Hamiltonian Field Theory

Transcription

1 Hamiltonian Field Theory August 31, Introduction So far we have treated classical field theory using Lagrangian and an action principle for Lagrangian. This approach is called Lagrangian field theory and is best suited for quantizing a field theory using the pathintegral aka functional integral approach. However, a quicker route to quantizing a classical field theory is thru the canonical quantization program which is what you will find in most introductory treatments on quantum field theory. So for us, the Lagrangian is just a crutch to extract the Hamiltonian. However there is a price to pay when we switch from a Lagrangian approach to a Hamiltonian approach, that is manifest Lorentz invariance. Recall that a field theory is a system where we have a degree of freedom located at each point in space. If this degree of freedom is a Lorentz scalar, we will denote the degree of freedom located at position x by Φx and its conjugate momenta, Πx. These degrees of freedom of course fluctuate in time, so when we include this time-dependence, the notation becomes, Φx, t and Πx, t. The job of Hamiltonian field theory is describe time-evolution of Π and Φ in equations as follows, Π..., Φ... The dot is time derivative, i.e. we have to choose a time axis or equivalently preferred reference frame which breaks manifest Lorentz invariance. Recall that earlier, in the Lagrangian/action approach we did not have to choose a time-frame because the equation of motion looked Lorentz invariant, e.g. the Euler-Lagrange equation for scalar field, L µ L µ Φ Φ 0. Here all Lorentz indices are contracted and the only quantities that appear are Lagrange densityl and Φ which are both Lorentz invariants/scalars - so the whole equation is manifestly Lorentz invariant. Similarly the action functional approach leads to a equation of motion of the schematic form, I Φx 0, which is also manifestly Lorentz invariant as both the numerator and denominator are Lorentz invariants themselves. In this note we review Hamiltonian field theory with an eye towards canonical quantization of classical fields. Since in most texts in classical mechanics, the dependence on time for the canonical pair of variables p and q are omitted from most formulas, we too shall choose not to display the time dependence, i.e. instead of Φx, t we will use, Φx and instead of Πx, t we will use, Πx. Just as in case of q, p it would be implicitly understood that Φx, Πx are functions of time. 1

2 Hamilton s principle and Hamilton s equation for field theory Hamilton s principle for classical mechanics states that the equations of motion of a physical system described by the generalized coordinate and momenta, q, p, can be obtain by extremizing i.e. setting the first order variation to zero of the following functional, I dt [p q Hp, q. 1 The function, Hp, q is called the Hamiltonian function. action that the equation of motion for this system are, One can easily check by varying the above q H p, ṗ H q. These equations are the well known Hamilton s equation. Note that in contrast with the Lagrangian approach where the Euler-Lagrange equations involve second order time-derivatives in time, the equations of motion in the Hamilton s approach are first order differential equations in time. Of course the price to pay is that we have twice the number of equations, one set for q and one set for p. One can easily generalize Hamilton s framework for classical mechanics for a single degree of freedom to field theory i.e. infinite degrees of freedom. To accomplish that first we write down the action for N number of degrees of freedom, q i, p i, i 1,,..., N. The action and Hamilton s equation for this case involving N degrees of freedom are: [ N I dt p i q i Hp i, q i 3 and, i1 q i H p i, ṗ i H q i. 4 Now in order to go to a continuum limit, we set N, then the discrete label, i turns into a continuum label such as the position coordinates, x x, y, z which are real number/continuous valued not discrete/integer valued, q i Φx, p i Πx. The summation of course is turned into an integration over the continuum position coordinate, N i1.

3 Finally one should also realize that the Hamiltonian function would change into a global functional of the fields, Φx, Πx. Hp i, q i H [Πx, Φx HΠx, Φx. The local functional H is quite naturally called the Hamiltonian density since its volume integral is the Hamiltonian. Making these changes to the Eq.s 3, 4, we get the the action and Hamilton s equation for a field theory, [ I [Φx, t, Πx, t dt Πx Φx H [Πx, Φx, 5 and, Φx Πx, Πx Φx. 6 However the rules of functional derivatives are different from what we had in Lorentz invariant action functional approach. The new rules of functional integration can be easily deduced by generalizing the following formula for N degrees of freedom to continuum limit, q i q j ij, p i p j ij. Using the prescribed rules, we now replace the i, j labels by position coordinates, x, y. To wit, Φx Φy 3 x y, Πx Πy 3 x y. 7 3 Hamiltonian from the Lagrangian Starting with a Lagrangian for a physical system, one can extract the Hamiltonian thru a Legendre transform. Two functions, fx and gx are said to be Legendre transforms of each other if their first derivatives are functional inverse of each other, i.e. x f g x dfy dy yg x where primes denote first derivative of the function wrt their respective arguments. Solving this condition one gets, x g x fy So the Hamiltonian is defined to be the Legendre transform of the Lagrangian, Now here qp means we have inverted the relation, Hp, q p qp Lq, qp. p Lq, q q 3

