Path Integrals from QM to QFT

The quantum-mechanical path integral sums over coordinate histories,

$$x(t),$$

while a field-theory path integral sums over field configurations,

$$\phi(\mathbf x,t).$$

That replacement is the short bridge:

$$\text{particle path} \quad \longrightarrow \quad \text{field history}.$$

The long bridge is everything this replacement changes. A field has infinitely many degrees of freedom in the continuum, local operator products are singular, perturbation theory needs regularization and renormalization, gauge theories contain redundant variables, and Euclidean continuation carries analytic assumptions. This page is a map of the translation, not a full field-theory construction.

Coordinate Paths in Quantum Mechanics

For one nonrelativistic particle, the real-time propagator is

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

The path-integral representation is formally

$$K(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal D x(t)\, e^{iS[x]/\hbar},$$

where

$$S[x] = \int_{t_i}^{t_f}dt\, \left[ \frac{m}{2}\dot x^2-V(x) \right].$$

The boundary conditions are part of the object being computed: this is a fixed-endpoint transition amplitude. The time-sliced construction in Time Slicing explains where the formal measure and normalization factors come from.

For $N$ particles or $N$ generalized coordinates $q^a(t)$, the same idea becomes a path integral over vector-valued histories:

$$\int \mathcal D q^1(t)\cdots \mathcal D q^N(t)\, e^{iS[q]/\hbar}.$$

This finite-dimensional many-coordinate version is the cleanest stepping stone to fields.

Field Configurations in QFT

A classical scalar field assigns a number to each point in space and time:

$$\phi=\phi(\mathbf x,t).$$

One way to see the field path integral is to discretize space into lattice sites $\mathbf x_a$. The field values become many coordinates,

$$q_a(t)=\phi(\mathbf x_a,t),$$

so the path integral resembles a coupled many-oscillator path integral:

$$\int \prod_a \mathcal D q_a(t)\, e^{iS[q]/\hbar}.$$

The continuum notation is the formal limit

$$\int \mathcal D\phi\, e^{iS[\phi]/\hbar}.$$

The field-theory integration variable $\phi(\mathbf x,t)$ is not the trajectory of one particle through space. It is an entire field configuration over spacetime. This distinction prevents many wrong interpretations of QFT propagators and diagrams.

Action Functionals

The particle action is an integral over time. A field action is an integral over spacetime. For a real scalar field, a common schematic Lorentzian action is

$$S[\phi] = \int dt\,d^d x\, \left[ \frac12(\partial_t\phi)^2 - \frac12(\nabla\phi)^2 - \frac12m^2\phi^2 - V_{\rm int}(\phi) \right],$$

using units and normalizations appropriate to the field-theory convention in use.

The structural analogy is:

Quantum mechanics	Field theory
coordinate $x(t)$	field $\phi(\mathbf x,t)$
velocity $\dot x(t)$	derivatives $\partial_\mu\phi(x)$
action $\int dt\,L$	action $\int d^{d+1}x\,\mathcal L$
endpoint data	boundary, vacuum, thermal, or source data
propagator kernel	correlation functions and transition amplitudes

The similarity is real, but it does not remove the extra questions created by continuum fields.

Sources and Generating Functionals

Sources turn path integrals into machines for producing correlation functions. In quantum mechanics, one can couple a source $J(t)$ to a coordinate:

$$Z[J] = \int\mathcal D x\, \exp\left[ \frac{i}{\hbar} \left( S[x]+\int dt\,J(t)x(t) \right) \right].$$

In a scalar field theory, the analogous expression is

$$Z[J] = \int\mathcal D\phi\, \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^{d+1}x\,J(x)\phi(x) \right) \right].$$

Functional derivatives with respect to $J$ bring down insertions of $x(t)$ or $\phi(x)$. For example,

$$\frac{\delta Z[J]}{\delta J(y)} = \frac{i}{\hbar} \int\mathcal D\phi\, \phi(y) \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^{d+1}x\,J(x)\phi(x) \right) \right].$$

The notation is compact, but the convention matters. A real-time source functional usually generates time-ordered objects. A Euclidean source functional generates Euclidean correlation functions. Normalization by $Z[0]$ removes vacuum-bubble factors in many perturbative conventions.

