III. Classical Field Theory and Canonical Quantization Hamilton's

III. Classical Field Theory and Canonical Quantization

Hamilton’s Principle and the Euler-Lagrange Equations 1 Noether’s Theorem and Symmetries 3 Poincar´eSymmetries 6 Space-Time Translations: Canonical Energy-Momentum Tensor 7 Lorentz Transformations: Improved Energy-Momentum Tensor 8 Canonical Quantization 14

1 III. Classical Field Theory and Canonical Quantization

In the previous chapter, we saw how to construct Lorentz covariant free field equations using group theoretic techniques applied to the Lorentz group. However, we still lack a dynamical principle which allows for the construction of the Poincarégenerators including the Hamiltonian in terms of the operator fields nor a means to give a physical interpretation to the solution to the these field equations. In this chapter, we begin to address these issues. The canonical quantization approach to constructing a quantum field theory starts with a classical field theory whose dynamics is governed by classical field equations which are obtained by applying (a generalized) Hamilton’s principle to an action functional of a Lagrange density (Lagrangian). The classical fields are then promoted to become quantum operator fields satisfying certain equal time canonical commutation or anti-commutation relations. Using this canonical formalism, the various symmetry generators, including the Hamiltonian, can be identified and constructed in terms of the field operators. As we shall see, a major advantage of basing a dynamical principle on an action is that it is Lorentz invariant while the Hamiltonian is not.

Hamilton’s Principle and the Euler-Lagrange Equations

The classical Lagrangian (density) characterizing the underlying dynamics is assumed to be a function of the classical ﬁelds, ϕs(~r, t), and their ﬁrst space-time derivatives, ∂µϕs(~r, t):

L = L(ϕs(~r, t), ∂µϕs(~r, t)) . (1) The action functional for the classical field theory is then defined as Z 4 S[ϕ] = d xL(ϕs(t, ~r), ∂µϕs(t, ~r)) , (2) Ω where Ω is the space-time volume. Here the subscript s is used to distinguish different fields. The (generalized) Hamilton principle states that the action is stationary with respect to variations of classical field, δϕs(~r, t), which vanish on the boundary of Ω but are otherwise arbitrary.1

1Rather than employing this generalized Hamilton’s principle, one could divide 3-space

2 Using functional derivatives, Hamilton’s principle takes the form

δS[ϕ] 0 = δϕr(x) δϕr(x) Z y ! 4 ∂L δϕs(y) ∂L δ∂µϕs(y) = δϕr(x) d y + y Ω ∂ϕs(y) δϕr(x) ∂∂µϕs(y) δϕr(x) Z ! 4 ∂L δϕs(y) ∂L y δϕs(y) = δϕr(x) d y + y ∂µ Ω ∂ϕs(y) δϕr(x) ∂∂µϕs(y) δϕr(x) Z ! 4 ∂L y ∂L δϕs(y) = δϕr(x) d y − ∂µ y Ω ∂ϕs(y) ∂∂µϕs(y) δϕr(x) Z ! 4 y ∂L δϕs(y) + δϕr(x) d y∂µ y Ω ∂∂µϕ(y) δϕr(x) Z ! 4 ∂L y ∂L 4 = δϕr(x) d y − ∂µ y δrsδ (x − y) Ω ∂ϕs(y) ∂∂µϕs(y) Z ! 4 y ∂L 4 + δϕr(x) d y∂µ y δrsδ (x − y) Ω ∂∂µϕ(y) Z ! 4 ∂L y ∂L 4 = δϕr(x) d y − ∂µ y δrsδ (x − y) Ω ∂ϕs(y) ∂∂µϕs(y) ∂L 4 + δϕr(x)nµ y δrsδ (x − y)|y∈Σ ∂∂µϕ(y) ∂L ∂L ! = δϕr(x) − ∂µ (3) ∂ϕr(x) ∂∂µϕr(x)

In obtaining the penultimate equality, we used the divergence theorem with µ µ Σ the boundary of Ω having unit (space-like) normal n (n nµ = 1), where the last equality follows since the ﬁeld variations vanish on Σ. Since, other than being constrained to vanish on the boundary, each variation into discrete cells (a lattice) and decompose the ﬁeld which depends on the continuum of spatial variables into a discretely numerated number of degrees of freedom, one associated with each lattice cell, all depending only on time. The Lagrange function and its associated action can then be constructed. The ordinary Hamilton’s principle which states that the action is stationary with respect to variations which vanish at the temporal endpoints, but are otherwise arbitrary, can then be applied. The resultant Euler-Lagrange equations so obtained reproduces the Euler-Lagrange equations obtained with the generalized Hamilton’s principle once the continuum limit is taken.

