Field Theory Seminar

March 10, 2010

James T. Wheeler: Relativistic Hamiltonian field theory

Lecture: Relativistic Hamiltonian field theory

Hamiltonian mechanics

Starting from the action we find the Hamiltonian formulation of mechanics by first finding the conjugate momenta. The conjugate momenta are the partial derivatives of the Lagrangian with respect to the velocities. The Hamiltonian is then H = pv - L. Writing the Hamiltonian H as a function of momentum and position variables only, we vary these independently to arrive at Hamilton’s equations.

We may recast this formulation in terms of differential forms. Multiplying by dt the expression for the Hamiltonian may be written as the “Hilbert integral”, L dt = p dx – H dt. Taking the exterior derivative of this 1-form, we see that it factors, with each factor being one of Hamilton’s equation. Therefore, the 1-form is closed when Hamilton’s equations are satisfied.

Hamiltonian field theory: c

The converse to this also holds. Using the converse to the Poincaré lemma, when the Hilbert integral is closed, it is exact. Substituting this exact form into the action shows that the action is constant, hence extremal. Therefore, Hamilton’s equations must be satisfied.

Hamiltonian field theory: d

There is considerable power in the use of differential forms, so we would like to extend their applicability to field theory. With this in mind, we carry out the same steps for field theory that we just did for mechanics. We would like the formulation to be fully relativistic.

The Lagrangian, the Hamiltonian, and the conjugate momenta now become densities, to be integrated over all space. The action becomes a full spacetime integral. A Klein-Gordon field provides a simple example.

Hamiltonian field theory: e

Hamiltonian field theory: f

Now try to implement the conversion to differential forms. The densities lead us to write 4-forms. As a result, two (related) problems become evident.

The first problem is that we do not know which particular 3-form to take for the conjugate momentum , and, correspondingly, we do not know how to define the conjugate momentum without choosing a particular time coordinate.

We might solve this first problem by extending the conjugate momentum to a vector-valued variable, and taking its dual to get a 3-form. The 4-vector replaces the conjugate momentum.

Hamiltonian field theory: g

This works to give us a relativistic 4-form. Now the second problem is evident: the exterior derivative, which we hoped would provide the field equations, vanishes identically.

To solve this problem we extend the manifold. One approach [Dirac, De Donder, Rovelli] is to recall that the field may be thought of as a fiber on a bundle. Considering the 4-form to be on this 5-dim bundle solves the problem. The differential of the field now possesses an additional degree of freedom.

Hamiltonian field theory: h

To complete the picture, we need a relativistic Hamiltonian. Since any non-vanishing Hamiltonian in the usual sense implies a choice of time, we take the super-Hamiltonian to vanish. The super-Hamiltonian is built from the DeDonder Hamiltonian by adding a new degree of freedom. This degree of freedom is immediately removed when we set the super-Hamiltonian to zero.

The remaining piece, the DeDonder Hamiltonian, is built from the momentum 4-vector and the field in a way that gives us the result we seek. We now have a 4-form in a 5-dim space.

Hamiltonian field theory: i

Now demand the vanishing of the exterior derivative of the 4-form. The unknown fields are assumed to have derivatives in the fiber direction as well as the spacetime directions. Since these derivative s are independent, we get two sets of equations, which determine the momentum 4-vector and combine to give the correct field equation.

Hamiltonian field theory: j

Now we present an alternative approach. Let the whole structure take place in a relativistic, 1-particle phase space. This space may be taken as the result of the biconformal gauging of the conformal group, which yields an 8-dim symplectic manifold, or we may introduce it by fiat. In either case, it is an 8-dim manifold, so the exterior derivative of the Hamiltonian 4-form does not automatically vanish. We expand the exterior derivative of the field in all 8 coordinates.

Hamiltonian field theory: k

When we demand the vanishing of the exterior derivative of the Hamiltonian 4-form, we get two equations. The first arises from a single term quadratic in the momentum differentials. It requires the momentum derivatives of the field and the momentum vector to be parallel, but with a four arbitrary functions giving the proportionality. Substituting this into the remaining equation, two terms become independent, giving the same equations as the previous method.

Hamiltonian field theory: l