Topology in Quantum Field Theory
The effects of Topology in Quantum Field Theory occur due to the nontrivial global structure of the configuration space of the field theory. This can be seen through either a conical approach to quantum field theory or the path integral approach. Other techniques of quantization should yield the same results but will not be considered here.
A few examples, chiral anomalies, theta states, symmetry breaking, and non uniqueness of a vacuum can be described by the effects of topology. These effects are present and easily seen in the case of quantum mechanics.
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Classical Phase Space
A classical phase space is symplectic manifold \((P, \omega )\) where \(P \) is a manifold and \(\omega \) a symplectic form.
Definition A symplectic form is a 2-form on \(P \) which obeys two conditions
i) It is non-degenerate which means \(\omega (X,Y)=0\) for all X means Y=0
ii) It is closed which means \(d\omega =0\).
A classical hamiltonian system is a symplectic manifold \((P, \omega )\) with Hamiltonian \(H\) which is a function \(H: P\rightarrow {\mathbb R }\).
If the Hamiltonian H is differentiable a vector field \(X_H \) satisfying \(dH(Y)=\omega(X_H,Y) \) exists and is the unique. \(X_H \) is called the Hamiltonian vector field. Moreover, the integral curves of \(X_H \) are solutions to Hamilton's equations.
The classical algebra of observables are real functions \(f: P\rightarrow {\mathbb R }\), one can take observables to be smooth functions on P denoted by \({C^{\infty } (P) }\). The algebra structure is given the Poisson bracket \(\{\ , \}\) is related to the symplectic form via the expression\[\{f ,g \}\equiv \omega (X_f,X_g)\] where f and g are two observables, and \(X_f=\nabla f \) and \(X_g=\nabla g \). The algebra of classical of observables is \({C^{\infty } (P) }\) with the algebra structure given by \(\{\ , \}\). All of the standard algebra structure of the Poisson bracket comes from the properties of the symplectic form. The form is a coordinate independent representation of the canonical structure and the canonical transformations are symplectic diffeomorphims, namely, diffeomorphisms such that \(\phi ^*\omega=\omega\).
Although, \((P, \omega )\) can be globally very complicated the locally the structure is relatively simple due to a theorem of Darboux which allows one to find local coordinates \((q_i,p_i)\) about each point such that \(\omega = \sum ^{n}_{i=1} dq_i\wedge dp_i \). Thus, locally the Poisson brackets have the usual form. In fact, the coordinates from the application of Darboux's Theorem are just the usual canonical coordinates, however, the global behaviour will totally different than that given by the standard Poisson bracket in canonical coordinates due to the global structure of \(P \) and \(\omega \). In finite dimensions, the phase space or a symplectic manifold is even dimensional. Thus one should think of the Poisson bracket as globally being defined by \(\{f ,g \}=\omega (X_f,X_g)\) when working on non-trivial phase spaces.
For most systems arising from standard physical systems, the phase space is the cotangent bundle of the configuration space of the system. Usually, one considers generalized coordinates and conjugate momentua. These form a set of coordinates for the classical phase space. The poisson bracket associated with the generalized coordinates and conjugate moment are local representations of the symplectic form. Given a physically system and its configuration \({\cal Q }\) the phase space \(P \) is the cotangent bundle \(T^*{\cal Q }\) of the configuration space.
A simple example is \({\cal Q }={\mathbb R } \) then \(P=T^*{\mathbb R }={\mathbb R }^{2}\). Pick \(\omega = dq\wedge dp \), and \(H=\frac{p^2}{2m}+V(q)\). In this case, the Hamiltonian vector field is \(X_H=-\frac{\partial H}{\partial q}\frac{\partial }{\partial p}+\frac{\partial H}{\partial p}\frac{\partial }{\partial q}\). The integral curve of this vector field is the curve \( (q(t),p(t))\) satisfying \( \dot q=\frac{\partial H}{\partial p}\) and \( \dot p=-\frac{\partial H}{\partial q}\). Illustrating the fact that the integral curves of the Hamiltonian vector field are solutions to Hamilton's equations. Moreover, \(\{f ,g \}=\omega (X_f,X_g)\) implies the usual Poisson bracket for a particle on \({\mathbb R } \). One can extend this to \(P={\mathbb R }^{2n}\) by using \(\omega = \sum ^{n}_{i=1} dq_i\wedge dp_i \).
In general for physical systems, the configuration space \({\cal Q }\) will not be \({\mathbb R }^{n}\) but a general manifold. Now, many systems can treated with \(P= T^*{\cal Q }\), however, some systems the phase space may be a space other than the cotangent bundle of the configuration space.