3 δϕr is arbitrary within Ω, it follows that ∂L ∂L 0 = − ∂µ (4) ∂ϕr(x) ∂∂µϕr(x) which are the Euler-Lagrange equations.

The canonically conjugate momentum classical ﬁeld, πs(x), is deﬁned as ∂L πs(x) = . (5) ∂∂0ϕs(x) Noether’s Theorem and Symmetries

Poincar´einvariance restricts the form of the ﬁeld equations and consequently the form of the actions from which they are derived. In addition, there may be additional symmetries of the system which will also restrict the form of the action. We shall formalize these restrictions and also establish the con- nection between the presence of a continuous symmetry, the existence of a conserved current and its associated time independent charge. This is em- bodied in Noether’s theorem.

Let ϕr(x) denote a multicomponent ﬁeld labeled by r. A transformation which changes the space-time point (eg. translations, Lorentz transformations) is referred to as a space-time transformation while a transformation which leaves the space-time point unchanged is referred to as an internal transformation.

Consider the transformation parametrized by Υ which is a symmetry of the action and is continuously connected to the identity. The classical ﬁeld theory transformation law is

Υ Υ ϕr(x) → ϕr (x ) = Drs(Υ)ϕs(x) (6) so that

Υ Υ−1 ϕ (x) = Drs(Υ)ϕs(x ) (7) where D(Υ) is a matrix representation of the transformation. Here xΥ is the transformed space-time point (if the transformation is internal, then

4 Υ xµ = xµ).

For inﬁnitesimal transformations, parametrized by real υA (cryptically Υ = 1 + υ) with |υA| << 1, the space-time point changes as

µ υ µ µ X µ ∆(υ)x = (x ) − x = υA∆Ax (8) A If the transformation is internal transformation, then ∆(υ)xµ = 0.

For inﬁnitesimal transformations, the matrix representation can be written

X A Drs(I + υ) = δrs + i υA(d )rs (9) A

(A) with drs being υ independent numbers. The intrinsic transformation of the ﬁeld is ϕr(x) → ϕr(x) + δ(υ)ϕr(x) (10) where

υ δ(υ)ϕr(x) = ϕr (x) − ϕr(x) = Drs(I + υ)ϕs(x − ∆(υ)x) − ϕ(x) ! X A = δrs + i υA(d )rs ϕs(x − ∆(υ)x) − ϕr(x) A µ X A = −∆x (υ)∂µϕr(x) + i υA(d )rsϕs(x) A X µ A = υA −δrs∆Ax ∂µ + i(d )rs ϕs(x) A X = υAδAϕr(x) (11) A which identiﬁes

µ A δAϕr(x) = −δrs∆Ax ∂µ + i(d )rs ϕs(x) (12) while δ(υ)∂µϕr(x) = ∂µ δ(υ)ϕr(x) ⇒ δA∂µϕr(x) = ∂µδAϕr(x) (13)

5 Since L = L(ϕr, ∂µϕr), it follows that ∂L ∂L δ(υ)L = δ(υ)ϕr + δ(υ)(∂µϕr) ∂ϕr ∂∂µϕr ∂L ∂L = δ(υ)ϕr + ∂µδ(υ)ϕr ∂ϕr ∂∂µϕr ∂L ∂L ∂L = δ(υ)ϕr + ∂µ δ(υ)ϕr − ∂µ δ(υ)ϕr ∂ϕr ∂∂µϕr ∂∂µϕr ∂L ∂L ! ∂L ! = − ∂µ δ(υ)ϕr + ∂µ δ(υ)ϕr ∂ϕr ∂∂µϕr ∂∂µϕr δS ∂L ! = δ(υ)ϕr + ∂µ δ(υ)ϕr (14) δϕr ∂∂µϕr or δS ∂L ! δ(υ)ϕr = δ(υ)L − ∂µ δ(υ)ϕr(x) . (15) δϕr ∂∂µϕr(x) which is an algebraic identity and constitutes Noether’s theorem. This identity holds for an ﬁeld transformation where it is a symmetry of action or not.

