Topic

Is QFT based on Riemannian Geometry?

QFT Geometry Foundations Fiber Bundles

Short Answer

Quantum Field Theory (QFT) is not primarily based on Riemannian geometry. Standard QFT uses flat Minkowski spacetime. When we do QFT on a curved background (like near a black hole), we use pseudo-Riemannian (Lorentzian) geometry—but the spacetime is treated classically, not quantized. The geometric heart of QFT lies in fiber bundle theory, where gauge fields are connections and matter fields are sections.

Important clarification: “QFT on curved spacetime” means quantum fields on a classical gravitational background. This is not quantum gravity—the metric is fixed, not dynamical. Full quantum gravity (making spacetime itself quantum) remains an open problem. See Why can’t gravity be unified with Quantum Mechanics?

Spacetime Geometry

Standard QFT: Minkowski Spacetime

In conventional QFT (as used in the Standard Model), the underlying spacetime is Minkowski space $\mathbb{R}^{1,3}$ with the metric:

$\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$

This is a flat pseudo-Riemannian manifold with signature $(1,3)$ or $(3,1)$ depending on convention.

QFT on Curved Spacetime

When combining QFT with General Relativity, we work on a Lorentzian manifold $(M, g)$ where:

$M$ is a 4-dimensional smooth manifold
$g$ is a metric tensor with signature $(-,+,+,+)$

This is not Riemannian (which requires positive-definite metric) but pseudo-Riemannian.

Riemannian vs Pseudo-Riemannian

Property	Riemannian	Pseudo-Riemannian (Lorentzian)
Metric signature	$(+,+,+,+)$	$(-,+,+,+)$
Distance	Always positive	Can be positive, negative, or zero
Physical meaning	Space only	Spacetime with causality
Example	$\mathbb{R}^4$ Euclidean	Minkowski space

Fiber Bundles: The Geometric Heart of QFT

The mathematical framework of QFT relies fundamentally on fiber bundle theory. This is where the true geometry of gauge theories lives.

What is a Fiber Bundle?

A fiber bundle is a space $E$ that locally looks like a product $B \times F$ , where:

$B$ is the base space (spacetime)
$F$ is the fiber (internal space at each point)
$\pi: E \to B$ is the projection map

Locally: $E|_U \cong U \times F$ for open sets $U \subset B$

Principal Bundles

A principal bundle $P \xrightarrow{\pi} M$ has a Lie group $G$ as its fiber, with $G$ acting freely on $P$ .

$P \times G \to P, \quad (p, g) \mapsto p \cdot g$

Examples in physics:

Gauge Theory	Structure Group $G$	Principal Bundle
Electromagnetism	$U(1)$	Circle bundle over spacetime
Weak force	$SU(2)$	$SU(2)$ -bundle
Strong force (QCD)	$SU(3)$	$SU(3)$ -bundle
Standard Model	$SU(3) \times SU(2) \times U(1)$	Product bundle

Connections on Principal Bundles

A connection on a principal bundle is a way to define “horizontal” directions—how to parallel transport along the base space.

Mathematically, a connection is a Lie algebra-valued 1-form:

$A \in \Omega^1(M, \mathfrak{g})$

where $\mathfrak{g}$ is the Lie algebra of $G$ .

This is exactly what physicists call a gauge field!

The photon field $A_\mu$ is a $\mathfrak{u}(1)$ -connection
The gluon field is an $\mathfrak{su}(3)$ -connection
The W and Z bosons come from an $\mathfrak{su}(2) \times \mathfrak{u}(1)$ -connection

Curvature as Field Strength

The curvature of a connection measures its failure to be flat:

$F = dA + A \wedge A$

In components:

$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$

This is the field strength tensor!

For electromagnetism ( $U(1)$ , abelian): $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$

This gives the electric and magnetic fields: $F_{0i} = -E_i, \quad F_{ij} = \epsilon_{ijk}B_k$

For non-abelian theories (like QCD), the commutator $[A_\mu, A_\nu]$ is non-zero, leading to gluon self-interactions.

Associated Vector Bundles

Matter fields live in associated vector bundles. Given a principal bundle $P$ and a representation $\rho: G \to GL(V)$ :

$E = P \times_\rho V = (P \times V) / G$

Sections of $E$ are the matter fields:

Electron field: section of a bundle associated to the spinor representation
Quark field: section of a bundle associated to the fundamental rep of $SU(3)$

The connection on $P$ induces a covariant derivative on sections:

$D_\mu \psi = \partial_\mu \psi + A_\mu \psi$

Spinor Geometry

Fermions require additional geometric structure: spin geometry.

The Spin Group

The rotation group $SO(n)$ has a double cover:

$Spin(n) \xrightarrow{2:1} SO(n)$

For Lorentzian signature:

$Spin(1,3) \xrightarrow{2:1} SO(1,3)$

The group $Spin(1,3) \cong SL(2,\mathbb{C})$ is the universal cover of the Lorentz group.

Spin Structures

A spin structure on a manifold $M$ is a lift of the frame bundle:

$\begin{array}{ccc} & & Spin(M) & \nearrow & \downarrow M & \leftarrow & SO(M) \end{array}$

Not every manifold admits a spin structure! The obstruction is the second Stiefel-Whitney class $w_2(M)$ .

Spinor Bundles

Given a spin structure, we can form the spinor bundle:

$S = Spin(M) \times_\rho \mathbb{C}^4$

where $\rho$ is the spinor representation of $Spin(1,3)$ .

Sections of $S$ are spinor fields—this is where the Dirac field $\psi$ lives.

