44.1 Internal symmetries

The relativistic real scalar field theory of chapter 43 lacks one important feature of the non-relativistic theory, which is an action of the group $U (1)$ by phase changes on complex fields. This is needed to provide a notion of $^{66} charg e^{35}$ and allow the introduction of electromagnetic forces into the theory (see chapter 45). In the real scalar field theory there is no distinction between states describing particles and states describing antiparticles. To get a theory with such a distinction we need to introduce fields with more components. Two possibilities are to consider real fields with $m$ components, in which case we will have a theory with $S O (m)$ symmetry, or to consider complex fields with $n$ components, in which case we have a theory with $U (n)$ symmetry. Identifying C with $R^{2}$ using the standard complex structure, we find $S O (2) = U (1)$ , and two equivalent ways of getting a theory with $U (1)$ symmetry, using two real or one complex scalar field.

44.1.1 SO(m) symmetry and real scalar fields

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · unitary representation · irreducible representation · group action

Starting with the case $m = 2$ , and taking as dual phase space $M$ the space of pairs $ϕ_{1}, ϕ_{2}$ of real solutions to the two-component Klein-Gordon equation, elements $g (θ)$ of the group $S O (2)$ will act on the fields by

(Φ _{2} ( x ) Φ _{1} ( x )) \to g (θ) \cdot (Φ _{2} ( x ) Φ _{1} ( x )) = (cos θ - sin θ sin θ cos θ) (Φ _{2} ( x ) Φ _{1} ( x ))

(Π _{2} ( x ) Π _{1} ( x )) \to g (θ) \cdot (Π _{2} ( x ) Π _{1} ( x )) = (cos θ - sin θ sin θ cos θ) (Π _{2} ( x ) Π _{1} ( x ))

Here $Φ_{1} (x), Φ_{2} (x), Π_{1} (x), Π_{2} (x)$ are the continuous basis elements for the space of two-component Klein-Gordon solutions, determined by their initial values at $t = 0$

This group action on $M$ breaks up into a direct sum of an infinite number (one for each value of $x)$ of identical copies of the case of rotations in a configuration space plane, as discussed in section 20.3.1. We will use the calculation there, where we found that for a basis element $L$ of the Lie algebra of $S O (2)$ the corresponding quadratic function on the phase space with coordinates $q_{1}, q_{2}, p_{1}, p_{2}$ was

μ_{L} = q_{1} p_{2} - q_{2} p_{1}

For the case here, we take

q_{1}, q_{2}, p_{1}, p_{2} \to Φ_{1} (x), Φ_{2} (x), Π_{1} (x), Π_{2} (x)

and integrate the analog of $μ_{L}$ over $R^{3}$ to get an appropriate moment map for the field theory case. This gives a quadratic functional on the fields that will have the desired Poisson bracket with the fields for each value of x. We will denote the result by $Q .$ , since it is an observable that will have a physical interpretation as electric charge when this theory is coupled to the electromagnetic field (see chapter 45):

Q = \int_{R^{3}} (Π_{2} (x) Φ_{1} (x) - Π_{1} (x) Φ_{2} (x)) d^{3} x

One can use the field Poisson bracket relations

{Φ_{j} (x), Π_{k} (x^{'})} = δ_{j k} δ (x - x^{'})

to check that

{Q, (Φ _{2} ( x ) Φ _{1} ( x ))} = (Φ _{1} ( x ) - Φ _{2} ( x )), {Q, (Π _{2} ( x ) Π _{1} ( x ))} = (Π _{1} ( x ) - Π _{2} ( x ))

Quantization of the classical field theory gives a unitary representation $U$ of $S O (2)$ on the multi-particle state space, with

U^{'} (L) = - i Q = - i \int_{R^{3}} (Π_{2} (x) Φ_{1} (x) - Π_{1} (x) Φ_{2} (x)) d^{3} x

The operator

U (θ) = e^{- i θ Q}

will act by conjugation on the fields:

U (θ) (Φ _{2} ( x ) Φ _{1} ( x )) U (θ)^{- 1} = (cos θ - sin θ sin θ cos θ) (Φ _{2} ( x ) Φ _{1} ( x ))

U (θ) (Π _{2} ( x ) Π _{1} ( x )) U (θ)^{- 1} = (cos θ - sin θ sin θ cos θ) (Π _{2} ( x ) Π _{1} ( x ))

It will also give a representation of $S O (2)$ on states, with the state space decomposing into sectors each labeled by the integer eigenvalue of the operator $Q$ (which will be called the “charge” of the state).

