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Home , Canonical transformation, Hamiltonian vector field, Symplectic manifold

Aspects of in Physics Josh Powell

1 Symplectic Geometry In

We seek here to use geometry to gain a more solid understanding of physics. This process may seem frighteningly mathematical, but we shouldn’t expect much gain in insight with only a modicum of effort. Following Alexander Pope: “A little learning is a dangerous thing; Drink deep, or taste not the Pierian spring: There shallow draughts intoxicate the brain, And drinking largely sobers us again.”

1.1 Hasty Review of Classical Mechanics Consider a mechanical system with both position degrees of freedom, qi, and momentum degrees of freedom, i pi. The Hamiltonian H(pi, q ) is a function on that governs the dynamics of the system, and in particular of these degrees of freedom. That is, it describes time evolution by specifying their equations of motion. The equations of motion are

i ∂H ∂H q˙ = andp ˙i = − i ∂pi ∂q

We have a system of differential equations which are all first order in time derivatives. Since we know the i expression for the phase space variables’ time derivatives, any function f(pi, q ) of these variables will have a time derivative given by

df X ∂f ∂f  = q˙i + p˙ dt ∂qi ∂p i i i X ∂f ∂H ∂f ∂H  = − =: {f, H} ∂qi ∂p ∂p ∂qi i i i where in the last line we have defined the of two functions on phase space. The function i also been explicitly dependent on time, f = f(pi, q , t), then we would have written df ∂f = {f, H} + dt ∂t in order to capture the total time derivative. We have thus seen how the Hamiltonian gives the time evolution of all dynamical quantities in our system.

What if we had chosen another coordinate system on our phase space? Let’s call this new coordinate i i system (Pi,Q ) to distinguish from the previous (pi, q ). The dynamics in the old system are described by varying the Z X i  δ pidq − Hdt = 0

and those in the new come from varying Z X i 0  δ PidQ − H dt = 0

The same dynamics are described in terms of these two coordinate systems if these two integrands differ by i a total differential dF of some function F = F (pi, q , t). We write this as

X i X i 0 dF = pidq − PidQ + (H − H)dt

i i 0 whence we get ∂F/∂q = pi, ∂F/∂Q = −Pi and ∂F/∂t = H − H. We call the function F the generating function of this . If F has no explicity time dependence, it’s an element of F(Q×Q). If it is time dependent, then F ∈ F(Q × Q × R), and the two Hamiltonians differ in a way more than just

1 rewriting the old Hamiltonian in terms of the new coordinates.

There is another common formalism in which the dynamics of a system can be described. Imagine we had i i i i wanted to work not in terms of the variables (pi, q ) but instead (q , q˙ )? Since ∂H/∂pi =q ˙ , we recognize i that what this amounts to is a Legendre transform from pi toq ˙ . Let’s say that the Legendre transform of i i i the Hamiltonian H(pi, q ) is some function L(q , q˙ ). This transformation and its inverse are given by

i ˙i X i i i L(q , q ) = piq˙ − H(pi, q ) with pi = ∂L/∂q˙

i X i i i i H(pi, q ) = piq˙ − L(q , q˙ ) withq ˙ = ∂H/∂pi

i i i The first canonical equation of motion, ∂H/∂pi =q ˙ , is transformed into the tautological statementq ˙ =q ˙ . This comes as no surprise, since it was the equation which specified in the first place the transformation we i are making. The second canonical equation of motion, ∂H/∂q = −q˙i, is transformed into

∂L d ∂L − = 0 ∂qi dt ∂q˙i

which is known as the Lagrange equation of motion. Note that it is second order in time derivatives, unlike the dynamical equations in the Hamiltonian formalism which were both first order. We’ve obtained the Lagrangian formulation of classical mechanics. The two are equivalent descriptions, up to certain technical issues related to when the is not defined.

1.2 Symplectic Geometry Let M be a smooth of even dimensionality and let Ω be a closed, non-degenerate 2-form on M. We call the pair (M, Ω) a , and Ω its symplectic structure. The condition that Ω be non-degenerate means that the only vector field X on M satisfying Xy Ω = 0 is the one vanishing uniformly. A non-zero X satisfying this condition would correspond to a zero-mode of the symplectic structure, and would prevent Ω from being invertible. If such a zero-mode exists, it’s possible that it can be projected out and a lower dimensional system thus obtained is a symplectic manifold with non-degenerate symplectic structure. This procedure is known as reduction. One familiar example of this process comes from gauge theories, in which it is sometimes necessary to project out non-physical degrees of freedom. In this case, it’s known as gauge fixing.

