From Path Integrals in QM to Field Path Integrals

The central replacement from quantum-mechanical path integrals to field-theory path integrals is

$$q(t) \quad\longrightarrow\quad \phi(\mathbf x,t).$$

In words: instead of integrating over histories of finitely many coordinates, a field path integral integrates over histories of field configurations. A point in the configuration space of a field theory is already a whole function of space. A history of that point is a function of both space and time.

This page is narrower than Path Integrals from QM to QFT. It focuses on the configuration-space translation and the technical warnings that immediately follow from it: measures, regulators, perturbation theory, and gauge redundancy.

Configuration Space in Mechanics

For a single coordinate $q$, the fixed-endpoint quantum-mechanical propagator is formally

$$K(q_f,t_f;q_i,t_i) = \int_{q(t_i)=q_i}^{q(t_f)=q_f} \mathcal Dq\, e^{iS[q]/\hbar}.$$

The path variable is a map from a time interval into the classical configuration space $Q$:

$$q: [t_i,t_f]\to Q.$$

For one particle on a line, $Q=\mathbb R$. For $N$ generalized coordinates, $Q$ may be locally described by coordinates $q^a$, $a=1,\ldots,N$, and a path is

$$t\mapsto \bigl(q^1(t),\ldots,q^N(t)\bigr).$$

The corresponding formal measure is a product over coordinate histories,

$$\mathcal Dq^1\cdots\mathcal Dq^N,$$

with normalizations fixed by the time-sliced construction. This idea is derived in From Propagators to Path Integrals and Time Slicing.

Two features will matter in the field-theory translation:

• the path integral is over histories of configuration-space points, not over probabilities;
• the action $S[q]$ is a functional of the whole history, not a function of one instant.

Configuration Space in Field Theory

For a real scalar field on a spatial slice $\Sigma$, a classical configuration is a function

$$\phi: \Sigma\to\mathbb R.$$

Classically this is a space of functions; quantum continuum measures generally live on less regular generalized fields, so smooth classical configurations do not describe typical fluctuations. Write the configuration space schematically as

$$\mathcal F = \{\phi(\mathbf x)\}.$$

A field history is a time-dependent point of this configuration space:

$$t\mapsto\phi_t(\mathbf x), \qquad \phi_t(\mathbf x)=\phi(\mathbf x,t).$$

Equivalently, on spacetime $M=\Sigma\times[t_i,t_f]$, it is a function

$$\phi: M\to\mathbb R.$$

This is the most important interpretive correction: a field path integral does not sum over the trajectory of one particle moving through space. It sums over possible spacetime field histories. Particle language can emerge from field correlations and asymptotic states, but it is not the primitive integration variable.

The cleanest finite-dimensional approximation is a spatial lattice. If the lattice sites are $\mathbf x_a$, define

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

Then a field history becomes a large set of coupled coordinate histories:

$$\phi(\mathbf x,t) \quad\leadsto\quad \{q_a(t)\}_{a=1}^{N_{\rm sites}}.$$

The lattice path integral has the familiar many-coordinate form

$$\int \prod_a\mathcal Dq_a(t)\, e^{iS_{\rm lat}[q]/\hbar}.$$

The continuum notation

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

is the formal limit of this idea as the spatial regulator is removed.

Actions and Lagrangian Densities

In mechanics, the action is an integral over time:

$$S[q] = \int_{t_i}^{t_f}dt\, L(q,\dot q,t).$$

In a local relativistic field theory with spacetime dimension $D=d+1$, the action is an integral over spacetime:

$$S[\phi] = \int d^D x\, \mathcal L(\phi,\partial_\mu\phi).$$

Use $c=1$, the mostly-minus metric, and let $m$ denote an inverse length (the physical rest energy is $\hbar m$). A real scalar example is

$$\mathcal L = \frac12\partial_\mu\phi\,\partial^\mu\phi - \frac12m^2\phi^2 - \frac{\lambda}{4!}\phi^4.$$

The Euler–Lagrange equation is the field analogue of the mechanical variational equation:

$$\partial_\mu \frac{\partial\mathcal L} {\partial(\partial_\mu\phi)} - \frac{\partial\mathcal L}{\partial\phi} = 0.$$

For the scalar example, this gives

$$\partial_\mu\partial^\mu\phi + m^2\phi + \frac{\lambda}{3!}\phi^3 = 0.$$

The analogy with mechanics is exact at the level of stationary action, but the meaning of the degrees of freedom has changed. The label $\mathbf x$ is not a particle coordinate in the same sense as $q(t)$; it labels the position at which the field is evaluated.

