Bridge to Quantum Field Theory

Quantum field theory keeps the Hilbert-space and symmetry principles of quantum mechanics and adds local fields, relativistic spacetime structure, and potentially infinitely many degrees of freedom. Familiar tools then acquire new uses: angular momentum helps classify particles, local currents describe charge flow, and symmetry relations constrain correlation functions. This chapter explains those connections before the full machinery of field quantization and renormalization.

Helpful background. Symmetry principles, continuous symmetries, and spinors supply the quantum-mechanical starting points. The Dynamics bridge explains why correlation functions become central.

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Question	Read next
What changes when the degrees of freedom are fields?	Why symmetry becomes central
How does charge conservation become local?	Generators and Noether currents
How do spacetime transformations classify particles?	Spin and relativistic representations and helicity
How do field components transform under boosts?	SU(2) and Lorentz spinors
What is added when the gauge field becomes dynamical?	Phase symmetry and gauge theory
How can geometric phase enter an action?	Berry phase and topological terms
How does symmetry constrain many-operator averages?	Selection rules and Ward identities
Which relativistic assumptions lead to CPT?	Discrete symmetries and CPT
How does broken symmetry produce low-energy particles?	Symmetry breaking and Goldstone modes
How can quantization change a symmetry’s implementation?	Projective representations and anomalies

A local current contains more information than a conserved number

In quantum mechanics, a time-independent generator with $[G,H]=0$ is conserved, with the usual domain qualifications. In a local field theory, a continuous variational symmetry of a suitable action gives a Noether current satisfying a local conservation equation on the equations of motion. An arbitrary conserved operator does not automatically come with such a local current.

Write the continuity equation as

$$\partial_t\rho+\nabla\cdot\mathbf j=0.$$

For a fixed spatial region $R$,

$$Q_R=\int_R\rho\,d^3x, \qquad \frac{dQ_R}{dt}=-\int_{\partial R}\mathbf j\cdot d\mathbf S.$$

Charge can leave a subregion without being destroyed. Conservation of the charge over all space needs controlled flux at infinity; defining the quantum charge also needs a suitable operator limit. In a spontaneously broken phase, formal spatial integrals of the current can be subtle even though local current relations remain useful.

Particle states and field components are different spaces

Proper orthochronous Poincaré transformations act unitarily on physical states. In a massive one-particle representation, the transformations leaving a rest momentum fixed form the rotation little group, whose double cover supplies the familiar spin labels. A massless particle has no rest frame. Its familiar finite-helicity representations require a different little-group analysis; continuous-spin representations are not included in this elementary account.

Fields also have component indices. A Weyl spinor has two components mixed by finite Lorentz matrices. For one chirality a boost along $z$ involves a matrix of the form $e^{\eta\sigma_z/2}$, up to the choice of active convention. Its eigenvalues $e^{\pm\eta/2}$ are not unit-modulus phases. That does not violate unitarity: the component matrix is not the Hilbert-space operator acting on quantum states. A relativistic field transformation combines the component matrix, the transformed spacetime argument, and a unitary state-space action.

For nonzero momentum, dimensionless helicity is

$$\Lambda=\frac{\mathbf J\cdot\mathbf P}{\hbar|\mathbf P|}.$$

It is a useful invariant label under proper orthochronous transformations for massless finite-helicity particles. For a massive particle, a change of inertial frame can reverse its momentum without reversing its spin; helicity can change while the particle’s spin classification does not. Chirality instead labels Lorentz-spinor structure. Equality between helicity and chirality for particular massless solutions does not make them identical definitions.

Symmetry constrains correlation functions

Suppose a well-defined charge obeys $[Q,O_k]=\hbar q_kO_k$ and the state $|\Omega\rangle$ is a charge eigenstate. Using the product rule for a commutator,

$$0=\langle\Omega|[Q,O_1\cdots O_n]|\Omega\rangle =\hbar\left(\sum_kq_k\right) \langle\Omega|O_1\cdots O_n|\Omega\rangle.$$

The expectation vanishes unless the inserted charges sum to zero. The charge eigenvalue of $|\Omega\rangle$ need not itself be zero: it cancels between the two terms of the commutator. A symmetry-breaking state, external source, or ill-defined charge can invalidate this argument.

Local Ward identities contain more information. For two bosonic insertions, differentiating the time-ordering step functions gives schematically

$$\partial_t\langle T\,\rho(t,\mathbf x)O(t')\rangle ={}\langle T\,\partial_t\rho(t,\mathbf x)O(t')\rangle +\delta(t-t')\langle[\rho(t,\mathbf x),O(t')]\rangle.$$

The last term is concentrated where the insertion times meet. Together with spatial current conservation, such terms express how the inserted operators transform. Renormalized products can require further contact-term care. Dropping them would erase part of the symmetry relation, rather than merely simplify it.

Phase freedom does not by itself create electromagnetism

Three uses of phase must be distinguished. Multiplying one state vector by a common phase leaves its ray unchanged. A global internal transformation can instead act differently on states with different charges and has physical consequences. A local gauge transformation changes fields and connections together as an equivalence of descriptions, with possible physical boundary transformations treated separately.