In general, any diffeomorphism of M which preserves the symplectic structure is known as a sym- plectomorphism. That is, a is a diffeomorphism between two symplectic ∗ f :(M, ΩM ) → (N, ΩN ) for which f ΩN = ΩM . We can generate diffeomorphisms (M, Ω) → (M, Ω) by exponentiating a vector field on M. If this vector field, call it X, is to generate a symplectomorphism, then it must satisfy LX Ω = 0. Vector fields satisfying this condition are known as symplectic vector fields. Furthermore, given a function f ∈ F(M), there is Xf ∈ X(M) satisfying

Xf y Ω + df = 0 (∗)

The invertibility of Ω guarantees the existence of Xf so defined. In local coordinates, this vector field has a ab the components (Xf ) = Ω ∂bf. Such a vector field is automatically symplectic:  LXf Ω = Xf y dΩ + d Xf y Ω = d(−df) = 0

We call the function f the generating function and the vector field Xf a Hamiltonian vector field. Conversely, given a symplectic vector field X, we can try to find the corresponding generating function by working back- wards. We can of course do this locally, but in order to find a globally defined function that generates X,

2 we need the first de Rham group to be trivial H1(M) = 0. Therefore, obstructions which keep a symplectic vector field from being Hamiltonian arise from H1(M).

Requiring that dΩ = 0 allows one to write Ω = dΘ, at least locally. Depending on the topology of M, this may hold for a globally define 1-form Θ. Θ is known as the symplectic potential. Imagine we have a Hamiltonian vector field Xf generated by f. What is the change in Θ under an infinitessimal flow along Xf ?  LXf Θ = Xf y dΘ + d Xf y Θ  = Xf y Ω + d Xf y Θ  = d Θ(Xf ) − f =: dΛ

where we used the relation (∗). Then Θ is an abelian gauge potential and Ω = dΘ is the corresponding field strength. From this point of view, the flow Ft(Xf ): M → M which preserves Ω but sends Θ → Θ + dΛ is a (symplectomorphic) gauge transformation.

1.2.1 Poisson Brackets

If we have two Hamiltonian vector fields Xf and Xg, their Lie bracket is also Hamiltonian.

L[Xf ,Xg ]Ω = [LXf , LXg ]Ω = 0

What is the corresponding generating function?

[Xf ,Xg]y Ω = (LXf Xg)y Ω

= LXf (Xgy Ω) − Xgy (LXf Ω)   = Xf y d Xgy Ω + d Ω(Xg,Xf )  = −d Ω(Xf ,Xg)

Therefore, we define the Poisson bracket of the functions f and g as

{f, g} := Ω(Xf ,Xg) [Xf ,Xg]y Ω + d{f, g} = 0

We could have stopped midway in our derivation of the Poisson bracket and ended up with other equivalent expressions: {f, g} = df(Xg) = LXg f and also {f, g} = −dg(Xf ) = −LXf g. In local coordinates, the Pois- −1 ab a b son bracket takes the form {f, g} = (Ω ) ∂af∂bg. Note in particular that if ξ and ξ are two coordinate functions on phase space, we have {ξa, ξb} = (Ω−1)ab.

The salient features of the Poisson bracket – namely, anti-symmetry and the Jacobi identity – follow readily from the definition.

- Anti-symmetry: {g, f} = Ω(Xg,Xf ) = −Ω(Xf ,Xg) = −{f, g}

- Jacobi Identity: From the second boxed equation above, we see that X{f,g} = −[Xf ,Xg]. From here, we find

{f, {g, h}} = LX{g,h} f = −L[Xg ,Xh]f = −[LXg , LXh ]f

= LXh (LXg f) − LXg (LXh f)

= LXh {f, g} − LXg {f, h} = {{f, g}, h} + {g, {f, h}}

3 Hence, the Poisson bracket and its properties are entirely geometric notions defined on any symplectic man- ifold.

Let Xf be a Hamiltonian vector field on M with generating function f. Then, Xf generates a one- parameter family of diffeomorphisms, its flow, through exponentiation, and is the first-order term in the expansion of an infinitessimal flow. The corresponding infinitessimal change in a function G ∈ F(M) under this infinitessimal diffeomorphism is δG = LXf G = {G, f}.

1.2.2 Theorem of Darboux Any two symplectic manifolds of the same dimension are locally symplectomorphic, so all their distinguishing features must manifest as global properties. These are the famed Gromov-Witten invariants.

Compare this situation to Riemannian geometry, where symmetries of the manifold must preserve the Riemann curvature tensor. This is a much more restrictive condition and leads to the result that Riemannian manifolds with large symmetry groups are rare.

1.3 Theory of Dynamics Using Geometry Everything up until now was common to all symplectic manifolds. We will now begin to add some physics to the mix. It will be our goal to rephrase the ideas of mechanics in the language of differential geometry on symplectic manifolds. Given an n-dimensional configuration space Q of a system, both TQ and T ∗Q are even-dimensional manifolds on which we will place symplectic structures. We will call these manifolds the system’s velocity phase space and the momentum phase space, respectively. Putting an atlas on the manifold Q allows us to give coordinates to a point q = (qa). We inherit an induced atlas on TQ, whose points are now labelled (q, q˙) withq ˙ ∈ TqQ. But these are the natural variables for the Lagrangian. So we’ve found that the Lagrangian is a function on TQ, the velocity phase space. Likewise, we can denote points on T ∗Q ∗ ∗ by (p, q) with p ∈ Tq Q and recognize that T Q is the manifold on which the Hamiltonian function is defined.