If the transformation is a symmetry of the classical action so that the action is invariant under the transformation,,2 then the Lagrangian is either invariant or transforms into a total divergence. As we shall see, for space-time symmetries, the Lagrangian transforms into a total divergence so that

µ δ(υ)L = ∂µΛ (υ) (18)

2The action is invariant provided it algebraically (independent of ﬁeld equations) sat- isﬁes

δ(υ)S[ϕ] = 0 (16) where Z 4 δ δ(υ) = d x (δ(υ)ϕr(x)) δϕr(x) Z ! 4 X µ A δ = d x υA −δrs∆Ax ∂µ + i(d )rs ϕs(x) (17) δϕr(x) A

6 while for internal symmetries, it is invariant (Λµ(υ) = 0). In either case, the RHS of Noether’s theorem is the divergence of a current

µ µ ∂L X µ J (υ) = Λ (υ) − δ(υ)ϕr(x) υAJA(x) (19) ∂∂µϕr(x) A and Noether’s theorem takes the form

δS[ϕ] µ δ(υ)ϕr(x) = ∂µJ (υ) (20) δϕr(x) or equivalently

δS[ϕ] µ δAϕr(x) = ∂µJA(x) (21) δϕr(x)

After application of the Euler-Lagrange ﬁeld equations: δS[ϕ] = 0, we see δϕr(x) µ µ µ µ that that the current J (υ) or JA(x) is conserved: ∂µJ (υ) = 0 = ∂µJA(x).

For such a conserved current, we can define charges Z 3 0 X Q(υ) = d xJ (υ) = υAQA (22) A such that dQ(υ) Z Z dQ = d3x∂ J 0(υ) = − d3x∂ J i(υ) = 0 ⇒ A = 0 (23) dt 0 i dt where we have used the divergence theorem and set the surface terms at infinity to zero. Thus, after application of the Euler-Lagrange field equations, the Noether constructed charges Q(υ) or QA are time independent. These charges will be identified with the generators of the associated symmetries.

Poincar`eSymmetries

We now apply Noether’s theorem to the space-time symmetries associated with the Poincar´etransformations.

7 Space-Time Translations: Canonical Energy-Momentum Tensor

The inﬁnitesimal space-time translation is given by

∆()xµ = µ (24)

µ µ P µ µ ν µ with | | << 1. Using ∆()x = A υA∆Ax = = gν , we can identify µ µ µ (c.f. Eq.(8)) A → µ, υA → and ∆Ax → gν . The classical ﬁeld inﬁnitesimal intrinsic variation is

µ δ()ϕr(x) = ϕr(x − ) − ϕr(x) = − ∂µϕr(x) (25) Similarly, for a translation invariant action3, the Lagrangian transforms as

µ µν δ()L = − ∂µL = ∂µ(−νg L) (28) so that

µ µν Λ () = −νg L (29) Combining these results gives the conserved Noether current µ µ ∂L J () = Λ () − δ()ϕr(x) ∂∂µϕr(x) ! µν ∂L = −ν g L − ∂νϕr(x) ∂∂µϕr(x) µν = −νT (30) where we have introduced the canonical energy-momentum tensor as

µν µν ∂L ν T = g L − ∂ ϕr (31) ∂∂µϕr

3The action is invariant under space-time translations provided it algebraically (i.e. not using ﬁeld equations) satisﬁes

δ()S[φ] = 0 . (26) where Z Z 4 δ µ 4 δ δ() = d xδ()φr(x) = − d x(∂µφr(x)) . (27) δφr(x) δφr(x)

8 which, after applications of the ﬁeld equations, is conserved as a consequence of Noether’s theorem: δS ν µν 0 = ∂ ϕr = ∂µT (32) δϕr The canonical energy-momentum tensor is the Noether ”current” associated µ µν with space-time translation invariance (JA → T ). Note the conservation is on the ﬁrst index of the canonical energy-momentum tensor.

