Theta Parameters, Theta States, and Sector Sums

A theta parameter, a character of large gauge transformations, a phase in a sector sum, and a theta-labeled state are equivalent descriptions only after the charge lattice and transform conventions have been fixed. For integral charge, the canonical relation is a Fourier transform: large transformations translate the winding label, theta states diagonalize that translation, and the Euclidean functional integral weights charge-$Q$ histories by $e^{i\theta Q}$. The branch index of a vacuum energy is a different label and must not be called theta.

Required background. Topological sectors, boundary data, and global form establishes the charge lattice and the restricted integrals $Z_Q$ used below.

Helpful background. Theta terms, periodicity, and vacuum sectors gives the primary coupling definition. Theta dependence in Yang–Mills and QCD develops the model-specific strong-interaction consequences.

Shared comparison. The sector and periodicity comparison fixes the global data behind the Fourier convention, while the sector–theta–branch map follows that transform into finite- and infinite-volume vacuum physics.

Theta as a character of large transformations

Suppose semiclassical gauge configurations have integer winding labels $n\in\mathbb Z$, and let $\mathcal U_\nu$ represent a large gauge transformation of winding $\nu$:

$$\mathcal U_\nu\lvert n\rangle=\lvert n+\nu\rangle.$$

The Fourier superposition

$$\lvert\theta\rangle=\mathcal N_\theta \sum_{n\in\mathbb Z}e^{-in\theta}\lvert n\rangle$$

then satisfies

$$\mathcal U_\nu\lvert\theta\rangle =e^{i\nu\theta}\lvert\theta\rangle.$$

Thus $e^{i\nu\theta}$ is a one-dimensional unitary representation—a character—of the large-transformation group. The sign in the coefficient $e^{-in\theta}$ is conventional; changing it changes the eigenvalue phase and the Euclidean Fourier convention together. Coleman develops this canonical construction in Coleman 1985, ch. 7, § 3.3, pp. 291–295, and Weinberg gives the gauge-theory derivation in Weinberg 1996, §§ 23.5–23.6, pp. 450–457.

The infinite sum is Bloch-like and is generally not a normalizable vector with an ordinary finite constant $\mathcal N_\theta$. It is understood through a finite regulator or a delta normalization in theta, just as a plane wave is. Physical predictions use normalized wave packets, finite-volume traces, or ratios of amplitudes, so the formal character construction does not create an infinite probability.

The states $\lvert n\rangle$ here are localized semiclassical reference states, not generally exact energy eigenstates. The exact statement is that physical states can transform in character sectors of the allowed large transformations. Which characters exist depends on the global form and matter content.

Fixed charge and fixed theta

In Euclidean signature use the weight

$$e^{-S_E+i\theta Q}.$$

For $Q\in\mathbb Z$, the fixed-theta partition function and its fixed-charge components obey

$$\begin{aligned} Z(\theta)&=\sum_{Q\in\mathbb Z}e^{i\theta Q}Z_Q,\\ Z_Q&=\frac1{2\pi}\int_0^{2\pi}\!\mathrm d\theta\, e^{-i\theta Q}Z(\theta). \end{aligned}$$

The first line is a sector-weighted functional integral. The second is Fourier inversion. Neither line says that theta is dynamically minimized: $\theta$ is a coupling held fixed when the theory is specified, unless another dynamical field such as an axion is explicitly introduced.

The relation also proves $2\pi$ periodicity under its assumptions. If the charge lattice is fractional, or if a $2\pi$ shift changes discrete theta data, the transform domain and the meaning of periodicity change; the dependence on global form and line-operator data is developed in Aharony, Seiberg, and Tachikawa 2013, §§ 1.2 and 6.1. The precise examples are compared on Topological Sectors, Boundary Data, and Global Form.