Since we’re doing physics, let’s introduce some terminology common to the physics literature. The following is a “Mathematician ↔ Physicist dictionary” for the relevant terms. Math Term Physics Term symplectomorphism canonical transformation symplectic structure canonical 2-form symplectic potential canonical 1-form physics without purpose mathematics without rigor physics

1.4 Geometry of Canonical Mechanics A Hamiltonian system (T ∗Q, Ω,H) is a momentum phase space together with a special function H, the Hamiltonian, which governs the time evolution of the system. We’ve already established a natural language in which to describe time evolution: first find the vector field XH generated by the Hamiltonian and then watch the flow of our phase space under this vector field. Recall that the vector field will satisfy Eqn. (∗):

XH y Ω + dH = 0 We’ve already seen how the canonical 1-form transforms under a Hamiltonian vector field’s flow:  LXH Θ = d Θ(XH ) − H = dΛ

4 1 where we’ve defined the phase function Λ := Θ(XH ) − H. If Ft is the flow generated by XH , this last line reads ∗ Ft Θ − Θ) LX Θ = lim = dΛ H t→0 t It follows that the difference between Θ and its flow is related to the integral of Λ. Z t ∗ 0 ∗ Ft Θ − Θ = dAt with At := Λ dt ∈ F(T Q) 0 ∗ where the integration is performed along the integral curve of XH from m ∈ T Q to Ft(m). Upon substituting the definition for Λ, we get Z t Z t  0 0 At = Θ(XH ) − H dt = Θ − H dt 0 0 0 The last equality follows from the fact that dt merely represents a step along the integral curve of XH 0 2 passing through the current point, and so Θ(XH )dt = Θ.

What if we evaluated this integral not along an integral curve of XH but along a curve close to it? By how much will the function S change its value? This variational calculation will be facilitated by picking b b coordinates and writing Θ = Θbdξ where the ξ are phase space coordinates. The variation will change our path from {t 7→ ξ(t}) to {t 7→ ξ(t) + η(t)}. To first order in η, the change in the integral will be Z t  b a a a 0 δAt = (∂aΘb)dξ η + Θadη − (∂aH)η dt 0 Z t  b a a a 0 = (∂aΘb − ∂bΘa)dξ η + d(Θaη ) − (∂aH)η dt 0 t Z t dξb ∂H  = Θ ηa − Ω − ηa dt0 a ab 0 a 0 0 dt ∂ξ

Requiring that δAt = 0 for any variation η implies ∂H ∂H Ω ξ˙b = ←→ ξ˙a = (Ω−1)ab , ab ∂ξa ∂ξb a a a a and Θaη (t) − Θaη (0) = 0 ←→ Θaη (t) = Θaη (0)

a These are statements of the equations of motion and conservation of the quantity Θaη . The surface terms vanish if we hold our endpoints fixed – ie, if η(0) = η(t) = 0. But by keeping the surface terms, we can deduce what form Θ must take. This is a commonly used trick and one which we will employ later.

We must have just varied the action. Upon comparison to §1.1, we find that our previous unwitting P a choice of canonical 1-form was Θ = pa dq and so

X a Ω = dΘ = dpa ∧ dq is our canonical 2-form on the momentum phase space. Note the symplectic structure given above will give the following values of the Poisson brackets of phase space coordinates:

{qa, qb} = 0

{pa, pb} = 0 a {q , pb} = δab

1Secretly, this is the pullback of the Langrangian L ∈ F(TQ) under the Legendre transform τ : T ∗Q → TQ, so I give this function a name close to L but not quite! 2In , to be introduced shortly, this equality reads: pqdt˙ 0 = p dq.

5 In this coordinate system on T ∗Q, the expression for Ω is particularly simple.   0 I (Ωab) = −I 0

∗ ∗ At is a function on T Q. Given a point m ∈ T Q, we can flow it under XH for a given time t and integrate the value of Θ − H dt0 along this curve. This statement amounts to saying that the dynamical history of a mechanical system can be determined by the initial configuration and conjugate momenta to the configuration coordiantes. But there is another view in some situations. Much like a “shooting problem” in the study of differential equations, we might be able to specify our initial and final configurations and determine which (unique) history bridges the two in time t. Such a view would make the action a function on the space Q × Q. Viewed as such, it’s known as the Hamilton two point characteristic function. In some cases, the configuration space allows many different dynamical histories to interpolate from the ini- tial to the final configuration in the given time. This situation is more complicated and will be discussed later.

We will see in §1.4.3 why these expressions for Θ and Ω are standard, at least within the context of non-relativistic mechanics. Coordinates which render the symplectic structure’s matrix expression in the above form are known as canonical coordinates. Looking at which changes of basis give rise to another set of canonical coordinates leads one to consider the symplectic groups Sp(2n; R).

