 of J, the quotient Pμ=J-1(μ)/Gμ of the preimage J-1(μ) by the isotropy group Gμ of μ has a natural symplectic structure. A dynamical interpretation of the Marsden-Weinstein reduction theorem gives the following. If a given Hamiltonian H∈C∞(P) is invariant under the action of the group G, then it projects to a reduced Hamiltonian Hμ on the reduced space Pμ. The integral curves of the Hamiltonian vector field XH project to the integral curves of XHμ. In this sense, one has reduced the system by symmetries. This reduction procedure unifies many methods and results concerning the use of symmetries in classical mechanics, some dating back to the time of Euler and Lagrange.In [60], Glimm generalizes this result to the setting of supermanifolds, using the analytic construction of supercalculus and supergeometry due to DeWitt [62] and Tuynman [61]. The general idea of supermanifolds and superanalysis is to do geometry and analysis over a graded algebra of even supernumbers rather than ℝ. In “supermathematics,” we have even and odd variables. Two variables a,b are called even, or commuting, or bosonic if a·b=b·a, whereas ξ,χ are called odd, or anticommuting, or fermionic if ξ·χ=-χ·ξ. The problem is doing analysis with even and odd variables.Many classes of differential equations have extensions that involve odd variables. These are calledsuperized versions, or supersymmetric extensions. An active area of research is in particular the construction of supersymmetric integrable systems.As an example, consider the Korteweg-de Vries equationut=-uxxx+6uux. A possible supersymmetric extension is the following system of an even variable u(t,x) and an odd variable ξ(t,x):(6.1)ut=-uxxx+6uux-3ξξxx,ξt=-ξxxx+3uξx+3ξux.There is a different way of writing system (6.1). For this, one considers the so-called 2|1-dimensional superspace. This is a space with coordinates (x,t,ϑ), where x and t are even numbers as before and ϑ is an odd number. We can now gather the components u(x,t) and ξ(x,t) into a superfieldΦ(t,x,ϑ) defined on superspace via(6.2)Φ(t,x,ϑ)=ξ(t,x)+ϑ·u(t,x).
The function Φ(t,x,ϑ) takes as values odd numbers; it is thus called an odd function. Note how the right-hand side can be seen as a Taylor series in ϑ; indeed, all higher powers of ϑ are zero because ϑ is anticommuting. One defines the odd differential operator D=∂/∂ϑ+ϑ·(∂/∂x), acting on superfields. A computation yields that system (6.1) is equivalent to(6.3)Φt=-D6Φ+3D2ΦDΦ.
(Note that D2=∂x.) System (6.1) is called the component formulation; (6.3) is called the superspace formulation. In our example at hand, a justification for calling the system an “extension” of KdV would be that if one takes (6.3) and writes it in component form (6.1), one recovers the “usual” KdV by setting ξ to zero. Also, it can be shown that (6.3) is invariant under transformations Φ(t,x,ϑ)↦Φ(t,x-ηϑ,ϑ+η), where η is an odd parameter. The infinitesimal version of this transformation is δΦ=η(∂ϑ-ϑ∂x)Φ. This transformation is called a “supersymmetry’’ in the present context. In components, it reads(6.4)δu=ηξx,δξ=ηu.
These equations illustrate a characterization of supersymmetries: supersymmetries (as opposed to regular symmetries) “mix” even and odd variables.One needs some concept of supermanifold even if one only works with the component formulation. For example, one has implicitly in system (6.1) the space of all (u(t,x),ξ(t,x)) on which the equations are defined; this is some kind of superspace itself. Also, there is some kind of supersubmanifold of those (u,ξ) which solve the equations.In [60], Glimm proves a comprehensive result on supersymplectic reduction. He uses an analytic-geometric approach to the theory of supermanifolds, and not the Kostant theory of graded manifolds [49]. The Poisson bracket induced by odd supersymplectic forms is not a super Lie bracket on the space of supersmooth functions. This stands of course in contrast to both the usual ungraded case and the super case with even supersymplectic forms. While this makes the algebraic approach conceptually more difficult, no such problems arise in the analytic approach. Also, we do not require that the action be free and proper, but have the weaker requirement that the quotient space only has a manifold structure.There are different approaches to supermanifolds. TheAlgebraic approach (Kostant [49], Berezin-Leĭtes, late 1970s) takes “superfunctions” as fundamental object. A graded manifold is a pair (M,A), where M is a conventional manifold and U↦A(U) is the following sheaf over M:(6.5)A(U)≃C∞(U)⊗⋀ℝn,U⊆M.
