Coherent-State Path Integrals

A coherent-state path integral rewrites operator evolution using an overcomplete family of states rather than an orthogonal coordinate basis. Its distinctive first-order term is not guessed from a classical field equation. It comes from the phase of adjacent coherent-state overlaps.

Helpful background. Oscillator coherent states supply the overlap and identity resolution; thermal coherent-state integrals explain the finite-slice trace and operator ordering.

At finite slice number, a transition amplitude has the skeleton

$$K = \int \prod_{j=1}^{M-1} d\mu(\xi_j) \times \prod_{j=0}^{M-1} \langle\xi_{j+1}| e^{-i\epsilon H/\hbar} |\xi_j\rangle.$$

The short-time Hamiltonian matrix element supplies an operator symbol. The resolution of identity supplies the measure; the adjacent overlap supplies its phase and normalization, including endpoint factors. Only after preserving all four pieces may one use the compact continuum action

$$S[\xi] = \int dt\, \left[ \mathcal A_i(\xi)\dot\xi^i - H_\sigma(\xi) \right].$$

Here $\mathcal A$ is a Berry one-form and $H_\sigma$ is the Hamiltonian symbol selected by the slicing convention. The notation is geometric, but the construction remains anchored to a regulated Hilbert-space amplitude.

Coherent-State Amplitudes on a Time Lattice

Continuum coherent-state notation suppresses data that can change the answer. A trustworthy calculation first records:

1. the Hilbert space and coherent-state family;
2. normalization and the resolution of identity;
3. the finite time lattice and direction of adjacent overlaps;
4. the operator symbol evaluated on adjacent slices;
5. endpoint or thermal closure conditions;
6. the limiting procedure used to remove the time lattice.

For a single bosonic mode, for example, the short-time ratio

$$\frac{ \langle z_{j+1}| e^{-i\epsilon H/\hbar} |z_j\rangle }{ \langle z_{j+1}|z_j\rangle }$$

is the primary object. Replacing it immediately by an arbitrary diagonal function of one smooth variable can lose ordering constants and endpoint terms. A continuum action is shorthand for a specified limit, not an independent definition.

Geometry from Overlaps

Let $|\xi\rangle$ be a smooth normalized coherent-state family. Its local Berry one-form is

$$\mathcal A = i\hbar \langle\xi|d\xi\rangle.$$

Normalization implies that $\langle\xi|d\xi\rangle$ is purely imaginary, so $\mathcal A$ is real on an ordinary real parameter manifold. Its curvature is

$$\Omega = d\mathcal A.$$

For the canonical boson and nonzero-spin families below, this curvature is nondegenerate and supplies a symplectic form. An arbitrary parameterized state family can instead have redundant coordinates or a degenerate pullback curvature.

The phase convention of the state is a gauge choice. Under

$$|\xi\rangle \longmapsto e^{i\chi(\xi)}|\xi\rangle,$$

the one-form changes as

$$\mathcal A \longmapsto \mathcal A - \hbar\,d\chi,$$

while $\Omega$ is unchanged. For a closed path $C$ contained in a valid gauge patch,

$$\exp\left( \frac{i}{\hbar} \oint_C\mathcal A \right)$$

is invariant under single-valued gauge changes. If $C=\partial\Sigma$, then

$$\oint_C\mathcal A = \int_\Sigma\Omega$$

whenever one patch covers the chosen surface. Multiple patches require transition functions, as the spin sphere will illustrate.

The overlap gives an operational derivation. For nearby normalized states,

$$\langle\xi+d\xi|\xi\rangle = \exp\left[ -\langle\xi|d\xi\rangle + \mathcal O(d\xi^2) \right].$$

Multiplying these phases along the discretized path produces the line integral of $\mathcal A$. The quadratic part of the overlap controls the local metric and short-step suppression. Phase and magnitude are therefore two aspects of the same overlap kernel.

The Canonical Bosonic Plane

For one oscillator mode, use normalized coherent states

$$|z\rangle = e^{-|z|^2/2} e^{z a^\dagger}|0\rangle, \qquad a|z\rangle = z|z\rangle.$$

Their overlap and identity resolution are

$$\begin{aligned} \langle z'|z\rangle &= \exp\left[ -\frac{|z'|^2}{2} -\frac{|z|^2}{2} +\bar z'z \right], \\ I &= \int\frac{d^2z}{\pi} |z\rangle\langle z|. \end{aligned}$$

Differentiating the overlap gives

$$\mathcal A_{\mathrm B} = \frac{i\hbar}{2} \left( \bar z\,dz - z\,d\bar z \right),$$

and hence

$$\Omega_{\mathrm B} = i\hbar\, d\bar z\wedge dz.$$

Introduce scaled real canonical coordinates, both with units of the square root of action, through

$$z = \frac{q+ip}{\sqrt{2\hbar}}.$$

Then

$$\begin{aligned} \mathcal A_{\mathrm B} &= \frac12 \left( p\,dq-q\,dp \right), \\ \Omega_{\mathrm B} &= dp\wedge dq. \end{aligned}$$

For an oscillator of mass $m$ and frequency $\omega$, these can be $q=\sqrt{m\omega}\,x$ and $p=p_x/\sqrt{m\omega}$; one must not add an unscaled physical position to a physical momentum. This differs from the common canonical potential $p\,dq$ by a total derivative. Both produce the same curvature and bulk Hamilton equations, but their endpoint terms differ.

