From Quantum Dynamics to Field Theory

Quantum field theory describes local fields and their excitations. A continuum field has infinitely many spatial degrees of freedom; a finite-volume spatial lattice temporarily reduces this to finitely many field variables. This is a different distinction from Hilbert-space dimension: even one particle on a line already has an infinite-dimensional Hilbert space. Yet much of its working language grows directly from quantum dynamics:

$$\begin{gathered} \text{evolution operators} \longrightarrow \text{time-ordered products} \\ \longrightarrow \text{propagators, sources, and field histories} \\ \longrightarrow \text{correlation functions and observable data}. \end{gathered}$$

This chapter makes those continuities precise without pretending that QFT is merely ordinary quantum mechanics with more coordinates. Locality, relativistic covariance, particle creation, operator-valued distributions, inequivalent representations, ultraviolet renormalization, spin–statistics, gauge redundancy, and the construction of asymptotic observables introduce additional structure. Relativistic field theory requires the corresponding locality, covariance, and spin–statistics conditions; nonrelativistic field theories need not share all of them.

The pages here are bridges. They identify which quantum-mechanical idea survives, what notation changes, which new assumptions enter, and where a full QFT treatment must take over.

Helpful background. Time evolution supplies operators, path integrals supply histories and sources, and phase space supplies canonical variables. For a broader preparation path, see Bridge to QFT.

From coordinates to local fields

Quantum-mechanical structure	Field-theory continuation	New issue
state vector or density operator	vacuum, thermal state, in/out states, or state functional	inequivalent representations and state dependence
coordinate $q(t)$	field $\phi(t,\mathbf x)$	infinitely many local degrees of freedom
momentum $p(t)$	momentum field $\pi(t,\mathbf x)$	distributions and constraints
$[q,p]=i\hbar$	equal-time field algebra	delta distributions, grading, gauge constraints
$U(t,t_0)$	interaction-picture field evolution	renormalized interaction densities
time ordering	ordered field products	fermionic signs and contact terms
propagator kernel	several QFT two-point functions	state, ordering, support, and pole prescription
source $J(t)$	spacetime source $J(x)$	local composite operators and renormalization
path $q(t)$	field history $\phi(x)$	regulated functional measure and gauge redundancy
imaginary time	Euclidean spacetime or thermal circle	reflection positivity and analytic continuation
spectral sum	poles, cuts, thresholds, and spectral densities	multiparticle continua and unstable resonances
transition amplitude	on-shell field-theory amplitude	LSZ, relativistic normalization, infrared structure

The left and middle columns show a durable analogy. The right column explains why translating a symbol is only the start of a field-theory calculation.

Two Complementary Routes

Operator and Hamiltonian route

The operator route starts from the interaction-picture expression below. In a regulated theory with a well-defined Hamiltonian it describes evolution; in continuum interacting QFT its Dyson expansion is generally used perturbatively, with renormalization. One cannot assume that free and interacting fields share a global interaction-picture Hilbert-space representation without further conditions:

$$U_I(t,t_0) = \mathcal T \exp\left[ - \frac{i}{\hbar} \int_{t_0}^{t}dt'\, H_I(t') \right].$$

From Evolution Operators to Time-Ordered Products explains how its Dyson expansion produces ordered field insertions. From Phase Space to Canonical Quantization supplies the complementary equal-time route from field momenta and Poisson brackets to operator algebras.

This route is especially useful when the Hamiltonian, Hilbert space, constraints, symmetries, or asymptotic states are central.

Functional and Euclidean route

The functional route begins with kernels and sources:

$$Z[J] \sim \int\mathcal D\phi\, \exp\left[ \frac{i}{\hbar} \left( S[\phi] + \int d^D x\,J(x)\phi(x) \right) \right].$$

From Propagators in QM to Propagators in QFT distinguishes the two-point objects. From Sources in QM to Generating Functionals in QFT develops field insertions. From Path Integrals in QM to Field Path Integrals changes the integration variable from coordinate paths to field histories. From Euclidean Time to Euclidean QFT treats vacuum projection, the thermal circle, and analytic continuation.

This route is especially useful for perturbation theory, Euclidean methods, lattice formulations, statistical mechanics, and generating functional identities.

Where the routes meet

Both routes meet at correlation functions:

$$\langle \mathcal O_1(x_1)\cdots\mathcal O_n(x_n) \rangle.$$

From Correlation Functions to QFT Observables explains how ordering, poles, residues, retarded support, on-shell limits, and Euclidean decay convert these functions into physical information.

The operator and functional routes are not rival theories. When both are well defined with matching states, boundary conditions, and conventions, they are complementary representations of the same quantum dynamics.