So a superfunction f∈A(U) can be written as (6.6)f=f∅+∑if{i}ϑi+∑i<jf{i,j}ϑiϑj+⋯.TheGeometric approach (DeWitt [62]) takes points as fundamental objects. Let 𝒜 be the ring generated by L generators θ1,θ2,…,θL (Grassmann numbers) with relations θiθj=-θjθi.An element is written as(6.7)a=a∅︸=Ba(“Body")+∑ia{i}θi+∑i<ja{i,j}θiθj+⋯︸nilpotent part(aω∈ℝ),a∈𝒜0⇔a=a∅+∑a{i,j}θiθj+termswitheven#ofθs,a∈𝒜1⇔a=∑a{i}θi+∑a{i,j,k}θiθjθk+termswithodd#ofθs.For calculus we “replace” realsℝ by Grassmann numbers 𝒜. A DeWitt supermanifold is a topological space M which is locally superdiffeomorphic to 𝒜m∣n=def(𝒜m⊕n)0=𝒜0m×𝒜1n.The two approaches are equivalent. There is a one-to-one correspondence between isomorphism classes ofm|n-dimensional DeWitt supermanifolds whose body is a fixed smooth m-dimensional manifold X and isomorphism classes of m|n-dimensional graded manifolds over X.The DeWitt topology of𝒜m∣n is defined as follows.U⊆𝒜m∣n is open if and only if B(U) is open in B𝒜m⊕n=ℝm,U=B-1(B(U)).Smooth functions𝒜m∣n→𝒜 are defined as follows. Let U⊆𝒜m∣n be an open set. A function f:U→𝒜 is called smooth if there is a collection of smooth real functions defined on BU⊆ℝm:(6.8)fi1…in:BU→ℝ,fori1,…,in=0,1
such that (6.9)f(x1,…,xm,ξ1,…,ξn)=∑i1,…,in=01ξ1i1…ξnin·Zfi1…in(x1,…,xm),
where Zg is defined as follows. If x∈𝒜m∣0 has the decomposition x=Bx+n, then(6.10)Zg(x)=∑k=1∞1k!Dkg(Bx)[n,…,n].Note.
Smooth functions map the body to the body!Superdifferential Geometry
Versions of the inverse function theorem and implicit function theorem are still valid. Concepts of tangent space, vector fields, flows, and Lie groups can be developed as in the ungraded case.Lie Supergroups
A Lie supergroup is a supermanifoldG that is a group and for which the group operations of multiplication and inversion are smooth. If a Lie supergroup G acts freely and properly on a supermanifold M, then the quotient M/G can be given the structure of a supermanifold such that the projection π:M→M/G is a surjective submersion. The structure of T(M/G) is given by the following.Theorem 6.1.
Letφ:M×G→M be an action. Suppose that M/G has a supermanifold structure such that π:M→M/G is a surjective submersion; that is, Tπ(p) is onto for every p∈M. For any p∈M, one has
(6.11)kerTπ(p)={𝔛M(p)∣𝔛∈𝔤}
and this is a proper subspace of TpM. In particular,
(6.12)T[p](M/G)≃TpM/{𝔛M(p)∣𝔛∈𝔤}.Remark 6.2.
There are examples whereM/G does not have a supermanifold structure and {𝔛M(p)∣𝔛∈𝔤} fails to be a free submodule.Supersymplectic Structures
A supersymplectic supermanifold(M,ω) is a supermanifold M together with a closed (i.e., dω=0) nondegenerate homogeneous left 2-form ω∈ΩL2(M).Examples 6.3.