Plane and Sphere Geometry

For $z=r e^{i\phi}$ traversed with increasing $\phi$, the bosonic phase is $\oint\mathcal A_{\mathrm B}/\hbar=-2\pi r^2$. For the highest-weight spin state at fixed polar angle $\theta$, the same orientation gives $-s\Omega_{\mathrm{cap}}=-2\pi s(1-\cos\theta)$. Both follow from adjacent overlaps; the sphere additionally needs gauge patches. The shaded areas use their stated orientations.

Symmetric and Holomorphic Actions

The normalized-state Berry form leads to the symmetric real-time kinetic term

$$S_{\mathrm B} = \int_0^T dt\, \left[ \frac{i\hbar}{2} \left( \bar z\dot z - \dot{\bar z}z \right) - H_\sigma(\bar z,z) \right].$$

Many-body calculations often use the holomorphic form

$$S_{\mathrm{hol}} = \int_0^T dt\, \left[ i\hbar\bar z\dot z - H_\sigma(\bar z,z) \right] + S_\partial.$$

The two bulk kinetic terms obey

$$i\hbar\bar z\dot z = \frac{i\hbar}{2} \left( \bar z\dot z - \dot{\bar z}z \right) + \frac{i\hbar}{2} \frac{d}{dt} (\bar z z).$$

The total derivative vanishes for a smooth closed path, but not for an open propagator. It must be combined with the coherent-state normalization and the boundary term inherited from the finite-slice product.

The bulk equations are first order:

$$\begin{aligned} i\hbar\dot z &= \frac{\partial H_\sigma}{\partial\bar z}, \\ -i\hbar\dot{\bar z} &= \frac{\partial H_\sigma}{\partial z}. \end{aligned}$$

The original bosonic integration at each slice is over two real variables, with $\bar z=z^*$ on that contour; treating them independently during variation does not double that integration dimension. On a real classical trajectory $\bar z=z^*$, but a saddle satisfying mixed coherent-state endpoint data can be genuinely complex.

Mixed Endpoint Data

Consider the Bargmann kernel

$$K(\bar z_f,z_i;T) = (z_f| e^{-iHT/\hbar} |z_i),$$

where $|z)=e^{za^\dagger}|0\rangle$ is unnormalized. One useful action convention is

$$\mathcal S_{\mathrm{hol}} = -i\hbar\bar z_f z(T) + \int_0^Tdt\, \left[ i\hbar\bar z\dot z - H_\sigma(\bar z,z) \right].$$

The variational problem fixes

$$z(0)=z_i, \qquad \bar z(T)=\bar z_f.$$

It does not fix $\bar z(0)$ or $z(T)$. Varying the endpoint term cancels the remaining variation at $T$. Other normalizations distribute endpoint terms differently, but the finite-slice kernel fixes the complete answer.

This mixed boundary problem is the coherent-state analogue of fixing the initial coordinate and final conjugate coordinate. Imposing both $z$ and $\bar z$ at both ends generally overdetermines the first-order equations.

Euclidean Continuation

Under $t=-i\tau$, the real-time weight becomes a Euclidean weight,

$$e^{iS/\hbar} \longmapsto e^{-S_{\mathrm E}/\hbar}.$$

For a normal-ordered bosonic thermal generator $\mathcal K=H-\mu N$, the common holomorphic shorthand is

$$S_{\mathrm E} = \int_0^{\beta\hbar} d\tau\, \left[ \hbar\bar z\partial_\tau z + \mathcal K_{\mathrm N}(\bar z,z) \right],$$

with periodic closure for an untwisted trace over the full bosonic Fock space. A fixed-number trace needs the projection or phase-twist integral described in Path Integrals for Many-Body Systems; number-changing fields cannot simply be cycled within one number sector. The symmetric Euclidean kinetic term is

$$\frac{\hbar}{2} \left( \bar z\partial_\tau z - (\partial_\tau\bar z)z \right).$$

For a smooth periodic path the difference is a total derivative. At finite slice number or with open endpoints, that observation does not license discarding the boundary data. The thermal trace derivation and its convergence conditions remain in Coherent-State Path Integrals Preview.