Using the definitions of $Φ$ and $\hat{Π}$ (43.19 and 43.20), $Q$ can be computed in terms of annihilation and creation operators, with the result

Q = i \int_{R^{3}} (a_{2}^{†} (p) a_{1} (p) - a_{1}^{†} (p) a_{2} (p)) d^{3} p (44.1)

One expects that since the time evolution action on the classical field space commutes with the $S O (2)$ action, the operator $Q$ should commute with the Hamiltonian operator $H$ . This can readily be checked by computing $[H, Q]$ using

H = \int_{R^{3}} ω_{p} (a_{1}^{†} (p) a_{1} (p) + a_{2}^{†} (p) a_{2} (p)) d^{3} p

Note that the vacuum state |0⟩ is an eigenvector for $Q$ and $H$ with both eigenvalues 0: it has zero energy and zero charge. States $a_{1}^{†} (p) ∣0 ⟩$ and $a_{2}^{†} (p) ∣ 0$ are eigenvectors of $H$ with eigenvalue and thus energy $ω_{p}$ , but these are not eigenvectors of $Q .$ , so do not have a well-defined charge.

All of this can be generalized to the case of m $> 2$ real scalar fields, with a larger group $S O (m)$ now acting instead of the group $S O (2)$ . The Lie algebra is now multi-dimensional, with a basis the elementary antisymmetric matrices $ϵ_{j k}$ 2 with $j, k = 1, 2, \dots, m$ and $j < k$ , which correspond to infinitesimal rotations in the $j k$ planes. Group elements can be constructed by multiplying rotations $e^{θ ϵ_{j k}}$ in diferent planes. Instead of a single operator $Q .$ , we get multiple operators

- i Q_{j k} = - i \int_{R^{3}} (Π_{k} (x) Φ_{j} (x) - Π_{j} (x) Φ_{k} (x)) d^{3} x

and conjugation by

U_{j k} (θ) = e^{- i θ Q_{j k}}

rotates the field operators in the $j k$ plane. These also provide unitary operators on the state space, and, taking appropriate products of them, a unitary representation of the full group $S O (m)$ on the state space. The $Q_{j k}$ commute with the Hamiltonian (generalized to the m-component case) so the energy eigenstates of the theory break up into irreducible representations of $S O (m) (a$ subject we haven’t discussed for $m > 3)$ .

44.1.2 U(1) symmetry and complex scalar fields

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · unitary representation · charge operator · U(1) · complexification

Instead of describing a scalar field system with $S O (2)$ symmetry using a pair $Φ_{1}, Φ_{2}$ of real fields, it is sometimes more convenient to work with complex scalar fields and a $U (1)$ symmetry. This will also allow the use of field operators and annihilation and creation operators for states with a definite value of the charge observable. Taking as $M$ the complex vector space of complex solutions to the Klein-Gordon equation however is confusing, since the Bargmann-Fock quantization method requires that we complexify $M$ , and the complexification of a complex vector space is a notion that requires some care. More simply, here one can think of the space $M$ of solutions to the Klein-Gordon equation for a pair of real fields as having two diferent complex structures:

• The relativistic complex structure $J_{r}$ , which is + $i$ on positive energy solutions in $M \otimes C$ and − $i$ on negative energy solutions in $M \otimes C$

• The “charge” complex structure $J_{C}$ , which is $+ i$ on positive charge solutions and − $i$ on negative charge solutions.

The operators $J_{r}$ and $J_{C}$ will commute, so we can simultaneously diagonalize them on $M \otimes C$ , and decompose the positive energy solution space into $\pm i$ eigenspaces of $J_{C}$ , so

H_{1} = M_{J_{r}}^{+} = H_{1}^{+} \oplus H_{1}^{-}

where $H_{1}^{+}$ will be positive energy solutions with $J_{C} = + i$ and $H_{1}^{-}$ will be positive energy solutions with $J_{C} = - i$ . Taking as before $α_{1} (p), α_{2} (p)$ for the momentum space initial data for elements of $H_{1}$ , we will define

α (p) = \frac{1}{2} (α_{1} (p) - i α_{2} (p)) \in H_{1}^{+}, β (p) = \frac{1}{2} (α_{1} (p) + i α_{2} (p)) \in H_{1}^{-}

The negative energy solution space can be decomposed as

M_{J_{r}}^{-} = \overline{H}_{1} = \overline{H}_{1}^{+} \oplus \overline{H}_{1}^{-}

and

\overline{α} (p) = \frac{1}{2} (\overline{α}_{1} (p) + i \overline{α}_{2} (p)) \in H_{1}^{+}, \overline{β} (p) = \frac{1}{2} (\overline{α}_{1} (p) - i \overline{α}_{2} (p)) \in H_{1}^{-}

We will write $A (p), B (p)$ for the solutions $α, β$ with initial data deltafunctions at $p, \overline{A} (p), \overline{B} (p)$ for their conjugates, and quantization will take

A (p) \to a^{†} (p) = \frac{1}{2} (a_{1}^{†} (p) - i a_{2}^{†} (p))

B (p) \to b^{†} (p) = \frac{1}{2} (a_{1}^{†} (p) + i a_{2}^{†} (p))

\overline{A} (p) \to a (p) = \frac{1}{2} (a_{1} (p) + i a_{2} (p))

\overline{B} (p) \to b (p) = \frac{1}{2} (a_{1} (p) - i a_{2} (p))

with the non-zero commutation relations between these operators given by

[a (p), a^{†} (p^{'})] = δ (p - p^{'}), [b (p), b^{†} (p^{'})] = δ (p - p^{'})