(1) 𝒜2m∣n with coordinates (qi,pi,ξj),
(6.13)ω=∑dqi∧dpi+12∑dξj∧dξj
defines an even supersymplectic form.
(2) On 𝒜m∣m with coordinates (xi,ξj),
(6.14)ω=∑dxi∧dξi
defines an odd supersymplectic form.Letω∈ΩL2(M) be a 2-form on the supermanifold M and p∈M. Then ω(p)∈AltL2(TpM) is nondegenerate if and only if the real 2-form Bω(Bp)=ω(Bp)∣B(TBpM)∈Alt2(B(TBpM)) is nondegenerate.Hamiltonian Supermechanics
A smooth vector fieldX∈𝔛(M) is called (globally) Hamiltonian if there is some function H∈C∞(M,𝒜) such that iXω=dH. For f,g∈C∞(M,𝒜) define the Super-Poisson bracket by
(6.15){f,g}=〈Xf,Xg∣ω〉∈C∞(M,𝒜).Fact 6.
Ifω is even, then C∞(M,𝒜) is a Lie superalgebra with respect to {·,·}. This is false if ω is odd.Momentum Maps
Letφ:M×G→M be an action of the Lie supergroup G on the supersymplectic manifold (M,ω) which preserves ω. Recall that, in the ungraded case, a momentum map is an ℝ-linear map Ĵ:𝔤→C∞(M) such that XĴ(𝔛)=𝔛M.Superversion.Ĵ∈C∞(𝔤×M,𝒜) is a momentum map for the action of G on M if Ĵ is leftlinear in the first argument and (6.16)〈(0,v)∣dĴ(𝔛,x)〉=〈v,𝔛M(x)∣ω(x)〉
for all x∈M, for all v∈TxM. Instead of Ĵ∈C∞(𝔤×M,𝒜), one can consider J:M→𝔤* defined through 〈𝔛∣J(x)〉=Ĵ(𝔛,x).Theorem 6.4.
LetH∈C∞(M,𝒜) be a Hamiltonian with vector field XH such that [XH,XH]=0. If H is G-invariant, then J is preserved by the flow ϕ of XH; that is,
(6.17)J∘ϕt,τ=J∀(t,τ)∈𝒜1∣1(wheredefined).Suppose that the momentum map isAd*-equivariant. Let μ∈B𝔤p(ω)* be a regular value of J. Let Gμ be the isotropy group of μ. Suppose that the quotient space Pμ=J-1(μ)/Gμ can be given a supermanifold structure such that the projection πμ:J-1(μ)→Pμ is a surjective submersion.Superreduction Theorem
The supermanifoldPμ has a unique supersymplectic form ωμ∈ΩL2(Pμ) with the property
(6.18)πμ*ωμ=ıμ*ω,
where πμ:J-1(μ)→Pμ is the projection and ıμ:J-1(μ)→M is the inclusion. The form ωμ has the same parity as ω.Example 6.5 (The Bose-Fermi Oscillator).
We consider the phase spaceP=𝒜2∣2 with supersymplectic form ω=dp∧dq+(1/2)(dξ1∧dξ1+dξ2∧δξ2). The Hamiltonian H(q,p,ξ1,ξ2)=(1/2)(p2+q2)-ξ1ξ2 defines the Hamiltonian vector field XH=p∂q-q∂p-ξ2∂ξ1+ξ1∂ξ2.The SUSY algebra of the Bose-Fermi oscillator is given as follows. The Lie supergroup of invertible2|2-supermatrices acts on 𝒜2∣2 via (6.19)𝒜2∣2×GL(2|2)→𝒜2∣2,(q,G)↦qSTG.The algebra of Bose-Fermi supersymmetry is the intersection of the Lie superalgebras of the stabilizer ofω and the stabilizer of H:(6.20)𝔟𝔣(2|2)=𝔰𝔱𝔞𝔟(H)∩𝔰𝔱𝔞𝔟(ω).The SUSY algebra of the Bose-Fermi oscillator𝔟𝔣(2|2) is generated by A1,A2,C1,C2, where (6.21)Ai=(00eiTγ000-eiTei-γ0ei0)fori=1,2,C1=12(-γ000γ0),C2=12(γ000γ0),γ0=(0-110).Momentum Map and Quotient
The action ofBF(2|2) on 𝒜2∣2 admits a momentum map
(6.22)Ĵ:𝔟𝔣(2|2)×𝒜2∣2→𝒜,(X,q)↦12qSTXΩq.
