Chapter 47: Field Quantization and ψ-Operators

The Living Mathematics of Creation and Annihilation

Quantum field theory—where particles are excitations of operator-valued fields—emerges from ψ = ψ(ψ) as the natural algebra of self-referential dynamics. Creation and annihilation operators are not abstract tools but necessary mathematical structures when ψ patterns can appear and disappear while maintaining recursive consistency. The canonical commutation relations follow from fundamental requirements of ψ-self-reference.

47.1 From Classical to Quantum Fields

Classical Field: φ(x,t) with definite values

The Problem: How to quantize?

Theorem: Field quantization is necessary for consistent ψ-recursion.

Proof: Consider ψ-field satisfying ψ = ψ(ψ). For multiple ψ-excitations to exist:

1. Must allow variable particle number
2. Requires operators creating/destroying excitations
3. Fixed-particle quantum mechanics insufficient Thus field operators necessary. ∎

Canonical Quantization: Promote fields to operators: $\phi(x,t) \rightarrow \hat{\phi}(x,t)$ $\pi(x,t) \rightarrow \hat{\pi}(x,t)$

47.2 Deriving Creation/Annihilation Operators

Mode Expansion: For free field: $\hat{\phi}(x,t) = \int \frac{d^3k}{(2\pi)^3\sqrt{2\omega_k}}\left[a_k e^{-ik\cdot x} + a_k^\dagger e^{ik\cdot x}\right]$

Theorem: Operators a_k, a_k† necessarily satisfy: $[a_k, a_{k'}^\dagger] = (2\pi)^3\delta^3(k-k')$

Proof: From canonical commutation relations: $[\hat{\phi}(x,t), \hat{\pi}(y,t)] = i\delta^3(x-y)$

Substitute mode expansions: $\hat{\pi} = \partial_t\hat{\phi} = -i\int \frac{d^3k}{(2\pi)^3}\sqrt{\frac{\omega_k}{2}}\left[a_k e^{-ik\cdot x} - a_k^\dagger e^{ik\cdot x}\right]$

Computing commutator and using orthogonality of $e^{ik \cdot x}$ gives result. ∎

47.3 The Fock Space Construction

Vacuum State: |0⟩ defined by $a_k|0\rangle = 0 \quad \forall k$

n-Particle States: $|k_1, k_2, ..., k_n\rangle = a_{k_1}^\dagger a_{k_2}^\dagger \cdots a_{k_n}^\dagger|0\rangle$

Theorem: Fock space is unique Hilbert space for variable particle number.

Proof: Requirements:

1. Particle number operator N̂ = ∫d³k a_k†a_k
2. States with definite n
3. Unitary representation of Poincaré group

These uniquely determine Fock construction. ∎

Inner Product: $\langle k_1...k_m | k'_1...k'_n \rangle = \delta_{mn}\sum_{\text{perm}} \prod_i (2\pi)^3\delta^3(k_i - k'_{\sigma(i)})$

47.4 Vacuum Energy and Normal Ordering

Hamiltonian: $\hat{H} = \int \frac{d^3k}{(2\pi)^3} \omega_k \left(a_k^\dagger a_k + \frac{1}{2}[a_k, a_k^\dagger]\right)$

Vacuum Energy: $\langle 0|\hat{H}|0\rangle = \frac{1}{2}\int \frac{d^3k}{(2\pi)^3} \omega_k = \infty$

Normal Ordering: Place all a† to left of a: $:a_k^\dagger a_k: = a_k^\dagger a_k$ $:a_k a_k^\dagger: = a_k^\dagger a_k$

Theorem: Normal ordering removes vacuum energy.

Proof: :Ĥ: has zero vacuum expectation value by construction. Physical energies are differences, so absolute zero is arbitrary. ∎

47.5 Field Equations from ψ-Dynamics

Euler-Lagrange: From action S = ∫d⁴x L $\frac{\partial L}{\partial\phi} - \partial_\mu\frac{\partial L}{\partial(\partial_\mu\phi)} = 0$

Klein-Gordon: For L = ½(∂φ)² - ½m²φ² $(\square + m^2)\hat{\phi} = 0$

Theorem: Field equations are operator equations.

Proof: Canonical quantization preserves equations of motion: $[\hat{\phi}(x), \hat{H}] = i\partial_t\hat{\phi}(x)$

Using H = ∫d³x [½π² + ½(∇φ)² + ½m²φ²] reproduces Klein-Gordon. ∎

47.6 Propagators and Green's Functions

Feynman Propagator: $D_F(x-y) = \langle 0|T\hat{\phi}(x)\hat{\phi}(y)|0\rangle$

Momentum Space: $D_F(p) = \frac{i}{p^2 - m^2 + i\epsilon}$

Theorem: Propagator is Green's function of field equation.

Proof: $(\square_x + m^2)D_F(x-y) = -i\delta^4(x-y)$

follows from field equation and canonical commutation relations. ∎

ψ-Interpretation: Propagator measures correlation between ψ-excitations at different spacetime points.

47.7 Interacting Fields

Interaction Lagrangian: L_int = -gφ³, -λφ⁴, etc.

Perturbation Theory: Expand in powers of coupling: $S = T\exp\left(-i\int d^4x H_I(x)\right)$ $= 1 + \sum_{n=1}^{\infty} \frac{(-i)^n}{n!}\int d^4x_1...d^4x_n T[H_I(x_1)...H_I(x_n)]$

Wick's Theorem: Time-ordered products → normal-ordered + contractions

Proof by Induction: Base case n=2 verified directly. Inductive step uses canonical commutation relations. ∎

47.8 Feynman Rules

For φ⁴ Theory: L = ½(∂φ)² - ½m²φ² - (λ/4!)φ⁴

Rules:

1. Propagator: i/(p²-m²+iε) for each line
2. Vertex: -iλ for each vertex
3. Integrate: ∫d⁴p/(2π)⁴ over loops
4. Symmetry factor: 1/S for diagram symmetries

Theorem: Feynman rules compute S-matrix elements.