The state space of this theory is a tensor product of two copies of the state space of a real scalar field. The operators $a^{†} (p), a (p)$ act on the state space by creating or annihilating a positively charged particle of momentum p, whereas the $b^{†} (p), b (p)$ create or annihilate antiparticles of negative charge. The vacuum state will satisfy

a (p) ∣0 ⟩ = b (p) ∣0 ⟩ = 0

The Hamiltonian operator for this theory will be

H = \int_{R^{3}} ω_{p} (a^{†} (p) a (p) + b^{†} (p) b (p)) d^{3} p

and the charge operator is

Q = \int_{R^{3}} (a^{†} (p) a (p) - b^{†} (p) b (p)) d^{3} p

Using these creation and annihilation operators, we can define position space field operators analogous to the ones given by equations 43.19 and 43.20 in the real scalar field case. Now $Φ (x)$ will not be self-adjoint, but its adjoint will be a field $Φ^{†} (x)$ which will act on states by increasing the charge by 1, with one term that creates particles and another that annihilates antiparticles. We define

Definition (Complex scalar quantum field)

The complex scalar quantum field operators are the operator-valued distributions defined by
$Φ (x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} (a (p) e^{i p \cdot x} + b^{†} (p) e^{- i p \cdot x}) \frac{d ^{3} p}{2 ω _{p}}$ $Φ^{†} (x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} (b (p) e^{i p \cdot x} + a^{†} (p) e^{- i p \cdot x}) \frac{d ^{3} p}{2 ω _{p}}$ $Π (x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} (- i ω_{p}) (a (p) e^{i p \cdot x} - b^{†} (p) e^{- i p \cdot x}) \frac{d ^{3} p}{2 ω _{p}}$ $Π^{†} (x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} (- i ω_{p}) (b (p) e^{i p \cdot x} - a^{†} (p) e^{- i p \cdot x}) \frac{d ^{3} p}{2 ω _{p}}$

These satisfy the commutation relations

[Φ (x), Π^{†} (x^{'})] = [Π (x), Π^{†} (x^{'})] = [Φ (x), Π^{†} (x^{'})] = [Φ^{†} (x), Π (x^{'})] = 0

[Φ (x), Π (x^{'})] = [Φ^{†} (x), Π^{†} (x^{'})] = i δ^{3} (x - x^{'}) (44.2)

In terms of these field operators, the Hamiltonian operator will be

H = \int_{R^{3}} : (Π^{†} (x) Π (x) + (\nabla Φ^{†} (x)) (\nabla Φ (x)) + m^{2} Φ^{†} (x) Φ (x)) : d^{3} x

and the charge operator will be

Q = - i \int_{R^{3}} : (Π (x) Φ (x) - Π^{†} (x) Φ^{†} (x)) : d^{3} x

Taking $L = i$ as a basis element for $u (1)$ , one gets a unitary representation $U$ of $U (1)$ using

U^{'} (L) = - i Q

and

U (θ) = e^{- i θ Q}

$U$ acts by conjugation on the fields:

U (θ) Φ U (θ)^{- 1} = e^{- i θ} Φ, U (θ) Φ^{†} U (θ)^{- 1} = e^{i θ} Φ^{†}

U (θ) Π U (θ)^{- 1} = e^{i θ} Π, U (θ) Π^{†} U (θ)^{- 1} = e^{- i θ} Π^{†}

It will also give a representation of $U (1)$ on states, with the state space decomposing into sectors each labeled by the integer eigenvalue of the operator $Q .$

Instead of starting in momentum space with solutions given by $A (p), B (p)$ ， we could instead have considered position space initial data and distributional fields

Φ (x) = \frac{1}{2} (Φ_{1} (x) + i Φ_{2} (x)), Π (x) = \frac{1}{2} (Π_{1} (x) - i Π_{2} (x)) (44.3)

and their complex conjugates $\overline{Φ} (x), \overline{Π} (x)$ . The Poisson bracket relations on such complex fields will be

{Φ (x), \overline{Φ} (x^{'})} {Φ (x), Π (x^{'})} = {Π (x), \overline{Π} (x^{'})} = {Φ (x), \overline{Π} (x^{'})} = {\overline{Φ} (x), Π (x^{'})} = 0 = {\overline{Φ} (x), \overline{Π} (x^{'})} = δ (x - x^{'})

and the classical Hamiltonian is

h = \int_{R^{3}} (∣Π ∣^{2} + ∣\nablaΦ ∣^{2} + m^{2} ∣Φ ∣^{2}) d^{3} x

The charge function $Q$ would be given by

Q = - i \int_{R^{3}} (Π (x) Φ (x) - \overline{Π} (x) \overline{Φ} (x)) d^{3} x (44.4)

satisfying

{Q, Φ (x)} = i Φ (x), {Q, \overline{Φ} (x)} = - i \overline{Φ} (x)

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 473、474、475、476、477、478、479、480、481、482、483

来源版本：2025-10-20

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书架

44.1 Internal symmetries