The charges which are identiﬁed with the space-time translation generators have the Noether construction Z P ν = d3xT 0ν(t, ~r) (33)

dP ν and are time independent dt = 0. These charges are identiﬁed as the Hamiltonian Z H = P 0 = d3xT 00(t, ~r) Z 3 = d x(−L(x) + πr(x)∂0ϕr(x)) (34) which generates time translations and the linear momentum Z P i = d3xT 0i(t, ~r) Z 3 = − d xπr(x)∂iϕr(x) (35) which generates space translations.

Note that the Hamiltonian so constructed is a 3-space integral, H = R d3xH, of the Hamiltonian density H which is the Legendre transform of the La- grangian density L = L(ϕr, ∂µφr). As such, the Hamiltonian density, H = H(ϕr, πr), is a natural function of the ﬁelds, ϕr and their conjugate momenta ﬁelds, πr.

Lorentz Transformations: Improved Energy-Momentum Tensor

The inﬁnitesimal Lorentz transformation is

ν ∆(ω)xµ = ωµνx (36)

9 with ωµν = −ωνµ and |ωµν| << 1.

µ P µ µν 1 µρ σ µσ ρ Using ∆(ω)x = A υA∆Ax = ω xν = 2 ωρσ(g x − g x ), we can iden- 1 µ µρ σ µσ ρ tify A → µν and υA → 2 ωµν, while ∆Ax → g x − g x .

The representation matrices are denoted d(A) → Sµν = −Sνµ so that for infinitesimal Lorentz transformations For infinitesimal transformations, i D (I + ω) = δ + ω (Sµν) (37) rs rs 2 µν rs The classical field infinitesimal transformation is thus ωµν δ(ω)ϕ (x) = δ (x ∂ − x ∂ ) + i(S ) ϕ (x) (38) r 2 rs µ ν ν µ µν rs s Since the action is a Lorentz invariant4, the Lagrangian carries the trivial representation (Sµν = 0) and transforms as ωµν δ(ω)L = (x ∂ − x ∂ )L 2 µ ν ν µ µν = ∂µ(−ω xνL) µ = ∂µΛ (ω) (41) with

µ µν Λ (ω) = −ω xνL ωρσ = (gµx − gµx )L (42) 2 σ ρ ρ σ 4The action is Lorentz invariant provided that it algebraically (i.e. not using ﬁeld equations) satisﬁes δ(ω)S[φ] = 0 (39) where Z 4 δ δ(ω) = d xδ(ω)φr(x) δφr(x) Z µν 4 ω δ = d x (δrs(xµ∂ν − xν ∂µ) + i(Sµν )rs) φs(x) (40) 2 δφr(x)

10 The conserved Noether current for Lorentz transformations then takes the form

µ µ ∂L J (ω) = Λ (ω) − δ(ω)ϕr(x) ∂∂µϕr(x) ρσ ! ω µ µ ∂L = (gσ xρ − gρ xσ)L − δrs(xρ∂σ − xσ∂ρ) + i(Sρσ)rs ϕs(x) 2 ∂∂µϕr(x) ρσ ω µ µ ∂L = xρT σ − xσT ρ − i (Sρσ)rsϕs(x) (43) 2 ∂∂µϕr(x)

The term containing the Sρσ matrix can be rewritten by adding a term symmetric under the ρ ↔ σ interchange. Such a term gives no contribution since it is contracted with ωρσ = −ωσρ. Thus we can write ωρσ i ∂L (Sρσ) ϕ = ωρσ i(ϕ ∂L Sρσ) +i ∂L (Sσµ) ϕ +i ∂L (Sρµ) ϕ 2 ∂∂µϕr rs s 2 s ∂∂µϕr rs ∂∂ρϕr rs s ∂∂σϕr rs s

Then deﬁning hµρσ = i ∂L (Sρσ) ϕ + i ∂L (Sσµ) ϕ + i ∂L (Sρµ) ϕ = −hρµσ 2 ∂∂µϕr rs s 2 ∂∂ρϕr rs s 2 ∂∂σϕr rs s where the last equality follows from the antisymmetry of Sµν, we have

ωρσ i ∂L (S ) ϕ = ω hµρσ 2 ∂∂µϕr ρσ rs s ρσ and the conserved Noether current for Lorentz transformations reads ω J µ(ω) = ρσ xρT µσ − xσT µρ − hµρσ + hµσρ (44) 2 where we have antisymmetrized hµρσ in ρ ↔ σ since it is contracted with ωρσ.