Page Map

Reader’s question	Start here	What you learn
Why does quantum dynamics remain central in QFT?	Why Dynamics Matters for QFT	how the formulations connect
How does the Dyson series become ordered field products?	From Evolution Operators to Time-Ordered Products	ordered operator insertions
What does “propagator” mean after fields become primary?	From Propagators in QM to Propagators in QFT	dictionary among kernels and two-point functions
How do source derivatives generate field correlators?	From Sources in QM to Generating Functionals in QFT	source derivatives and connected correlators
What is integrated over in a field path integral?	From Path Integrals in QM to Field Path Integrals	field configuration space and regulated measures
How do ground-state projection and thermal traces become Euclidean QFT?	From Euclidean Time to Euclidean QFT	vacuum projection, thermal boundaries, and reconstruction
How do field momenta and brackets lead to canonical quantization?	From Phase Space to Canonical Quantization	scalar pipeline plus fermion and gauge-constraint preview
How do correlators become masses, response, scattering, or lattice data?	From Correlation Functions to QFT Observables	how physical information is extracted

One oscillator already illustrates a field correlator

For a harmonic oscillator, write $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$. Since $a(t)=a e^{-i\omega t}$, its vacuum correlation is

$$\langle0|x(t)x(0)|0\rangle =\frac{\hbar}{2m\omega}e^{-i\omega t}.$$

Only $\langle0|aa^\dagger|0\rangle=1$ contributes. Time ordering instead gives

$$\langle0|\mathcal T x(t)x(0)|0\rangle =\frac{\hbar}{2m\omega}e^{-i\omega|t|}.$$

These are two different functions, and neither is the coordinate kernel $\langle x_f|U|x_i\rangle$. A free scalar field can be decomposed into oscillator modes; summing their appropriately normalized correlations gives the field two-point function. Interactions add effects that independent oscillators cannot describe.

The many-body bridge develops occupation-state language. For scattering, the S-matrix bridge explains why an ordered correlator still needs asymptotic-state limits and normalization before it becomes a scattering amplitude.

Exercises

Kernels and correlations

Rewrite the statement “the propagator tells us how the particle moves from $x$ to $y$” so that it is accurate for both a nonrelativistic kernel and a scalar QFT two-point function.

Solution

For nonrelativistic quantum mechanics, one can say:

The kernel

$$K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i) \lvert x_i\rangle$$

is the position-space matrix element of the time-evolution operator and acts as an amplitude density under integration against the initial wavefunction. Position eigenstates are delta normalized, so its squared magnitude alone is not a normalized position probability.

For scalar QFT, one should instead say:

The chosen two-point function, for example

$$\langle0\rvert \mathcal T\phi(x)\phi(y) \lvert0\rangle,$$

is an ordered vacuum field correlation. Its poles and spectral density encode excitations that the field can create; it is not generally the probability amplitude for one classical particle trajectory from $y$ to $x$.

Two calculations of a free-field correlation

Outline how the operator route and the functional route obtain the free scalar two-point function. Identify the point at which their conventions must agree.

Solution

In the operator route:

1. derive the conjugate momentum and impose the equal-time commutator;
2. expand the free field in normalized oscillator modes;
3. choose the vacuum annihilated by the mode annihilation operators;
4. time order the product $\phi(x)\phi(y)$ and evaluate its vacuum expectation value.

In the functional route:

1. define the regulated quadratic field action and vacuum boundary prescription;
2. introduce a source $J$;
3. evaluate the Gaussian generating functional;
4. differentiate twice with respect to $J$ and set $J=0$.

The routes must agree on the field normalization, metric and Fourier conventions, vacuum prescription, and $i0$ boundary condition. If any of these differ, their displayed propagator formulas can differ by signs or factors even when the physical content agrees.

From a Euclidean correlator to a cross section

A calculation produces a Euclidean gauge-field two-point function at finite lattice spacing and calls it a measured scattering cross section. List the missing steps or possible obstructions.

Solution

At minimum, the claim must address:

1. whether the operator is gauge invariant or the result is gauge dependent;
2. how the lattice spacing and finite volume are controlled;
3. how the operator is renormalized;
4. whether a reflection-positive Euclidean correlator reconstructs the required Lorentzian object;
5. how analytic continuation or spectral extraction is performed and with what uncertainty;
6. whether stable asymptotic particle poles exist so that LSZ applies;
7. how external legs are amputated and states normalized;
8. how the amplitude is converted to a cross section with phase space, flux, quantum-number sums, and infrared-safe inclusiveness.

Without these steps, the Euclidean correlator is useful intermediate data, not a measured cross section.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1992.
• M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.