Particle on a circle

A particle with angular coordinate $\phi\sim\phi+2\pi$ is a clean $0+1$-dimensional, one-degree-of-freedom model. Its Hilbert space is nevertheless infinite-dimensional. Euclidean paths close up to a winding,

$$\phi(\beta)=\phi(0)+2\pi n, \qquad n=\frac1{2\pi}\int_0^\beta\!\mathrm d\tau\,\dot\phi\in\mathbb Z.$$

Adding the topological term means that the path of winding $n$ carries $e^{i\theta n}$. The Lorentzian Lagrangian and canonical momentum are

$$L_\theta=\frac I2\dot\phi^2+\frac{\theta}{2\pi}\dot\phi, \qquad p_\phi=I\dot\phi+\frac{\theta}{2\pi}.$$

The second term is a total derivative locally, so it does not change the classical equation of motion, but its integral distinguishes winding histories. Solving for $\dot\phi$ gives the Hamiltonian

$$H_\theta=\frac1{2I} \left(-i\frac{\partial}{\partial\phi}-\frac{\theta}{2\pi}\right)^2,$$

with spectrum

$$E_m(\theta)=\frac1{2I} \left(m-\frac{\theta}{2\pi}\right)^2, \qquad m\in\mathbb Z.$$

Poisson resummation makes the equality between the canonical trace and the winding-sector sum explicit:

$$\begin{aligned} Z_\beta(\theta) &=\operatorname{Tr}e^{-\beta H_\theta} =\sum_{m\in\mathbb Z} \exp\!\left[-\frac{\beta}{2I} \left(m-\frac{\theta}{2\pi}\right)^2\right]\\ &=\sqrt{\frac{2\pi I}{\beta}} \sum_{n\in\mathbb Z} \exp\!\left(-\frac{2\pi^2 I n^2}{\beta}+i\theta n\right). \end{aligned}$$

The last sign follows the winding orientation used above; reversing $n$ gives the equivalent Poisson convention. The Gaussian prefactor is the common fluctuation determinant about each classical winding path. For every finite $\beta>0$, the uniformly convergent trace is smooth and positive. Ground-state projection instead gives

$$-\lim_{\beta\to\infty}\frac1\beta\log Z_\beta(\theta) =\min_{m\in\mathbb Z}E_m(\theta),$$

so the cusp at $\theta=\pi$ appears only in the infinite-Euclidean-time lower envelope. This is a complete, solvable example of an order of limits producing a nonanalytic ground-state energy from analytic finite-$\beta$ data.

Although one level $E_m(\theta)$ is not periodic, the spectrum as a set is: $E_m(\theta+2\pi)=E_{m-1}(\theta)$. This is the simplest example of periodic physics produced by relabeling branches. It also shows why “theta state” and “branch $m$” are not synonyms.

The particle is an analogy, not a derivation of Yang–Mills dynamics. Its winding sectors and canonical momentum are explicit, but it has no four-dimensional gauge bundle, center one-form symmetry, or QCD phenomenology.

Four meanings that must remain separate

Phrase	Mathematical role	What fixes it	What it is not
Theta coupling	Coefficient of a topological term in the action	Theory definition, charge normalization, and global form	A variable minimized by the vacuum in ordinary fixed-$\theta$ QFT
Theta character	Phase $e^{i\nu\theta}$ under a large transformation	Character group of the allowed transformations	A perturbative gauge transformation
Theta-labeled state	State in a character sector	Hilbert space, boundary conditions, and transform convention	One semiclassical winding state $\lvert n\rangle$
Sector weight	Fourier phase $e^{i\theta Q}$ multiplying $Z_Q$	Euclidean continuation and charge convention	Evidence that a dilute instanton gas is controlled
Vacuum branch $k$	Locally smooth candidate energy exchanged under parameter shifts	Dynamics and the chosen approximation	The theta parameter itself

Callan, Dashen, and Gross established the physical role of tunneling sectors and theta dependence in the gauge vacuum Callan, Dashen, and Gross 1976, pp. 334–340. Mariño gives a modern fixed-sector and theta-vacuum construction in Mariño 2015, § 4.3, pp. 112–124. Their semiclassical language should be translated into the modern global-form specification before it is applied to a particular gauge theory.

Checks and limitations

Three checks should accompany any theta construction.

Fourier round trip. Transform $Z_Q$ to $Z(\theta)$ and back with the same sign and normalization. A sign error changes odd cumulants and the transformation law of $\lvert\theta\rangle$.