In charged-particle quantum mechanics, prescribed potentials make local phase covariance concrete. Turning them into dynamical electromagnetic fields requires additional input, including a field action, constraints, and a quantum treatment of the field. Neither global ray equivalence nor writing a covariant derivative uniquely determines all those dynamics.

A geometric contribution to phase can also become an action term. For example, if closed histories have an integer winding $w$, a term $S_\theta=\hbar\theta w$ weights them by $e^{i\theta w}$. The integer makes this closed-history phase periodic under $\theta\mapsto\theta+2\pi$. Berry phases prepare the idea, but a generic Berry phase is not quantized; field-theory topological terms need their own global and boundary analysis.

Relativistic theorems and quantum obstructions

The CPT theorem is a result about local relativistic quantum field theory under precise assumptions including Lorentz covariance, positivity and spectral conditions, and appropriate locality. It is not a theorem about an arbitrary finite-dimensional Hamiltonian. Separate $C$, $P$, or $T$ invariance is not required by CPT. See Streater and Wightman for an axiomatic treatment.

Goldstone’s theorem concerns a broken continuous global symmetry in an appropriate infinite-volume setting. Under the standard relativistic locality and vacuum assumptions, conserved-current relations imply gapless excitations. Nonrelativistic counting differs. A gauge-dependent field expectation value does not by itself prove a physical broken global symmetry or a separate massless particle.

Finally, the word anomaly covers distinctions worth preserving. A classical current can fail to be conserved after quantization, as in the chiral anomaly. An exact global symmetry can instead have a ’t Hooft anomaly: it cannot be consistently promoted to a gauge symmetry without additional structure. An uncanceled anomaly of a dynamical gauge symmetry obstructs a standalone gauge theory. A “global gauge anomaly” concerns large gauge transformations and is not simply an anomaly of a global physical symmetry.

Projective representations, such as spin one half under $SO(3)$, are consistent actions on rays. They are not automatically an inconsistency or current nonconservation. In suitable settings their phases can describe an obstruction to gauging a symmetry, but that conclusion requires the additional setup. The anomaly bridge explains the relationship.

For a first path, work through currents and the correlation-function selection rule. For particle physics, continue with spin, helicity, and Lorentz spinors. For geometric or many-body interests, follow the gauge, topological-term, and Goldstone discussions.

Exercises

1. Local conservation and boundary flux

Let a current satisfy

$$\partial_t j^0+\nabla\cdot\mathbf j=0$$

inside a spatial region $R$. Derive the time derivative of

$$Q_R(t)=\int_R d^3x\,j^0.$$

When is $Q_R$ conserved?

Solution

Integrating the continuity equation gives

$$\begin{aligned} \frac{dQ_R}{dt} &= -\int_R d^3x\, \nabla\cdot\mathbf j \\ &= -\int_{\partial R} d\mathbf S\cdot\mathbf j, \end{aligned}$$

where the divergence theorem was used. The charge in $R$ is conserved when the net outward flux through $\partial R$ vanishes. For all space, sufficient falloff of the current at infinity is the usual condition. Local conservation allows charge to move; it does not require the charge inside every subregion to be constant.

2. Nonunitary field matrices and unitary physics

Finite-dimensional Lorentz boost matrices acting on Weyl or Dirac components are not unitary in the ordinary Euclidean component norm. Explain why this does not violate the unitarity of quantum theory.

Solution

The two transformations act on different spaces. A finite matrix $D(\Lambda)$ mixes the Lorentz or spinor components of a field. The Lorentz group is noncompact, so its nontrivial finite-dimensional representations are not unitary in a positive-definite component norm.

The operator $U(\Lambda)$ acting on the physical Hilbert space of states is unitary for proper orthochronous Lorentz transformations. Probabilities and inner products are preserved by this Hilbert-space action. Covariant field transformation laws relate $U(\Lambda)$ and $D(\Lambda)$ without identifying them.

3. Helicity under parity

Using

$$\Lambda = \frac{\mathbf J\cdot\mathbf P} {\hbar|\mathbf P|},$$

determine how helicity transforms under parity. What does this imply for a massless particle sector containing only one nonzero helicity?

Solution

Momentum is a polar vector, so parity sends

$$\mathbf P\mapsto-\mathbf P.$$

Angular momentum is an axial vector, so

$$\mathbf J\mapsto\mathbf J.$$

Consequently

$$\Lambda\mapsto-\Lambda.$$

Parity pairs opposite helicities. A massless theory containing a state of nonzero helicity but no corresponding opposite-helicity state cannot realize parity as a symmetry within that particle sector.

References

• S. Coleman, Aspects of Symmetry, Cambridge University Press, 1985.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison–Wesley, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, revised paperback edition, Princeton University Press, 2000; originally W. A. Benjamin, 1964.
• D. Tong, Gauge Theory, lecture notes, University of Cambridge, 2018, chapter 3, “Anomalies.” Author’s notes.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.
• E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Mathematics 40, 149–204, 1939.