The Dirac Operator

The Dirac operator is a first-order differential operator on spinors:

$D!!!!/ = \gamma^\mu \nabla_\mu$

where $\gamma^\mu$ are the gamma matrices satisfying the Clifford algebra:

${\gamma^\mu, \gamma^\nu} = 2g^{\mu\nu}$

The Dirac equation is:

$(i D!!!!/ - m)\psi = 0$

Gauge Theory Geometry

Yang-Mills Theory

The dynamics of gauge fields is governed by the Yang-Mills action:

$S_{YM} = -\frac{1}{4} \int_M \text{Tr}(F \wedge *F)$

where $*$ is the Hodge star operator.

In components:

$S_{YM} = -\frac{1}{4} \int d^4x \, F^a_{\mu\nu} F^{a\mu\nu}$

The Euler-Lagrange equations give the Yang-Mills equations:

$D_\mu F^{\mu\nu} = J^\nu$

Gauge Transformations

A gauge transformation is a section $g: M \to G$ acting on:

Connections: $A \mapsto g^{-1}Ag + g^{-1}dg$
Matter fields: $\psi \mapsto \rho(g^{-1})\psi$

The curvature transforms covariantly:

$F \mapsto g^{-1}Fg$

This is why $\text{Tr}(F \wedge *F)$ is gauge-invariant.

Instantons and Topology

In Euclidean Yang-Mills theory, there exist non-trivial solutions called instantons—connections with self-dual curvature:

$F = *F$

Instantons are classified by the instanton number (second Chern class):

$k = \frac{1}{8\pi^2} \int_M \text{Tr}(F \wedge F) \in \mathbb{Z}$

This topological invariant has physical consequences (tunneling between vacua, the $\theta$ -term in QCD).

Electromagnetism as U(1) Gauge Theory

The simplest example: electromagnetism is a $U(1)$ gauge theory.

The Setup

Principal bundle: $P = M \times U(1)$ (trivial bundle)
Connection: $A = A_\mu dx^\mu$ (the electromagnetic potential)
Curvature: $F = dA$ (the field strength)

Matter Coupling

An electron field $\psi$ is a section of a line bundle with charge $e$ :

$D_\mu \psi = (\partial_\mu + ieA_\mu)\psi$

The Lagrangian:

$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

Gauge Invariance

Under $U(1)$ transformation $g(x) = e^{i\alpha(x)}$ :

$\psi \mapsto e^{-ie\alpha}\psi, \quad A_\mu \mapsto A_\mu + \partial_\mu \alpha$

The Lagrangian is invariant.

QCD as SU(3) Gauge Theory

Quantum Chromodynamics is an $SU(3)$ gauge theory.

The Setup

Structure group: $SU(3)$ (color symmetry)
Connection: $A_\mu = A_\mu^a T^a$ (8 gluon fields)
Generators: $T^a = \frac{1}{2}\lambda^a$ (Gell-Mann matrices)

Field Strength

$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c$

The structure constants $f^{abc}$ are non-zero, leading to gluon self-interactions.

Quarks

Quarks transform in the fundamental representation 3 of $SU(3)$ :

$\psi = \begin{pmatrix} \psi_r \ \psi_g \ \psi_b \end{pmatrix}$

The covariant derivative:

$D_\mu \psi = \partial_\mu \psi - ig A_\mu^a T^a \psi$

Comparison: Gauge Theory vs General Relativity

Both gauge theory and general relativity use the same geometric language: manifolds, connections, and curvature. In both, curvature is the force.

For a detailed comparison, see: Gauge Theory vs General Relativity: The Deep Unity

Wick Rotation and Euclidean QFT

The Problem with Minkowski Space

Path integrals in Minkowski space involve oscillatory integrands:

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

The factor $e^{iS}$ oscillates wildly and doesn’t converge.

Wick Rotation

We analytically continue to imaginary time:

$t \to -i\tau$

This transforms:

Minkowski metric $ds^2 = -dt^2 + d\vec{x}^2$ → Euclidean metric $ds^2 = d\tau^2 + d\vec{x}^2$
Action $iS$ → $-S_E$
Path integral $e^{iS}$ → $e^{-S_E}$

The Euclidean path integral converges:

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

Why It Works

The analytic continuation is justified when:

The action has suitable analyticity properties
Correlation functions can be continued back to Minkowski space

This is formalized by the Osterwalder-Schrader axioms, which give conditions for a Euclidean QFT to define a valid Lorentzian theory.

Applications

Lattice QFT: Discretize Euclidean spacetime, compute numerically
Instantons: Classical solutions in Euclidean space
Thermal field theory: Euclidean time becomes periodic with period $\beta = 1/T$

Summary: Mathematical Structures in QFT

Physics	Mathematics
Spacetime	Lorentzian manifold $(M, g)$
Gauge symmetry	Lie group $G$
Gauge field	Connection on principal $G$ -bundle
Field strength	Curvature of connection
Matter field	Section of associated vector bundle
Fermion	Section of spinor bundle
Gauge transformation	Bundle automorphism
Covariant derivative	Connection-induced derivative
Yang-Mills action	Integral of curvature squared
Path integral	Euclidean formulation via Wick rotation

References

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995)
M. Nakahara, Geometry, Topology and Physics, 2nd ed. (CRC Press, 2003)
T. Frankel, The Geometry of Physics, 3rd ed. (Cambridge University Press, 2011)
J. Baez and J. P. Muniain, Gauge Fields, Knots and Gravity (World Scientific, 1994)
S. Weinberg, The Quantum Theory of Fields, Vols. 1–2 (Cambridge University Press, 1995–1996)
R. W. R. Darling, Differential Forms and Connections (Cambridge University Press, 1994)
nLab: Gauge theory — Mathematical perspective