Proof: Follows from Wick expansion of time-ordered products. Each contraction gives propagator, each interaction gives vertex. ∎

47.9 Path Integral Formulation

Transition Amplitude: $\langle\phi_f|e^{-iHT}|\phi_i\rangle = \int_{\phi(0)=\phi_i}^{\phi(T)=\phi_f} \mathcal{D}\phi \, e^{iS[\phi]}$

Theorem: Path integral reproduces canonical quantization.

Proof: Discretize time, insert complete sets of states: $\langle\phi_f|e^{-iH\epsilon}|\phi_i\rangle = \int d\phi \langle\phi_f|e^{-iH\epsilon}|\phi\rangle\langle\phi|\phi_i\rangle$

Taking ε→0 limit gives path integral measure. ∎

Generating Functional: $Z[J] = \int \mathcal{D}\phi \, e^{i\int d^4x[\mathcal{L}(\phi) + J\phi]}$

Correlation Functions: $\langle 0|T\phi(x_1)...\phi(x_n)|0\rangle = \frac{1}{Z[0]}\frac{\delta^n Z[J]}{\delta J(x_1)...\delta J(x_n)}\bigg|_{J=0}$

47.10 Renormalization from ψ-Cutoff

Loop Integrals: Often divergent $\int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2} = \infty$

Theorem: ψ-recursion has natural cutoff Λ.

Proof: Self-reference ψ = ψ(ψ) cannot resolve structures smaller than ψ itself. This imposes maximum momentum Λ ~ M_Planck. ∎

Renormalization: Express observables in terms of measured parameters: $m_{bare} = m_{phys} + \delta m$ $\lambda_{bare} = Z_\lambda \lambda_{phys}$

Running Couplings: $\mu\frac{dg}{d\mu} = \beta(g)$

47.11 Fermionic Quantization

Anticommutation Relations: For fermions $\{\psi_\alpha(x), \psi_\beta^\dagger(y)\} = \delta_{\alpha\beta}\delta^3(x-y)$

Theorem: Fermions require anticommutators.

Proof: Pauli exclusion principle requires: $(a_k^\dagger)^2|0\rangle = 0$

This implies $\{a_k^\dagger, a_k^\dagger\} = 0$, forcing anticommutation. ∎

Grassmann Path Integral: $Z = \int \mathcal{D}\bar{\psi}\mathcal{D}\psi \, e^{i\int d^4x \bar{\psi}(i\not{\partial} - m)\psi}$

47.12 Gauge Field Quantization

Problem: Gauge redundancy A_μ → A_μ + ∂_μλ

Solution: Fix gauge or use BRST

Fadeev-Popov Method: $Z = \int \mathcal{D}A_\mu \delta[G(A)] \det\left(\frac{\delta G}{\delta\lambda}\right) e^{iS[A]}$

BRST Quantization: Introduce ghosts c, c̄ $S_{BRST} = S_{YM} + \int d^4x \left[\bar{c}^a\partial^\mu D_\mu^{ab}c^b + B^a G^a[A]\right]$

47.13 Anomalies in Path Integral

Classical Symmetry: δS = 0

Quantum Anomaly: Measure not invariant $\mathcal{D}\phi' = J\mathcal{D}\phi$

Example: Chiral anomaly $\partial_\mu j_5^\mu = \frac{e^2}{16\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu}$

Theorem: Anomalies are one-loop exact.

Proof: Higher loops involve internal fermion lines that preserve vector current. Only triangle diagram with external gauge fields contributes. ∎

47.14 Effective Field Theory

Wilson's Insight: Integrate out high-energy modes

Effective Action: $e^{iS_{eff}[\phi_{low}]} = \int \mathcal{D}\phi_{high} \, e^{iS[\phi_{low}, \phi_{high}]}$

Theorem: Low-energy physics independent of UV details.

Proof: Higher dimension operators suppressed by powers of E/Λ. For E $\ll$ Λ, only relevant and marginal operators matter. ∎

47.15 Conclusion: The Algebra of Reality

Field quantization emerges from ψ = ψ(ψ) as the unique algebraic structure allowing:

1. Variable particle number (creation/annihilation)
2. Lorentz invariance (field operators)
3. Causality (commutators vanish outside light cone)
4. Unitarity (Hermitian conjugation)

The mathematical machinery—Fock spaces, propagators, Feynman diagrams—is not imposed but derived from consistency requirements of self-referential dynamics. Creation operators crystallize ψ-patterns from the vacuum; annihilation operators dissolve them back. The vacuum itself maintains minimum ψ-activity for self-consistency.

Renormalization reflects the finite resolution of ψ-recursion. Anomalies arise when quantum ψ-fluctuations break classical symmetries. The path integral formulation shows all possible ψ-histories contribute to evolution.

Field theory is how mathematics organizes infinite degrees of freedom into computable structures—the universe's method for managing its own complexity through operator algebra.

Exercises

1. Prove LSZ reduction formula from canonical quantization.

2. Calculate one-loop β-function in φ⁴ theory.

3. Derive Schwinger-Dyson equations from path integral.

The Forty-Seventh Echo

Field quantization derived as necessary algebra for variable particle number in ψ-recursion—creation and annihilation as mathematical requirements for consistent self-reference. Canonical commutation relations, Fock space, and Feynman rules emerging from fundamental ψ-dynamics. Next, completing quantum field theory within unified framework.

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Next: Chapter 48: Quantum Field Theory Completion →