1.4.1 Rehash of Λ and H In general, given an action integral S = R dt Λ, the canonical 1-form Θ and the canonical 2-form Ω, we can determine the Hamiltonian H of the system by solving the two equations

XH y Ω + dH = 0 XH y Θ − H = Λ

a If we assume a canonical coordinate system (pa, q ) has been put on our , these become the following especially nice relationship

X ∂H Λ = pa − H ∂pa which becomes a Legendre transform just as soon as we dignify ∂H/∂pa to be new coordinates to take the place of the pa’s.

1.4.2 Liouville’s Theorem We can define a , ie a top form, on (T ∗Q, Ω) by noting the following. Since dim T ∗Q = 2n and 2 ∗ −n Vn 2n ∗ a Ω ∈ Ω (T Q), we can consider Ω := (2π) Ω ∈ Ω (T Q). Picking canonical coordinates (pa, q ) on T ∗Q, we see that the volume form is expressed as

n a ^ dpa ∧ dq  = Ω 2π a=1

This volume form is preserved under Ft, the flow generated by XH , since Ω is preserved.

n ∗ −n ^ ∗ Ft Ω = (2π) Ft Ω = Ω

6 1.4.3 An Intrinsic Definition of Θ We motivated the typical form Θ takes, up to gauge transformations, through comparison with the expression of the action stated in §1.1. The following is another characterization which explains why we dub this framework “canonical.” Θ ∈ Ω1(T ∗Q) is uniquely determined by the condition that for any 1-form α ∈ Ω1(Q) on our configuration space α∗Θ = α, where we view α as a map α: Q → T ∗Q to take the pullback. Define the basepoint projection π : T ∗Q → Q : (p, q) 7→ q. Since α is a of the contangent bundle over Q, we know that π ◦ α = idQ. We can invert ∗ ∗ the above expression to solve explicitly for Θ. At each point p ∈ T Q we have Θp = π p, or equivalently

 ∗ Θp(u) = p π∗u for any u ∈ Tp(T Q)

In this view, Θ uses p both as a basepoint at which to anchor a vector u on T ∗Q and as a covector with which to map π∗u, the downward projection of u, into the reals. What other choice of covector would make sense for a vector anchored at p? I imagine it’s for this reason that Bryant, et al call Θ the tautological ∗ a a ∗ 1-form on T Q. Coordinatizing Q as (q ) and inducing the coordinates (pa, q ) on T Q from there, one finds the sought after expression for Θ. X a Θ = pa dq Notice that no equivalent trick can be pulled to define a tautological 1-form on TQ, where the Lagrangian lives. In this sense, is less canonical. Indeed, the symplectic structure in a Lagrangian system depends on the particular choice of Lagrangian function, whereas no reference to any Hamiltonian function was made in the canonical case.

1.4.4 Example: Charged Particle In a Magnetic Field Consider a charged particle in a three dimensional space, which contains a magnetic field. Say the compo- nents of the field are Ba. The typical treatment of the dynamics involves constructing a vector potential A of the magnetic field. One then minimally couples the Hamiltonian of the particle to the field by sending 0 e ∗ H(p, q) → H (p, q) := H(p − c π A, q). But what if A cannot be globally defined?

Consider the following treatment instead. The configuration space Q is just the position of this single 2 particle. We construct a related 2-form componentwise: Fab := abcBc. That is, F ∈ Ω (Q), for which dF = 0 since the magnetic field is divergence-free. Thus, F = dA, at least locally. The basepoint projection π : T ∗Q → Q :(p, q) 7→ q gives a means of lifting F up to a 2-form on T ∗Q. We now have π∗F ∈ Ω2(T ∗Q). Let’s make a on the phase space T ∗Q which we define by

∗ ∗ e ∗ φA : T Q → T Q :(p, q) 7→ (p + c π A, q) The second copy of phase space is the one that we described above. Why? The change in coordinates e ∗ (p, q) 7→ (p + c π A, q) will be compensated by the aforementioned change in the Hamiltonian so that it attaches the same value to the same point in T ∗Q, though it’s now labelled by a different p value. We now find what the first system looks like. It has a Hamiltonian given by

∗ 0 0 e ∗ φAH (p, q) = H (p + c π A, q) = H(p, q) and a symplectic potential

∗ X ∗ a X e ∗  a e ∗ φAΘ = φA padq = pa + c π A dq = Θ + c π A

∗ e ∗ The symplectic structure is φAΩ = Ω + c π F . The symplectic structure has changed! The dynamical evolution of the system is determined by the integral curves of the field XH which depends both on H and on Ω. The typical procedure to minimally couple a system to a gauge field changes the Hamiltonian but we

7 see here we may instead change the symplectic structure. The latter has the advantage that one needs only to know F and not the gauge connection A which may not be globally defined.