Correlation Functions

In the operator language, a common field-theory object is the time-ordered vacuum correlation function

$$G_n(x_1,\ldots,x_n) = \langle 0\rvert \mathcal T\{ \phi(x_1)\cdots\phi(x_n) \} \lvert 0\rangle.$$

In the path-integral language, the corresponding schematic source formula is

$$G_n(x_1,\ldots,x_n) = \frac{1}{Z[0]} \left( \frac{\hbar}{i} \right)^n \frac{\delta^n Z[J]} {\delta J(x_1)\cdots\delta J(x_n)} \bigg\rvert_{J=0}.$$

This is one of the most important reasons ordinary quantum-mechanical path integrals are useful preparation for QFT. The same logic that says “differentiate with respect to a source to insert a coordinate” becomes the field-theory rule “differentiate with respect to a source to insert a field.”

The physical interpretation changes, however. A QFT two-point function is a field correlator. It may behave like a particle propagator in suitable regimes, but it is not simply a single-particle wavefunction kernel.

Wick Rotation

Real-time field integrals are oscillatory:

$$Z = \int\mathcal D\phi\, e^{iS[\phi]/\hbar}.$$

After Wick rotation in suitable theories and states, one obtains a Euclidean functional integral

$$Z_E = \int\mathcal D\phi\, e^{-S_E[\phi]/\hbar}.$$

The analogy with imaginary-time quantum mechanics is direct: $e^{-iHT/\hbar}$ becomes $e^{-H\tau/\hbar}$, and real-time phases become Euclidean damping weights. The Euclidean construction is central in statistical mechanics, lattice field theory, instanton estimates, and nonperturbative definitions.

But Wick rotation is not a universal symbol substitution. Singularities, boundary prescriptions, operator ordering, thermal conditions, and gauge choices determine whether and how the Euclidean object reconstructs Lorentzian physics. See Euclidean and Imaginary-Time Path Integrals for the quantum-mechanical version of this warning.

From Euclidean Time to Euclidean QFT gives the field-theory continuation, including thermal boundary conditions, Matsubara modes, and reflection positivity.

Perturbation Theory

Suppose the action splits into a quadratic part and an interaction:

$$S[\phi]=S_0[\phi]+S_{\rm int}[\phi].$$

The quadratic part defines the Gaussian theory. Its inverse operator gives the free propagator once boundary conditions and the $i0$ prescription are specified. The interaction is then expanded:

$$e^{iS_{\rm int}[\phi]/\hbar} = 1+\frac{i}{\hbar}S_{\rm int}[\phi] + \frac{1}{2!} \left( \frac{i}{\hbar}S_{\rm int}[\phi] \right)^2 +\cdots.$$

In a polynomial theory, source derivatives and Gaussian contractions organize this expansion into diagrams. The diagrams are bookkeeping for integrals and operator contractions; they are not literal pictures of particles moving along classical paths.

The quantum-mechanical ancestor is already visible in the interaction picture and Dyson expansion. QFT keeps that time-ordered perturbative structure but applies it to fields and their local interactions.

What Becomes Harder in QFT

The formal replacement $x(t)\to\phi(\mathbf x,t)$ hides several new difficulties:

• Continuum fields have infinitely many degrees of freedom.
• Local products of fields can be singular.
• Loop integrals often diverge and require regularization and renormalization.
• Gauge fields contain redundant descriptions of the same physical configuration.
• Vacuum structure can be nontrivial and state-dependent.
• Boundary conditions, thermal conditions, and real-time prescriptions are part of the definition.
• Euclidean and Lorentzian formulations are related by analytic continuation only under appropriate assumptions.
• Nonperturbative definitions may require lattices, constructive methods, algebraic methods, or other frameworks.

These are not small technical footnotes. They are why the QFT path integral is both powerful and delicate.