In components, we write X=∑i=1,2aiAi+ciCi. Then
(6.23)Ĵ(X,q,p,ξ)=(c2-c1)14(p2+q2)-(c2+c1)14ξTγ0ξ-aT(p𝕀2-qγ0)ξ,
where a=(a1,a2)T.Now letμ∈B(𝔟𝔣(2|2)0*)≃ℝ2. The isotropy group of μ is the whole group BF(2|2). The action of BF(2|2) on J-1(μ) is transitive; that is, the quotient space J-1(μ)/BF(2|2) is a single point.Example 6.6 (Wess-Zumino Model in2+1-Dimensional Spacetime).
Let𝒮(𝒜) be the Schwartz space of functions ℝ2→𝒜. Consider the phase space 𝒮(𝒜)2∣2=[𝒮(𝒜)0]×[𝒮(𝒜)0]×[𝒮(𝒜)1]2∋(ϕ,π,ψ) with supersymplectic form ω=dπ∧dϕ+(1/2)dψ∧dψ and Hamiltonian
(6.24)ℋ(ϕ,π,ψ)=12∫ℝ2(∥∇ϕ∥2+π2+ψ̅𝔤idiψ)d2x,
where 𝔤1=(0110), 𝔤2=(100-1), and ψ̅=ψTγ0.Then Hamilton’s equations are the following:(6.25)ϕ̇=δℋδπ=π,π̇=-δℋδϕ=Δϕ,ψ̇=δℋδψ=𝔤0𝔤i∂iψ.
This is equivalent to (∂02-∂12-∂22)ϕ=0, 𝔤μ∂μψ=0 which are the massless Klein-Gordon and Dirac equations in 2+1-dimensional spacetime.Remark 6.7.
The well-known SUSY algebra from the Lagrangian description can be “exported” to the Hamiltonian setup.We reduce by an Abelian subgroup of the SUSY group:(6.26)𝒮(𝒜)2∣2×𝒜2∣0→𝒮(𝒜)2∣2,(Φ,r)↦[Sr(Φ)](x)=∑k1k!DkΦ(x+Br)[n,…,n],
where r=Br+n, is a “superspatial shift”. The momentum map becomes(6.27)J(Φ)=(∫(π∂iΦ+12ψT∂iψ))i=1,2.We determine thereduced phase space. The center of mass of Φ=(ϕ,π,ψ)∈𝒮(𝒜)2∣2 is 𝒞(Φ)=(1/M(Φ))∫ℝ2x·|Φ|2d2x∈𝒜2∣0, where M(Φ)=∫|Φ|2d2x∈𝒜0. We may identify the quotient space J-1(μ)/𝒜2∣0 with the subset(6.28)Pμ={Φ̃∈J-1(μ)∣𝒞(Φ̃)=0}⊆𝒮(𝒜)2∣2.Finally thereduced equations become(6.29)ϕ̇=π+Fi(ϕ,π,ψ)∂iϕ,π̇=Δϕ+Fi(ϕ,π,ψ)∂iπ,ψ̇=𝔤0𝔤i∂iψ+Fi(ϕ,π,ψ)∂iψ,
where Fi(Φ)=(1/M(Φ))∫xi·(ϕπ+πΔϕ-ψT𝔤j∂jψ)d2x.See [41, 42, 49, 60–63].



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