If we pick the canonical coordinates (p, q) which give the original symplectic structure the matrix repre- sentation   0 I (Ωab) = −I 0 the “charged” symplectic structure is  e F  (Ω0 ) = c I ab −I 0 whose inverse is   0 −I e I c F Therefore, the Poisson brackets of these phase space variables are

{qa, qb} = 0 a {q , pb} = δab e {pa, pb} = c Fab

1.5 Making Time a Dimension Let’s now consider the extended configuration space in which time isn’t the parameter of the flow generated ˜ by XH but instead is a dimension in its own right. That is, we consider Q := Q × R whose points are coordinatized as (q, t). Dynamic evolution consists of watching the flow of points in T ∗Q˜ under the vector field ∂ X˜ := X + H ∂t The added term causes the point m ∈ T ∗Q˜ to flow forward in the time direction while the rest of the coordinates evolve as they did before. As it stands, the flow paremeter τ will differ from the time t by at most an additive constant t0. This “gauge condition” will be relaxed soon, and we will no longer fix the flow parameter to satisfy dt/dτ = 1. For example, we will have to drop this condition in a manifestly covariant treatment when we no longer pick a time direction.

The time coordinate t has its own conjugate momentum which we label s. Then, the phase space T ∗Q˜ has the symplectic structure and symplectic potential

Ω˜ = Ω + ds ∧ dt and Θ˜ = Θ + s dt

The Hamiltonian is seen to be H˜ = H +s, because this is needed to satisfy the canonical equations of motion.  ∂    X˜ Ω˜ + dH + s = X + Ω + ds ∧ dt + dH + s y H ∂t y   = XH y Ω − ds + d H + s = 0 In the extended configuration space, the action can be written as Z   Z   S = Θ − H dt = Θ˜ − H˜ dτ C C where C is the name given to the segment of the integral curve over which we are computing the action. We used the fact in the equation above that, so far, our gauge condition has made dt = dτ hold true. Let’s relax this stipulation and allow t = t(τ) to be a function which is not linear. Instead, put an arbitrary 2 √ Riemannian metric γ = γττ (dτ) on the integral curve. Defining η(τ) := det γ and notice that under a

8 change of coordinates η0(τ 0)dτ 0 = η(τ)dτ. The above action integral may now be recast for arbitrary (valid) relationship between time and the flow parameter τ. Z   S = Θ˜ − H˜ ηdτ C We actually want the action to be independent of the arbitrary metric γ we’ve chosen to put on the integral curve. This means we enforce the condition: δS = H˜ = 0 δγττ

If we take H˜ = 0, this constitutes restricting our system to a of the phase space. On this ˜ ˜ ˜ submanifold, the statement Xy Ω = 0 holds true. Therefore, X, whose flow generates dynamical evolution, is actually a zero-mode of the Ω,˜ whichs is therefore a presymplectic structure.3 The momentum conjugate to time is minus the Hamiltonian, s = −H, and so Ω˜ admits the coordinate representation:

X a Ω˜ = dpa ∧ dq − dH ∧ dt

d ˜ Note in passing that if we take dt f to be the deriative of a function f along an integral curve of X, we have df  ∂  ∂f = X˜(f) = X + f = {f, H} + dt H ∂t ∂t which we a result we knew from §1.1.

In a relativistic treatment of a problem, we do not pick a special coordinate on the (extended) configura- tion space which we designate as the time. Let’s drop the tildes to show that we’ve done this. We can still distinguish between position and momentum variables in T ∗Q. That is, we relabel t 7→ q0 and its conjugate momentum will be p0. The equations of motion are simple

Xy Ω = 0 and a constraint we must impose is H = 0. By τ we’ll denote the parameter of the flow under the dynamical vector field X ∈ X(T ∗Q). Orbits τ 7→ (t(τ), qa(τ)) in the configuration space Q are considered the same if they differ by a valid reparametrization, τ 7→ τ 0(τ), which must be injective and smooth. Depending on your point of view, this freedom to reparametrize is the cause or the consequence of the degeneracy of the presymplectic structure Ω. One can undo the introduced degeneracy by “gauge fixing” the parameter and reducing.

1.5.1 Example: Relativistic Point Mass For a single point mass in Minkowski space, Q will just be Minkowski space itself.

X α β Ω = ηαβdp ∧ dq α,β where we use a Greek index instead of a Latin one simply to remind ourselves we mean all four coordinates. Then, we find that the degenerate vectors of the presymplectic structure take the form ∂ X = p α ∂qα

2 2 α β 1 2 for arbitrary fixed pα satisfying H = p − m = 0. This is because Xy Ω = −ηαβp dp = − 2 d(p ) = 0. The α α corresponding integral curves of X are τ 7→ (pα, p τ + q0 ). The orbits in Q are the basepoint projection of α α α these integral curves: τ 7→ x (τ) = p τ + q0 . Notice that this entire treatment was fully covariant. 3The degeneracy prevents it from being a proper symplectic structure.