Many systems can treated with \(P= T^*{\cal Q }\), however, for some systems the phase space may be a space other than the cotangent bundle of the configuration space. For example, one can have constraints on the system which means one will consider \(P\subset T^*{\cal Q }\).
This is the case of Witten's approach to 2+1 gravity. The reason for this is the constraints are imposed classically before quantization. Thus, the phase space will only be a subset of the unconstrained phase space.
Everything mentioned above was considered to be finite dimensional system, however, all of the ideas can be generalized to infinite dimensions spaces to include the idea of infinite dimensional symplectic manifolds and the classical phase spaces of field theories. In the infinite dimensional case instead of modelling the manifold as locally \({\mathbb R }^{n}\) one models the space locally on a nice function space such as a Hilbert space or Banach space.
Quantization
The quantization of a system with classical phase space \((P , \omega )\), and Hamiltonian \(H \) is thought of as algebra isomorphism. The Poisson algebra given by \(\{f ,g \}\equiv \omega (X_f,X_g)\) where f and g are two observables, and \(X_f=\nabla f \) and \(X_g=\nabla g \) is mapped in a quantum algebra. This is done by using ideas proposed by Dirac in 1925. A Hilbert space \({\cal H(P) }\) is picked and the quantum observables are an algebra of self-adjoint operators \({\cal O} (P)\) with commutator \([ \ ,\ ] \) the quantization will be a mapping \({\hat {}}:C^{\infty }(P)\rightarrow {\cal O}(P) \). More precisely,
Definition: A quantization is an algebra morphism which maps \( f\) linearly to \(\hat f \) and \(g\) linearly to \(\hat g \) such that \({[{\hat f} ,{\hat g} ] }=i\hbar{\widehat {\{f ,g \} }}\). Constant classical observables are mapped to multiplication operators and the algebra of classical observables with Poisson bracket \(\{\ ,\ \} \) is mapped to an algebra of quantum observables with commutator \([ \ ,\ ] \).
The final condition is mapping a set of classical observables to a complete set of quantum observables.
The quantum observables form a set of self adjoint operators on a Hilbert space \({\cal H} \). Technically, these operators must self-adjoint operators and so domains must be chosen carefully and it is not possible to pick a complete isomorphism so some physical input is need to pick out a sub-algebra of the classical algebra for which the correspondence holds.
One can not just consider operator on a Hilbert space based on \((P , \omega )\) one must pick a polarization which is a choice of a subspace of \((P , \omega )\) of half of the dimension. One knows this from the behaviour of standard physical systems such a quantum free particle where the wave function does not depend on both the position and momentum. Mathematically, this is related to irreducibility of the representations of the position and momentum operators of the Hilbert space. A standard polarization to pick if \(P=T^*{\cal Q }\) is pick just to \({\cal Q }\) and work with operators on \({\cal Q }\). Thus, in many physical interesting cases one can avoid the polarization choices by just working \({\cal Q }\).
The example of \(P={\mathbb R }^{2n}\) by using \(\omega = \sum ^{n}_{i=1} dq_i\wedge dp_i \) with Hamiltonian H. The canonical coordinates give the usual quantization of particle on \({\mathbb R }^{n}\) in Schroedinger representation. The quantization in this case is in fact unique because Stone-von Neumann Theorem[1].
In general, all symplectic manifolds admit the canonical coordinates locally so one can locally write a Schroedinger representation, however, the Stone-von Neumann Theorem [2] only applies to \(P={\mathbb R }^{2n}\) so the quantization will no longer be unique due the global structure of the Phase space just as the case of an infinite number of degrees of freedom.
If one considers the infinite dimensional case such a scalar field one can still use the mathematical formalism of the symplectic manifold the Fock representation as the analog of the Schroedinger representation, however, there are infinite number irreducible representations in the case of an infinite number of degrees of freedom.
Quantum field theory
Topological Effects
Gauge Theories
In the case of gauge theories \({\cal Q }={\cal A }/{\cal G } \) where \({\cal A }\) is the space of gauge potentials and \({\cal G }\) is the group of all gauge transformations. The quotient space comes from identifying connections which differ by a gauge transformation. The space \({\cal A }\) is a vector space so it has no non-trivial topology or structure, however, \({\cal Q }\) can have non-trivial topology due the identifications.
Gravity
In the case of gauge theories \({\cal Q }={\rm Riem }/{\rm Diff } \) where \({\rm Riem }\) is the space of Riemann metrics and \({\rm Diff }\) is the group of all diffeomorphisms. The quotient space comes from identifying metrics which differ by a diffeomorphism. The space \({\rm Riem }\) is not a vector space, however, it has trivial topology because it is a convex subspace of a vector space. The configure space \({\cal Q }\) can have non-trivial topology due the identifications.