4 to express, q as a function of p and more generally a function of both p and q. For a system with N- degrees of freedom, the definition is, H p i q i Lq i, q i i Now we can follow the prescription mentioned in the previous section to take the continuum limit, N whereby we obtain the Lagrangian for a field theory, H Πx Φx L[Φx, Φx, Now since the Lagrangian itself can be expressed as a volume integral of Lagrangian density, L[Φx, Φx L, L LΦx, Φx, we have, H Πx Φx L, where first we need to express, Φx in terms of Πx by inverting, Πx 3.1 Example: Real free scalar field theory L Φx. The real scalar field theory is defined by the action, I[Φx d 4 x L, L 1 µφ µ Φ 1 m Φ. So the Lagrangian is, L The momentum conjugate to Φx is, Πx L, 1 µφ µ Φ 1 m Φ 1 Φ 1 Φ Φ 1 m Φ Φx 1 Φx d 3 y So Φ y 1 yφy y Φy 1 m Φ y d 3 1 y Φx Φ y d 3 Φy y Φy Φx d 3 y Φy 3 x y Φx.. 4

5 We can use this to eliminate Φ in terms of Π. The Lagrangian then becomes, 1 L Π 1 Φ Φ 1 m Φ, and the Hamiltonian is, H Πx Φx L Π x L 1 Π x Π 1 Φ Φ 1 m Φ 1 Π + 1 Φ Φ + 1 m Φ. 8 From this we can extract the Hamilton s equations. The first Hamilton s equation doesn t give us anything but the old relation, However the second equation is non-trivial,, Πx Φx Πx Πx. 9 Φx 1 d 3 y Φx Π + 1 yφ y Φ + 1 m Φ [ d 3 y y Φ y Φx Φy + m Φy Φy Φx d 3 y [ y Φ y 3 x y + m Φy 3 x y d 3 y yφ + m Φ 3 x y m Φx. 10 So equations are, Φx Πx, Πx m Φx. Replacing, Π Φ in the second equation we get the second equation to look like, t + m Φ 0. Φ m Φ, This is nothing but the well-familiar Klein-Gordon equation, + m Φ 0. Thus we have recovered the correct equation of motion using the field Hamiltonian, Eq. Hamilton s equations Eq and the 5

6 4 Poisson brackets, Charges and algebra of charges In general, the time evolution of a quantity, fp, q can be deduced from the Poisson brackets PB of that quantity with the Hamiltonian, H, dfp, q dt f dq q dt + f p dp }{{} dt f H q p f H p q }{{} {f, H}. In particular if the system has conserved charges, Q, since, dq dt {Q, H} 0. 0, one has for conserved charge, Q, Again, as done previously, we can generalize the Poisson bracket expression for single degree of freedom to field theory thru the intermediate step of going to N degrees of freedom. For N degrees of freedom, the Poisson bracket can be easily shown to be of the form, {A, B} i A q i B p i A p i B q i So in the continuum limit, N, we replace, q i, p i Φx, Πx and i, and get, A {A, B} d 3 B x Φx Πx A B. Πx Φx Here of course A, B are themselves functions or functional of Φ, Π. Now, evidently, This immediately implies if we set, B A, {A, B} {B, A}. {A, A} {A, A}, {A, A} 0, {A, A} Example: Scalar fields and Conserved charges We already know thru Noether s theorem that there are charges corresponding to Poincare symmetry, namely the energy-momenta, P 0, P i and the boost-rotation charges L µν. Let s check that these are conserved by taking the Poisson bracket with the Hamiltonian. First we need to express the charges in the canonical variables, Φ, Π i.e. by eliminating the velocity, Φ in lieu of the conjugate momenta, Πx. We start from the expression, P µ T µ0. [ µ Φ Φx η µ0 L. 6