9 2 Symplectic Geometry in Classical Field Theory

We now introduce the machinery to extend the canonical formalism to a field, which has an infinite number of degrees of freedom. There are many ways to approach the problem and I’ve tried to stick to the one that seems the most physical while still allowing for the insight provided by a geometrical treatment. The methods remain much the same, and require little that doesn’t have a direct analogue in the mechanical case.

2.1 Theoretical Development We posit the following three spaces

- A manifold M on which the field lives. This might be R corresponding to a time axis, R1,3 for spacetime, or any of a number of possibilities. We’ll define m := dim M. Frequently, we’ll also have a symmetric, bilinear, non-degenerate 2-form g which makes (M, g) a . -A configuration bundle ξ : Q → M which specifies the current arrangement of the system’s degrees of −1 freedom. The fibers Qx := ξ (x) will be the configuration space over x ∈ M. If X is the target space ∼ of the fields, ie the space in which the fields take their values, then the fibers are Qx = X and there are neighborhoods U ⊂ M covering spacetime on which ξ−1(U) =∼ U × X. That is, the bundle admits local trivializations. -A phase bundle η : P → M whose sections correspond to states of the system, which are essentially the possible dynamical histories. Exactly which histories are possible gets decided by the dynamical equations, first order differential equations which themselves come from the Hamiltonian. For a given q ∈ Q, define m−1 ∗ ^ ∗  Pq := Tq Qξ(q) ⊗ Tξ(q)M S Then, P = Pq is then the total space of a vector bundle π : P → Q. Then, η = ξ ◦ π and we call −1 Px := η (x) the phase space at x. Then, there is a restriction πx : Px → Qx. A point p ∈ Px can Vm−1 ∗ Vm−1 ∗ be viewed as a linear mapping p : Tπ(p)Qx → (Tx M). We can view (Tx M) as the space of vector densities at x.

∗ In mechanics, the space M is just the 1-dimensional time axis. Therefore, Pq = Tq Q so the phase space P was just T ∗Q, the cotangent bundle of the configuration space.

α α α Coordinatize M by (x ) and Q by (x , φa) for a = 1,...,N. The projection mapping is ξ :(x , φa) 7→ α p 1 m Vm−1 ∗ (x ). Define the volume element in M to be  := |det g| dx ∧ · · · ∧ dx . A basis on (Tx M) can be ∂αy .

We can take a point p ∈ Px and express it as

X α  p = pa δφa ⊗ ∂αy 

α α In this setup, P has coordinates (x , φa, pa ). We now make an analogy to the mechanical case and define a density-valued canonical 1-form. For each p ∈ Px,

m−1 ∗  ^ ∗ Θp = πx p −→ Θp(v) = p (πx)∗v ∈ (Tx M), for any v ∈ TpPx.

Then Ω = δΘ, and (P, Ω) is the symplectic manifold. In coordinates, the canonical 1-form is expressed as Z X α  Θ = pa δφa ⊗ ∂αy  α,a Σ

10 and the symplectic structure is Z X α  Ω = δpa ∧ δφa ⊗ ∂αy  α,a Σ α Σ ⊂ M is the codimension 1 hypersurface on which we specify the Cauchy data. pa (x) can take non-zero values in those directions for which the normal vector at x ∈ Σ has non-trivial components. We can introduce P α a more compact notation by defining pa := α pa ⊗ (∂αy ). In this notation, the symplectic potential and structure become Z X Θ = pa δφa Z X Ω = δpa ∧ δφa

These expressions for Θ and Ω are in a canonical coordinate system, which gives them a particularly simple α α form. The equations of motion are specified by the Hamiltonian density H = h(x , φa, pa ) ⊗ .

δh α δh ∂αφa(x) = α and ∂αpa (x) = − δpa (x) δφa(x) We can drop our coordinate patches on M by writing this in a geometric notation.

α β α δh δH dpa(x) = ∂βpa (x) ⊗ dx ∧ (∂αy ) = ∂αpa ⊗  = − ⊗  = − δφa(x) δφa(x) and α δh α δH dφa(x) = ∂αφa(x) ⊗ dx = α ⊗ dx = δpa (x) δpa(x) In this last equation, we had to allow ourselves the slightly unusual definition

α  α  = dx ∧ (∂αy ) −→ := dx ∂αy  Fortunately, this definition continues to make sense under valid change of coordinates because wedge products commute with pullbacks. ∗ ∗ ∗ ∗ φ  ∗ α φ  = φ ∂α ∧ φ (∂αy ) −→ ∗ = φ dx φ (∂αy  In general, the symplectic structure will take the form ZZ 0 X 0 0 Ω = dµ dµ ΩAB(µ, µ ) δζA(µ) ∧ δζB(µ ) A,B

α where the ζA(x)’s are the phase space variables – that is, φa(x) or pa (x) – at a point x ∈ M. I’m struggling with choosing good notation here. I want the labels A, B, . . . to carry information about which field we’re α talking about, which is contained in the labels a, as well as if the field is a φa or a pa . In the latter case, I was A also to specify α. In this language, the Poisson brackets of the phase space variables take the form

0 −1 0 {ζA(µ), ζB(µ )} = ΩAB(µ, µ )

−1 0 where ΩAB(µ, µ ) must satisfy Z 0 X 0 −1 0 00 00 dµ ΩAB(µ, µ )ΩBC (µ , µ ) = δAC δ(µ − µ ) B

11 2.2 Applications We will now turn to applying the formalism of the previous section to solving example problems. These will begin with equal-time and then light-cone treatment of a massive scalar field. The difference between these two amounts to choosing different hypersurfaces Σ.