Continue in Field Theory

The safest learning sequence is:

1. Understand propagators, time slicing, and Euclidean time in ordinary quantum mechanics.
2. Understand the harmonic oscillator as the model for independent field modes.
3. Learn sources and functional derivatives.
4. Learn free-field Gaussian integrals and correlation functions.
5. Add interactions perturbatively, then study regularization and renormalization.
6. Treat gauge symmetry, fermions, finite temperature, and nonperturbative definitions as additional structures, not as cosmetic variants.

The bridge pages Why Dynamics Matters for QFT, From Path Integrals in QM to Field Path Integrals, From Euclidean Time to Euclidean QFT, Path Integrals, and Green Functions are orientation maps. They should be read alongside a full field-theory treatment when the goal is calculation rather than translation.

Common Mistakes

• Thinking QFT path integrals sum over particle trajectories rather than field configurations.
• Treating $\mathcal D\phi$ as an ordinary finite-dimensional measure.
• Forgetting that source derivatives generate convention-dependent correlation functions.
• Ignoring normalization by $Z[0]$ when comparing connected and disconnected contributions.
• Treating Feynman diagrams as literal spacetime histories.
• Assuming Wick rotation is always reversible without spectral and boundary-condition data.
• Importing finite-dimensional intuition without checking ultraviolet behavior.

Cross-Links

• Time Slicing
• Path Integral Formulation
• Euclidean and Imaginary-Time Path Integrals
• From Euclidean Time to Euclidean QFT
• Common Pitfalls in Path Integrals
• Functional Derivatives
• Time Ordering
• Interaction Picture
• What Is a Green Function?
• Green Functions from QM to QFT
• From Path Integrals in QM to Field Path Integrals
• Harmonic Oscillator to Fields
• Path Integrals
• Green Functions

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.
• 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.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010.

Exercises

1. Explain why a spatial lattice makes the transition from quantum mechanics to field theory look like a many-coordinate path integral.

Solution

On a spatial lattice, a field value at each site becomes a coordinate:

$$q_a(t)=\phi(\mathbf x_a,t).$$

The field history is then a collection of coordinate histories $q_a(t)$, one for each lattice site. The path integral has the finite-dimensional-looking form

$$\int \prod_a \mathcal D q_a(t)\, e^{iS[q]/\hbar}.$$

The continuum field integral is the formal limit as the lattice spacing is taken to zero and the number of degrees of freedom becomes infinite. This limit is exactly where ultraviolet regularization and renormalization enter.

2. Show how a source derivative inserts a field in the generating functional.

Solution

For

$$Z[J] = \int\mathcal D\phi\, \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^{d+1}x\,J(x)\phi(x) \right) \right],$$

differentiate with respect to $J(y)$:

$$\frac{\delta Z[J]}{\delta J(y)} = \frac{i}{\hbar} \int\mathcal D\phi\, \phi(y) \exp\left[ \frac{i}{\hbar} \left( S[\phi]+\int d^{d+1}x\,J(x)\phi(x) \right) \right].$$

Thus source derivatives insert fields. Higher derivatives insert products of fields, with ordering determined by the convention used to define $Z[J]$.

3. Why is a QFT two-point function not automatically the same thing as a single-particle propagator kernel?

Solution

A single-particle kernel has the form

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

and evolves wavefunctions in a fixed-particle Hilbert space. A QFT two-point function is typically

$$\langle 0\rvert \mathcal T\{\phi(x)\phi(y)\} \lvert 0\rangle.$$

It is a vacuum correlation function of fields. It can encode propagation and particle poles, but its interpretation depends on the theory, the state, interactions, and the asymptotic particle concept.

4. In perturbation theory, why is the quadratic part of the action singled out?

Solution

Quadratic path integrals are Gaussian and can be evaluated exactly once the inverse operator and boundary conditions are specified. The inverse of the quadratic operator gives the free propagator. Interactions are then treated by expanding

$$e^{iS_{\rm int}/\hbar}$$

and evaluating the resulting Gaussian moments. This is the path-integral version of using a solvable free theory as the starting point for perturbation theory.