Now deﬁne the improved (Belinfante) energy-momentum tensor as

µν µν µλν Θ = T + ∂λh . (45)

µλν λµν µλν Since, by construction, h = −h , it follows that algebraically ∂µ∂λh = 0 and consequently, after application of the ﬁeld equations, that

µν µν ∂µΘ = ∂µT = 0 (46)

11 so the improved energy-momentum tensor is conserved on the ﬁrst index. Moreover, using that ρ µσ σ µρ ρ µσ σ µρ µρσ µσρ ρ µλσ σ µλρ x Θ − x Θ = x T − x T − h + h + ∂λ(x h − x h ) along with hµλσ = −hλµσ, it follows that ω ω ρσ ∂ (xρΘµσ − xσΘµρ) = ρσ ∂ (xρT µσ − xσT µρ − hµρσ + hµσρ) 2 µ 2 µ µ = ∂µJ (ω) (47) where J µ(ω) is the conserved Noether current for Lorentz transformations. Thus we can deﬁne the improved, conserved current for Lorentz transformations as ω M µ(ω) = ρσ ∂ M ρµσ (48) 2 µ with the conserved angular momentum tensor

M ρµσ = xρΘµσ − xσΘµρ = −M σµρ (49)

ρµσ satisfying ∂µM = 0 after application of the ﬁeld equations. Using this conservation law in conjunction with the conservation of the improved energy- momentum tensor gives

µλν µ λν ν λµ 0 = ∂λM = ∂λ(x Θ − x Θ ) µν µ λν νµ ν λµ = Θ + x ∂λΘ − Θ − x ∂λΘ = Θµν − Θνµ (50) so that the improved energy-momentum tensor is symmetric:

Θµν = Θνµ (51) and is thus conserved on both indices. Note that if only scalar ﬁelds are present, then Sµν = 0 ⇒ hµλν = 0. Conse- quently, in this case, Θµν = T µν and T µν = T νµ.

The angular momentum generating the Lorentz transformations is obtained via the Noether construction as Z J µν = d3xM µ0ν (52)

12 dJµν and is time independent: dt = 0.

We close this section by showing that for a classical field theory, the operator charges representing the Poincarégenerators can constructed using the improved energy-momentum tensor. To do so, define Z µ 3 0µ PΘ = d xΘ Z µ 3 0µ PT = d xT Z µν 3 µ 0ν ν 0µ JΘ = d x(x Θ − x Θ ) Z ! µν 3 µ 0ν ν 0µ ∂L µν JT = d x x T − x T − i (S )rsϕs (53) ∂∂0ϕr where

µν µν X ∂L ν T = g L − ∂ ϕa (54) a ∂∂µϕa and

µν µν µρν Θ = T + ∂ρh (55) with ! µρν i ∂L rρν ∂L ρµ ∂L νµ h = r Srs ϕs + Srs ϕs + Srs ϕs 2 ∂∂µϕr ∂∂νϕr ∂∂ρϕr (56) Now consider Z Z µ µ 3 0µ 0µ 3 0ρµ Pθ − PT = d x(Θ − T ) = d x∂ρh (57)

However since hµνρ = −hνµρ, it follows that h00ρ = 0. Therefore, Z µ µ 3 0iµ Pθ − PT = d x∂ih ; i = 1, 2, 3 (58)

Using the divergence theorem and assuming the fields fall sufficiently fast at spatial infinity, the RHS vanishes and hence

µ µ µ Pθ = PT ≡ P (59)

13 Next consider

Z ! µν µν 3 µ 0ρν ν 0ρµ ∂L µν JΘ − JT = d x x ∂ρh − x ∂ρh + i (S )rsϕs (60) ∂∂0ϕr

Integrate the ﬁrst two terms on the RHS by parts and drop the surface terms giving Z ! µν µν 3 0µν 0νµ ∂L µν JΘ − JT = d x −h + h + i (S )rsϕs (61) ∂∂0ϕr Substituting the expression for hµνρ gives