2.2.1 Equal-Time Treatment of a Free Scalar Field Pick a coordinate system (xα) = (t, x) on the spacetime manifold M, which is endowed with the Minkowski metric η = diag(+1, −1). We will choose our hypersurfaces Σt to be of the form {t} × R, ie time slices.A free scalar field has action within a spacetime region V equal to Z 1 α 2 2 S = dt dx 2 ∂αφ ∂ φ − m φ V

Let’s pick V = [ti, tf ] × R so that ∂V = Σtf − Σti . Then, Z α 2  δS = dt dx ∂αδφ ∂ φ − m φδφ V Z Z 0  α 2  = dx ∂ φ δφ − dt dx ∂α∂ φ + m φ δφ ∂V V The surface term at the “spatial boundary” vanishes, and the remaining surface terms specify the canonical 1-form Z Θ(t) = dx ∂0φ(t, x)δφ(t, x)

Σt whose variational gives the canonical 2-form Z Ω(t) = dx δ∂0φ(t, x) ∧ δφ(t, x)

Σt ZZ = dx dx0 δ(x − x0) δ∂0φ(t, x) ∧ δφ(t, x0)

Σt

α 0 0 Evidently the only non-vanishing momentum p is p = ∂ φ. This is the usual momentum π := δL/δ(∂0φ). The other component vanishes, p1 = 0. It’s comforting that the formalism described above reproduces the 0 results of an easy scenario when treated with more familiar techniques. Thus ΩAB(x, x ) takes the form  0 δ(x − x0) [Ω (x, x0)] = AB −δ(x − x0) 0 and the inverse is found to be  0 −δ(x − x0) [Ω−1 (x, x0)] = AB δ(x − x0) 0 so the equal time Poisson bracket relations are

{φ(t, x), φ(t, x0)} = 0 {φ(t, x), pα(t, x0)} = δα,0δ(x − x0) {pα(t, x), pβ(t, x0)} = 0

If we wanted to start doing quantum field theory, the prescription would tell us to ˆ ˆ obtain the corresponding equal time commutation relations by sending {ζA, ζB} → [ζA, ζB]/i~.

12 2.2.2 Lightcone Treatment of a Free Scalar Field

We take the same physical system√ from the previous√ example and look at it under a different set of coordinates on M. Define u := (x + t)/ 2 and v := (x − t)/ 2. Instead of having a hypersurface Σ given by time slices, we’ll take these surfaces to have constant u. This means we are now looking at how the system evolves with increasing u values. The action in a region V = [ui, uf ] × R is equal to Z 1 2 2 S = du dv −∂uφ ∂vφ − 2 m φ V so that the variation is Z 2  δS = du dv −∂uδφ ∂vφ − ∂uφ ∂vδφ − m φ δφ V Z u Z  f 2  = − dv −∂vφ(u, v) δφ(u, v) + du dv 2∂u∂vφ(u, v) − m φ(u, v) δφ ui V Hence, the canonical 1-form and canonical 2-form are Z  Θ(u) = dv −∂vφ(u, v) δφ(u, v) Z Z  0 0 0  Ω(u) = dv −δ∂vφ(u, v) ∧ δφ(u, v) = dv dv −δ∂vφ(u, v) ∧ δφ(u, v ) δ(v − v ) Z Z 0 0 0  1 0 0 0 = dv dv −∂vδ(v − v ) δφ(u, v) ∧ δφ(u, v ) = 2 dv dv Ω(v, v ) δφ(u, v) ∧ δφ(u, v )

0 0 We find the components of the canonical 2-form are Ω(v, v ) = −2∂vδ(v − v ) between two φ values and vanish otherwise. Hence, the inverse of this takes the form (  1 sgn (v − v0) 0 +1 v − v0 > 0 [Ω−1 (v, v0)] = 4 for sgn (v − v0) := AB 0 0 −1 v − v0 < 0

R 0  1 00  0 00 because dv −2∂vδ(v − v ) · 4 sgn (v − v ) = δ(v − v ). The lightcone Poisson bracket relations take the form

0 1 0 {φ(u, v), φ(u, v )} = 4 sgn (v − v ) {φ(u, v), pα(u, v0)} = 0 {pα(u, v), pβ(u, v0)} = 0

