[Paper Review] Causality in Quantum Field Theory with Classical Sources - Quantum Electrodynamics

Bo-Sture Skagerstam, Karl‐Erik Eriksson+1 more|arXiv (Cornell University)|Jan 30, 2018

Quantum Mechanics and Applications45 references3 citations

TL;DR

This paper demonstrates that in a second-quantized quantum electrodynamics framework with classical sources, space-time expectation values of the electromagnetic field automatically yield causal, retarded solutions that reproduce classical Maxwell's equations without assuming causality a priori. The derivation shows that gauge-invariant, observable field strengths emerge causally through exact time-evolution under the Schrödinger equation, resolving long-standing questions about causality and retardation in quantum field theory with classical sources.

ABSTRACT

In an exact quantum-mechanical framework, we show that expectation values of the second-quantized electro-magnetic fields in the Coulomb gauge, and in the presence of classical sources, automatically lead to causal and retarded electro-magnetic field strengths. The classical $\hbar$-independent Maxwell's equations naturally emerge from this fundamental quantum-mechanical approach in terms of expectation values of quantum fields, and are therefore also consistent with the special theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as, e.g., in classical optics, and interference effects of quantum states is clarified. The framework outlined also provides for a simple approach to, e.g., spontaneous photon emission and/or absorption processes as well as to the classical Vavilov-Cherenkov radiation. The inherent and necessary quantum fluctuations, limiting a precise space-time knowledge of expectation values of the quantum fields considered, are, finally, recalled.

Motivation & Objective

• To resolve foundational issues of causality and retardation in quantum field theory with classical sources.
• To show that gauge-invariant electromagnetic field strengths emerge causally from second-quantized quantum fields without pre-assigned causal order.
• To clarify the distinction between quantum interference and classical interference in linear Maxwell theory.
• To provide a framework for understanding spontaneous emission, absorption, and Cherenkov radiation within a quantum field-theoretic approach.
• To highlight the role of quantum uncertainty in limiting the precision of space-time field expectation values.

Proposed method

• Formalism uses second-quantized electromagnetic fields in the Coulomb gauge with a general classical conserved current as source.
• Time-evolution is governed by the Schrödinger equation, with no pre-assigned causal structure.
• The system is mapped to decoupled harmonic oscillators with space-time-dependent external forces via optical quadrature representation.
• Expectation values of the transverse vector potential are computed exactly in the interaction picture using time-ordered exponential evolution.
• The classical limit emerges as ħ → 0, yielding retarded Maxwell equations from quantum expectation values.
• Transverse current components are extracted via Fourier decomposition, ensuring gauge invariance and consistency with ∇·j_T = 0.

Experimental results

Research questions

• RQ1 How do space-time expectation values of quantum electromagnetic fields in the presence of a classical source ensure causality and retardation?
• RQ2 In what way does the classical Maxwell theory emerge from a fully quantum-mechanical framework without assuming causality a priori?
• RQ3 What is the fundamental difference between quantum interference in field expectation values and classical interference in linear Maxwell theory?
• RQ4 How can spontaneous emission and absorption processes be consistently described in this quantum framework?
• RQ5 How does quantum uncertainty limit the precision of space-time knowledge of field expectation values?

Key findings

• The expectation value of the transverse vector potential ⟨A_T(x,t)⟩ is exactly computed and shown to be causal and retarded, depending only on the source history up to time t.
• The second time-derivative of ⟨A_T(x,t)⟩ reproduces the inhomogeneous wave equation with a source term proportional to the transverse current j_T(x,t), confirming the emergence of classical Maxwell's equations.
• The classical limit (ħ → 0) yields gauge-invariant, retarded solutions that satisfy the special theory of relativity and are consistent with causality.
• The framework naturally reproduces the classical Thomson cross-section in the low-energy limit via a Born approximation, consistent with quantum electrodynamics.
• The Vavilov-Cherenkov radiation is reproduced exactly and straightforwardly in this quantum framework, confirming its consistency with classical radiation theory.
• Quantum uncertainty limits the precision of space-time localization of field expectation values, a fundamental constraint inherent in the quantum formalism.