Z µν µν 3 i ∂L µν ∂L µ0 ∂L ν0 JΘ − JT = d x − (S )rsϕs − (S )rsϕs − (S )rsϕs 2 ∂∂0ϕr ∂∂νϕr ∂∂µϕr ∂L νµ ∂L ν0 ∂L µ0 + (S )rsϕs + (S )rsϕs + (S )rsϕs ∂∂0ϕr ∂∂µϕr ∂∂νϕr ∂L µν + 2 (S )rsϕa ∂∂0ϕa Z 3 i ∂L µν νµ = d x S )rsϕs + (S )rsϕs (62) 2 ∂∂0ϕr But Sµν = −Sνµ and hence

µν µν µν JΘ = JT ≡ J (63)

14 Canonical Quantization

To canonically quantize the classical field theory, we replace the classical field ϕr(x) by operator fields φr(x) so that the quantum action functional is Z 4 S[φs] = d xL(φs(t, ~r), ∂µφs(t, ~r)) (64) Ω and the operator field equations are δS ∂L ∂L 0 = = − ∂µ (65) δφs(x) ∂φs(x) ∂∂µφs(x) In addition to the field equations, the operator fields and their canonically conjugate momentum operators are subjected to certain equal time commutation or equal time anti-commutation relations.

Focusing on bosonic ﬁelds, we impose the equal time commutation relations

[φr(t, ~x), φs(t, ~y)] = 0 [πr(t, ~x), πs(t, ~y)] = 0 3 [φr(t, ~x), πs(t, ~y)] = iδrsδ (~x − ~y) (66) where the canonical momentum operator is ∂L πr(t, ~x) = (67) ∂∂0φr(t, ~x) Note that as a consequence of these relations, it follows that

[∂iφr(t, ~x), φs(t, ~y)] = 0 [∂xi φr(t, ~x), ∂yj φs(t, ~y)] = 0 (68) We reiterate that the quantum operator Lagrangian is obtained from the classical field theory Lagrangian by simply replacing the classical fields by the quantum operator fields. In fact, this seemingly innocuous step is actually highly non-trivial. This follows since L contains products of operator fields and their derivatives at the same space-time point where certain equal time commutation relations are non-zero and singular. Thus great care is actually required in attempting to define these products of fields at coincident space- time points. We shall have a great deal to say about how to define products

15 of operator ﬁelds at coincident space-time points throughout the course. For the time being, we shall simply proceed in a somewhat cavalier fashion and when we encounter a problem resulting from this singular structure, we shall attempt to make a sensible interpretation.

In the canonical procedure, Noether currents and charges are also promoted to be quantum mechanical operators. The charges are identified with the generators of the respective symmetries. This identification needs to be checked model by model to insure that the Noether charges so constructed indeed have the proper commutation relations with the fields as required by the symmetry.

Since the momentum and Hamiltonian operators are time independent, we can evaluate all the ﬁelds appearing in their Noether construction to be at the same time as the ﬁeld with which we are commuting. Proceeding formally, we have Z 3 [Pi, φr(t, ~x)] = − d y[πs(t, ~y)∂yi φs(t, ~y), φr(t, ~x)] Z 4 3 = − d y(−i)δrsδ (~x − ~y)∂yi φs(t, ~y)

= i∂iφs(t, ~x) (69) which is precisely what we expect the momentum operator to do.

Next consider Z 3 [H, φr(t, ~x)] = d y[−L(φ(t, ~y), ∂µφ(t, ~y)) + πs(t, ~y)∂0φs(t, ~y), φr(t, ~x)] Z 3 ∂L = d y − [φs(t, ~y), φr(t, ~x)] ∂φs(t, ~y) ∂L y − y [∂i φs(t, ~y), φr(t, ~x)] ∂i φs(t, ~y) ∂L − [∂0φs(t, ~y), φr(t, ~x)] ∂0φs(t, ~y) + [πs(t, ~y), φr(t, ~x)]∂0φs(t, ~y) + πs(t, ~y)[∂0φs(t, ~y), φr(t, ~x)] Z 3 = d y − πs(t, ~y)[∂0φs(t, ~y), φr(t, ~x)]

16 + [πs(t, ~y), φr(t, ~x)]∂0φs(t, ~y) + πs(t, ~y)[∂0φs(t, ~y), φr(t, ~x)] Z 3 3 = d y(−i)δrsδ (~x − ~y)∂0φs(t, ~y)

= −i∂0φr(t, ~x) (70) which is recognized as the Heisenberg equation of motion and is precisely the relationship satisﬁed by the generator of time translations.