We now have deteremined the form of Θ from the action and can try to determine the Hamiltonian generating evolution in u. u Z Z f 1 2 2  S = − du dv ∂uφ ∂vφ + 2 m φ = du Θ(XH ) − H

V ui where this Hamiltonian is related to the Hamiltonian density H by

u Z Z Z f H = ∂uy H −→ H = du H

Σ Σ×[ui,uf ] ui We find δΛ 1 2 2 1 2 2 h = ∂vφ − Λ = 2 m φ −→ H = 2 m φ ⊗  δ(∂vφ)

13 Therefore, the Hamiltonian generating translations in u is given by Z 1 2 2 H = dv 2 m φ Σ 3 Geometric Quantization 4 Appendix

I have relegated some topics to this appendix so as to avoid interrupting the development of the physics with only tangentially related mathematics. Hopefully the ideas recorded here will still be of value, just lacking a good placement within the main text.

4.1 Notation and Terminology and Statement of Differential Geometry Results The following are a few bits of notation and terminology used in these notes.

- The of one vector field with respect to another is LX Y = [X,Y ]. - The inner product, insertion or interior derivative of a vector field X ∈ X(M) into the k-form ω ∈ Ω•(M) is the form denoted by Xy ω := ω(X,...) That is, we insert X into the first “slot” of the form ω. If ω is a 0-form, ie a function on M, then this insertion vanishes.

- Cartan’s Theorem: For any vector field X ∈ X(M) and any form ω ∈ Ω•(M), we know  LX ω = Xy dω + d Xy ω

4.2 Geometry of Lagrangian Mechanics This section is still rather undeveloped since it’s only tangentially related (har har). We saw earlier that there is a Legendre transform which sends us from the Hamiltonian formalism to the Lagrangian formalism. In terms of the manifolds on which these formalisms are described, this transformation can be denoted ρ: T ∗Q → TQ. Given a Hamiltonian system (T ∗Q, Ω,H), we obtain a corresponding Lagrangian system (T Q, ΩL,L) by pulling back the term dubbed Λ earlier.

∗  L = ρ Θ(XH ) − H

Note that this is not the pullback of the Hamiltonian. That is the energy function on TQ. ∂L E(q, q˙) = ρ∗H =q ˙a − L ∂q˙a We find the symplectic structure and symplectic potential by pulling back those of the canonical formalism.

∂2L ∂2L Ω = ρ∗Ω = dqa ∧ dqb + dq˙a ∧ dqb L ∂qa∂q˙b ∂q˙a∂q˙b ∂L Θ = ρ∗Θ = dqa L ∂q˙a

14 a b The Lagrangian is called regular if the function multiplying dq˙ ∧ dq in the expression for ΩL vanishes nowhere. Having defined these quantities, consider any orbit t 7→ (qa(t), q˙a(t)) through Lagrangian phase a ∂ a ∂ space. The tangent vectors along this solution will be X =q ˙ ∂qa +q ¨ ∂q˙a . Then, h ∂L ∂L ∂L i X Ω + dE = q˙b +q ¨b − dqa y L ∂q˙a∂qb ∂q˙a∂q˙b ∂qa h d  ∂L  ∂L i = − dqa dt ∂q˙a ∂qa

vanishes exactly when the orbit satisfies the classical equations of motion, just like Eqn. (∗) was a statement of the equations of motion in the canonical case as well.

Notice I said a corresponding Lagrangian system and not the corresponding Lagrangian system. This is because, given the equations of motion, it is not possible to determine the Lagrangian uniquely. Given a Lagrangian L and a smooth function f : R → R with a first derivative which never takes the value 0, then L0 = f ◦ L is another Lagrangian producing the same equations of motion. Another more important means of changing the Lagrangian without modifying the dynamics is to send X  L(q, q˙) −→ L(q, q˙) + q˙y βi(q) + Ki

where each βi(q) is a closed 1-form on Q. This observation is the starting point for the method of imposing holonomic constraints on the system. We will not pursue this topic.

5 References

For more information on the geometry of classical mechanics and classical field theories, refer to

- Arnold “Mathematical Methods of Classical Mechanics”

- Berndt “An Introduction to Symplectic Geometry”

- Bryant, et al “Exterior Differential Systems and Euler-Lagrange Differential Equations”

- Guillemin & Sternberg “Symplectic Techniques in Physics”

- Kijowski, Tulczyjew “A Symplectic Framework for Field Theories”

- Rovelli “A Note on the Foundation of Relativistic Mechanics” [arXiv:gr-qc/0111037]

For more on geometic quantization, see

- Puta “Hamiltonian Mechanical Systems and Geometric Quantization”

- Woodhouse “Geometric Quantization”

For additional reading about geometry in quantum field theories, check out

- Freed, Uhlenbeck “Geometry and Quantum Field Theory”

- Freed, Morrison, Singer “Quantum Field Theory, Supersymmetry, and Enumerative Geometry”

- Nair “Quantum Field Theory: A Modern Approach”

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