Worked Example: Lattice Scalar Field

Consider a real scalar field on a cubic spatial lattice with spacing $a$. At each lattice site $\mathbf n$, write

$$\phi_{\mathbf n}(t) = \phi(a\mathbf n,t).$$

A simple lattice Lagrangian is

$$L_{\rm lat} = a^d\sum_{\mathbf n} \left[ \frac12\dot\phi_{\mathbf n}^2 - \frac12m^2\phi_{\mathbf n}^2 - \frac{\lambda}{4!}\phi_{\mathbf n}^4 \right] - \frac{a^d}{2} \sum_{\mathbf n,i} \left( \frac{\phi_{\mathbf n+\hat i}-\phi_{\mathbf n}}{a} \right)^2.$$

This is a many-coordinate mechanical system. The coordinates are the lattice field values $\phi_{\mathbf n}$, and the gradient term couples neighboring coordinates. If the lattice has finitely many sites, the path integral is an ordinary many-coordinate path integral in formal disguise:

$$Z_{\rm lat} = \int \prod_{\mathbf n} \mathcal D\phi_{\mathbf n}(t)\, \exp\left[ \frac{i}{\hbar}S_{\rm lat}[\phi] \right].$$

The continuum field integral is obtained only after specifying how the limit $a\to0$ is taken. That limiting procedure is not a harmless notation change; it is where ultraviolet divergences and renormalization enter.

Measures and Regularization

For a time-sliced lattice field theory, the schematic measure looks like

$$\mathcal D\phi \sim \prod_{r,\mathbf n}d\phi_{\mathbf n}(t_r),$$

where $r$ labels time slices and $\mathbf n$ labels spatial sites. This expression is useful because it shows why the field integral resembles a product over many variables. It is not, by itself, a regulator-independent continuum measure.

Several choices are part of the definition:

• the spacetime regulator, such as a lattice spacing, momentum cutoff, heat-kernel cutoff, or dimensional regularization;
• the normalization of the Gaussian measure;
• the boundary conditions or state prescription;
• the treatment of zero modes, constraints, and infrared behavior;
• the renormalized parameters held fixed as the cutoff is removed.

In a continuum QFT, products such as $\phi(x)^4$ and propagators evaluated at coincident points are usually singular. A regulator makes intermediate expressions meaningful, and renormalization states how physical predictions are kept finite as the regulator is changed or removed.

This is why the notation

$$\int\mathcal D\phi$$

should be read as a compact notation for a regulated prescription, not as an ordinary Lebesgue integral over an infinite-dimensional space.

Boundary and State Prescriptions

A path integral is not fully specified by the action alone. In mechanics, the fixed-endpoint kernel uses boundary conditions

$$q(t_i)=q_i, \qquad q(t_f)=q_f.$$

In field theory, common prescriptions include:

• fixed boundary fields $\phi_i(\mathbf x)$ and $\phi_f(\mathbf x)$;
• vacuum-to-vacuum generating functionals;
• in-in or closed-time-path functionals for real-time expectation values;
• thermal traces in imaginary time;
• Euclidean path integrals with boundary conditions appropriate to the problem.

For example, a vacuum generating functional for a scalar field is often written schematically as

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

with the vacuum prescription and regulator understood. Source derivatives of this object generate time-ordered vacuum correlators in the convention being used. A different contour or trace generates a different class of correlators.

Perturbation Theory

Perturbative field theory usually begins by splitting the action into a solvable quadratic part and an interaction:

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

Write the quadratic action as $S_0=\frac12\int\phi D\phi$, including integrations in the operator notation. Specify boundary conditions and a regulator so that $D^{-1}$ exists. Completing the square then gives

$$Z_0[J] \propto \exp\left[ -\frac{i}{2\hbar} \int d^D x\,d^D y\, J(x)D^{-1}(x,y)J(y) \right],$$

The normalized time-ordered correlator is $\Delta_F=i\hbar D^{-1}$ in this Lorentzian convention. Thus the exponent is also $-\int J\Delta_F J/(2\hbar^2)$, agreeing with the source bridge. The inverse kernel and the correlator differ by $i\hbar$; calling both a propagator can hide this factor.

The interaction is then treated by replacing fields with source derivatives:

$$Z[J] = \exp\left[ \frac{i}{\hbar} S_{\rm int} \left[ \frac{\hbar}{i} \frac{\delta}{\delta J} \right] \right] Z_0[J].$$

This formula is schematic, but it explains why Feynman diagrams arise. Derivatives of the Gaussian $Z_0[J]$ produce pairings governed by the free propagator, while powers of $S_{\rm int}$ produce interaction vertices. The detailed source calculus is developed in From Sources in QM to Generating Functionals in QFT.

Gauge Redundancy Warning

The scalar-field story is already formal, but gauge fields add another layer. A gauge potential $A_\mu$ contains redundant variables: gauge-related configurations describe the same physical field configuration. Naively writing

$$\int\mathcal DA_\mu\, e^{iS[A]/\hbar}$$

integrates over gauge copies as well as physical configurations. This overcounting is not a small normalization issue in general; it can make the expression ill defined.