Period from the charge lattice. Derive the smallest $\Delta\theta$ satisfying $e^{i\Delta\theta Q}=1$ for every allowed $Q$, including bundle and defect sectors. Do not assume $\Delta\theta=2\pi$ from notation.

CP transformation. When the topological density is CP odd, CP sends $\theta\mapsto-\theta$. CP is a symmetry only where $-\theta$ is equivalent to $\theta$ after all continuous and discrete theta data are included. The dynamical realization at such points is treated on Theta Dependence, CP, and Model-Dependent Branches.

Common pitfalls

Treating theta as a chosen vacuum expectation value. In a fixed theory theta is a parameter. A dynamical axion or another field can relax an effective angle, but that is additional dynamics.

Confusing winding references with exact vacua. The $\lvert n\rangle$ basis is useful for constructing the character eigenstate. Tunneling generally mixes those reference configurations, and exact energy eigenstates need not have definite $n$.

Inferring instanton control from the Fourier sum. The decomposition into $Z_Q$ is exact once the regulator and configuration space are defined. Approximating $Z_Q$ by a few instantons is a separate semiclassical step.

Exercises

Starting from $\lvert\theta\rangle=\sum_n e^{-in\theta}\lvert n\rangle$, verify the large-transformation eigenvalue and show that $\lvert\theta+2\pi\rangle=\lvert\theta\rangle$ up to normalization.

Solution

Relabeling $m=n+\nu$ gives

$$\mathcal U_\nu\lvert\theta\rangle =\sum_m e^{-i(m-\nu)\theta}\lvert m\rangle =e^{i\nu\theta}\lvert\theta\rangle.$$

For integer $n$, $e^{-in(\theta+2\pi)}=e^{-in\theta}$ term by term, so the state is $2\pi$ periodic in this integral-lattice construction.

For the particle on a circle, locate the ground-state branch crossing in $0\leq\theta\leq2\pi$ and determine the one-sided derivatives of the ground-state energy there.

Solution

The branches $m=0$ and $m=1$ cross at $\theta=\pi$. The lower envelope is

$$E_0^{\mathrm{gs}}(\theta)= \begin{cases} \theta^2/(8\pi^2 I), & 0\leq\theta\leq\pi,\\ (2\pi-\theta)^2/(8\pi^2 I), & \pi\leq\theta\leq2\pi. \end{cases}$$

The left and right derivatives at $\pi$ are $+1/(4\pi I)$ and $-1/(4\pi I)$. The cusp belongs to the lower envelope; each quadratic branch is smooth.

Starting from $L_\theta$, perform a Legendre transform and verify both the sign of the momentum shift in $H_\theta$ and the relabeling $E_m(\theta+2\pi)=E_{m-1}(\theta)$.

Solution

Since $\dot\phi=(p_\phi-\theta/2\pi)/I$,

$$H=p_\phi\dot\phi-L_\theta =\frac1{2I}\left(p_\phi-\frac{\theta}{2\pi}\right)^2.$$

Single-valued wavefunctions have $p_\phi=m\in\mathbb Z$, which gives the displayed spectrum. Then

$$E_m(\theta+2\pi) =\frac1{2I}\left(m-1-\frac{\theta}{2\pi}\right)^2 =E_{m-1}(\theta).$$

The check ties the Euclidean phase, canonical momentum shift, and branch relabeling to one sign convention.

Continue from the transform to dynamics

Use Theta Dependence, CP, and Model-Dependent Branches to learn what the complete spectrum or energy envelope can do after the Fourier convention is fixed. Use Instantons, Fermion Zero Modes, and Tunneling when the task is to calculate the sector-changing amplitude rather than merely organize it.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. arXiv. DOI.
• Callan, Curtis G., Roger F. Dashen, and David J. Gross. “The Structure of the Gauge Theory Vacuum.” Physics Letters B 63, no. 3 (1976): 334–340. CERN record. DOI.
• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 7, § 3.3, pp. 291–295. DOI.
• Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015, § 4.3, pp. 112–124. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume II: Modern Applications. Cambridge University Press, 1996, §§ 23.5–23.6, pp. 450–457. DOI.