7 P 0 [ Φ x L [ Φ x 1 µ µ + m [ 1 Φ Φ x + 1 m + Φ [ 1 Π x + 1 m + Φ, 11 and, [ P i i Φx Φx η i0 L i Φx Φx i Φx Πx. 1 Here, we have used that due to the fact that the metric component, η ii 1, i i. Notice that, P 0 H, so its PB with the Hamiltonian i.e. itself will be zero due to antisymmetric nature of the PB, { P 0, H } {H, H} d 3 x Φx Πx 0. Πx Φx This proves, P 0 is conserved. The next case, i.e. P i given by the expression in Eq. 1. We need to evaluate, { P i, H } P d 3 i y Φy Πy P i. 13 Πy Φy Using Eq. 1 we compute, P i Φy i Φx Πx Φy iφx Πx Φy Φx i Πx Φy d 3 x i 3 x y Πx 3 x y i Πx Πy, 14 yi 7

8 P i Πy i Φx Πx Πy d 3 x i Φx Πy Πx i Φx 3 x y Φy. 15 yi Now recall from Eq. 9 and 10, Φy y m Φy., Πy. 16 Πy Now plugging Eq. 14, 15 and 16 in the expression for the PB in RHS of Eq. 13, we obtain, { P i, H } P d 3 i y Φy Πy P i Πy Φy d 3 y Πy Πy yi y i Φy y m Φy d 3 y Πy Πy yi y i Φy yφ m y i ΦyΦy d 3 y 1 y i Π y + 1 Φ + m Φ y + d 3 y i Φ Φ, 17 where we have used, i Φ Φ i Φ j j Φ j i Φ j Φ j i Φ j Φ j i Φ j Φ i j Φ j Φ 1 j i Φ j Φ i jφ j Φ 1 i Φ Φ i Φ. Thus both terms in the expression for{p i, H} i.e. the rhs of 17 are integral of a total derivative and after integration turns into surface terms at spatial infinity where of course they are zero we are assuming for all integrals to be finite that all fields decay to zero at spatial infinity. Thus {P i, H} 0 which means P i i.e. the linear momentum is conserved as well. For the boost and rotation symmetry combined i.e. Lorentz symmetry, the Noether charges are given by the expression, L µν M µν0 x µ T ν0 x ν T µ0, 8

9 In particular, these charges are, the angular momenta, L ij x i T j0 x j T i0, and the less familiar Boost charges or Boost generators L 0i x 0 T i0 x i T 00 t T i0 x i H, t P i x i H In the center of massmomentum frame, of course we have, P i 0, then the Boost charges have the intepretation of the angular energy or moments of energy, L i0 CM x i H CM. Lets now check that the angular momentum is conserved using the Poission Bracket with the Hamiltonian. First we express the angular momentum in Π, Φ variables by eliminating, Φ in favor or Π. L ij x i T j0 x j T i0 x i j Φx Πx + x j i Φx Πx. Then second step, we evaluate the functional derivaties, L ij d 3 y y i Φy Πy + yj Φy Πy Φx Φx yj yi d 3 y y i y j Φy Πy + yj Φx y i Φy Πy Φx d 3 y y i y j 3 x y Πy + y j y i 3 x y Πy [ d 3 y 3 x y y i y j Πy y j y i Πy j x i Πx i x j Πx x i j Πx x j i Πx. 18 9

10 and, L ij Πx d 3 y y i Φy Πy + yj Φy Πy Πx yj yi d 3 y y i y j Φy Πy + yj Πx y i Φy Πx Πy d 3 y y i y j Φy 3 x y + y j y i Φy 3 x y x i j Φx + x j i Φx. 19 Finally we compute the Poisson brackets by using Eq.s 16, 18 and 19 { L ij, H } L ij Φx Πx Lij Πx Φx [ x i j Πx x j i Πx Πx x i j Φx x j i Φx m Φx [ x i j x j Π x j + m Φ x x i j Φx x j i Φx Φx [ j x i Π x + x i m Φ x i x j Π x + x j m Φ x x i j Φx x j i Φx Φx. 0 The first two terms are of course integrals of total derivatives which would lead to surface terms at spatial infinity which in turn vanish. The last term needs to be massaged a littel bit. We have alrady seen that, i Φ Φ i Φ Φ i 1 Φ. So then, x i j Φ x j i Φ Φ x i j Φ Φ x j i Φ Φ x i j 1 Φ + x j i 1 Φ x i k j Φ k Φ x j k i Φ k Φ x i j 1 Φ + x j i 1 Φ k x i j Φ k Φ x j i Φ k Φ j x i 1 Φ + i x j 1 Φ. This is also a total derivative. Thus all three terms in the {L ij, H} are integrals of total derivatives which vanish on integration, {L ij, H} 0 dlij dt 0. 10