A schematic gauge-fixed expression has the form

$$Z = \int\mathcal DA\, \Delta_{\rm FP}[A]\, \delta(G[A])\, e^{iS[A]/\hbar},$$

where $G[A]=0$ is a gauge condition and $\Delta_{\rm FP}$ is the Faddeev–Popov determinant. This is a local perturbative construction where the gauge condition selects representatives without residual zero modes; it need not select one representative globally because of Gribov copies. In non-abelian gauge theories this is commonly represented using ghost fields and BRST symmetry. Field-theory treatments develop the details. The key warning is: field path integrals over gauge variables are integrals over redundant coordinates unless gauge symmetry is handled explicitly.

Translation Table

Quantum mechanics	Field theory
Coordinate $q(t)$	Field history $\phi(\mathbf x,t)$
Configuration space $Q$	Function space $\mathcal F=\{\phi(\mathbf x)\}$
Path $[t_i,t_f]\to Q$	History $M\to\mathbb R$ or another target space
Action $\int dt\,L$	Action $\int d^D x\,\mathcal L$
Time-sliced measure $\prod_r dq(t_r)$	Regulated measure $\prod_{r,\mathbf n}d\phi_{\mathbf n}(t_r)$
Kernel boundary data $q_i,q_f$	Boundary fields, vacuum data, thermal data, or contour data
Gaussian oscillator	Free field, a continuum of oscillator modes
Perturbing potential	Interaction density and vertices
Redundant coordinates in constrained mechanics	Gauge equivalence identifies physically redundant field configurations

A change of coordinates can relabel distinct configurations without identifying any of them. Gauge equivalence instead identifies configurations that represent the same physical state; its mechanical analogue requires a system with that redundancy.

Common Mistakes

• Thinking the field path integral sums over particle trajectories. It sums over field histories.
• Treating $\mathcal D\phi$ as a regulator-independent measure without specifying a construction.
• Forgetting that $\mathbf x$ labels the field value, not the location of one particle following a path.
• Mixing vacuum, thermal, fixed-boundary, and in-in generating functionals as if they generated the same correlators.
• Assuming the continuum limit of a lattice field integral is automatic once the formal notation has been written.
• Integrating over gauge potentials without gauge fixing or gauge-invariant reformulation.

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• 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.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.

Exercises

1. Explain why a spatial lattice turns a scalar field into a many-coordinate mechanical system.

Solution

On a spatial lattice, each field value $\phi(\mathbf x_a,t)$ becomes a coordinate $q_a(t)$. A field history is therefore a collection of coordinate histories, one for each lattice site:

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

The lattice path integral then has the form

$$\int \prod_a\mathcal Dq_a(t)\, e^{iS_{\rm lat}[q]/\hbar},$$

which is the usual many-coordinate path integral. The continuum field integral is the limit in which the number of lattice sites becomes infinite while the lattice spacing is handled by a regulator and renormalization prescription.

2. Derive the field equation for the scalar Lagrangian density

$$\mathcal L = \frac12\partial_\mu\phi\,\partial^\mu\phi - \frac12m^2\phi^2 - \frac{\lambda}{4!}\phi^4.$$

Solution

Compute

$$\frac{\partial\mathcal L} {\partial(\partial_\mu\phi)} = \partial^\mu\phi,$$

and

$$\frac{\partial\mathcal L}{\partial\phi} = -m^2\phi - \frac{\lambda}{3!}\phi^3.$$

Substitution into

$$\partial_\mu \frac{\partial\mathcal L} {\partial(\partial_\mu\phi)} - \frac{\partial\mathcal L}{\partial\phi} = 0$$

gives

$$\partial_\mu\partial^\mu\phi + m^2\phi + \frac{\lambda}{3!}\phi^3 = 0.$$

3. Why is the expression $\int\mathcal D\phi$ not enough to define a continuum QFT?

Solution

The symbol hides the regulator, normalization, boundary or state prescription, treatment of zero modes, and renormalization prescription. In continuum field theory, local products and coincident-point propagators are typically singular. A definition requires a regulated construction and a rule for relating bare parameters to finite physical quantities.

4. Translate the source derivative insertion from quantum mechanics to scalar field theory.

Solution

In quantum mechanics, differentiating

$$\exp\left[ \frac{i}{\hbar} \int dt\,J(t)q(t) \right]$$

with respect to $J(t_0)$ inserts $(i/\hbar)q(t_0)$. In scalar field theory, differentiating

$$\exp\left[ \frac{i}{\hbar} \int d^D x\,J(x)\phi(x) \right]$$

with respect to $J(y)$ inserts $(i/\hbar)\phi(y)$. Multiplying by $\hbar/i$ converts the derivative into a field insertion.

5. Why does a perturbative gauge-field Gaussian need gauge fixing or an equivalent reduction?

Solution

Gauge-related potentials represent the same physical configuration, and the unreduced quadratic kinetic operator has gauge zero modes, so it cannot simply be inverted. A naive integral over $A_\mu$ also integrates repeatedly over gauge copies. This overcounting can make the integral divergent or ill defined and can obscure the physical degrees of freedom. Gauge fixing introduces the determinant or ghost structure needed for this perturbative construction. It is not mandatory in every nonperturbative formulation: a compact lattice gauge theory can compute gauge-invariant observables using normalized group integrals without fixing a gauge.