Number–Phase Variables

Write a nonzero bosonic amplitude as

$$z = \sqrt n\,e^{-i\theta}.$$

Then

$$\mathcal A_{\mathrm B} = \hbar n\,d\theta, \qquad \Omega_{\mathrm B} = \hbar\,dn\wedge d\theta.$$

The action becomes

$$S = \int dt\, \left[ \hbar n\dot\theta - H_\sigma(n,\theta) \right],$$

so the equations are

$$\begin{aligned} \hbar\dot\theta &= \frac{\partial H_\sigma}{\partial n}, \\ \hbar\dot n &= - \frac{\partial H_\sigma}{\partial\theta}. \end{aligned}$$

Thus $\hbar n$ is conjugate to $\theta$. The phase is undefined at $n=0$, so polar variables do not provide one global coordinate chart on the complex plane. Vortices, zeros, and phase slips are precisely where this local chart requires care.

This relation is not the statement that the exact number operator has a self-adjoint phase operator with unrestricted canonical commutator. It is a classical relation on the coherent-state label manifold. Number quantization and phase-operator subtleties remain encoded in the original Hilbert space and finite-slice integral.

Exact Check: One Harmonic Mode

Take the normal-ordered Hamiltonian

$$H = \hbar\omega a^\dagger a.$$

The holomorphic symbol is

$$H_{\mathrm N}(\bar z,z) = \hbar\omega\bar z z.$$

The mixed-boundary saddle equations give

$$\begin{aligned} z(t) &= z_i e^{-i\omega t}, \\ \bar z(t) &= \bar z_f e^{i\omega(t-T)}. \end{aligned}$$

Along this path,

$$i\hbar\bar z\dot z - \hbar\omega\bar z z = 0.$$

Only the endpoint term remains, and therefore

$$(z_f| e^{-iHT/\hbar} |z_i) = \exp\left( \bar z_f z_i e^{-i\omega T} \right).$$

Restoring normalized endpoint states yields

$$\langle z_f| e^{-iHT/\hbar} |z_i\rangle = \exp\Bigg[ -\frac{|z_f|^2}{2} -\frac{|z_i|^2}{2} + \bar z_f z_i e^{-i\omega T} \Bigg].$$

This exact result checks the overlap convention, symbol, endpoint term, and equations simultaneously. A coherent-state derivation that misses any one of the three exponent terms has mishandled its boundaries or normalization.

Many Modes and Spatial Fields

For bosonic modes $a_r$, the geometric term adds over modes:

$$\mathcal A = \frac{i\hbar}{2} \sum_r \left( \bar z_r\,dz_r - z_r\,d\bar z_r \right).$$

The corresponding action is

$$S = \int dt\, \Bigg[ \frac{i\hbar}{2} \sum_r \left( \bar z_r\dot z_r - \dot{\bar z}_r z_r \right) - H_\sigma(\bar{\boldsymbol z},\boldsymbol z) \Bigg].$$

Choose a regulated spatial basis $u_r(\mathbf x)$ and define

$$\psi(\mathbf x,t) = \sum_r u_r(\mathbf x)z_r(t).$$

For an orthonormal basis, the kinetic term becomes a spatial integral exactly within the chosen mode space. If the Hamiltonian has a local density, write

$$S = \int dt \int d^d x\, \Bigg[ \frac{i\hbar}{2} \left( \psi^*\partial_t\psi - (\partial_t\psi^*)\psi \right) - \mathcal H_\sigma \Bigg].$$

In the continuum, or when the functional derivative already lies in the retained mode space, its stationary equation is

$$i\hbar\partial_t\psi = \frac{\delta H_\sigma}{\delta\psi^*}.$$

For a finite-mode restriction, project the right side onto that same mode space. Nonlocal interactions may instead require a double spatial integral. This is the geometric origin of the first-order time derivative in a nonrelativistic Schrödinger field action. The field $\psi$ is still an integration coordinate. It becomes a mean field only after a saddle or other controlled approximation is selected.

Density–Phase Fields

Where $\psi$ is nonzero, write

$$\psi(\mathbf x,t) = \sqrt{\rho(\mathbf x,t)} e^{-i\theta(\mathbf x,t)}.$$

The Berry term is

$$S_{\mathrm B} = \hbar \int dt \int d^d x\, \rho\,\partial_t\theta.$$

Consequently,

$$\hbar\partial_t\rho = - \frac{\delta H}{\delta\theta}.$$

For a local, phase-shift-invariant Hamiltonian density, $\theta$ enters through spatial derivatives and the equation takes a local continuity-equation form. A nonlocal Hamiltonian can depend on phase differences at separated points; global number conservation then does not by itself imply this local derivative-only form. The conjugacy between density and phase is therefore inherited from the coherent-state overlap, not added as a hydrodynamic guess.

Zeros of $\psi$ make $\theta$ singular. A density–phase effective theory must specify whether vortex sectors, compactness, winding, and amplitude zeros are retained or excluded. Those choices can carry physical information that a smooth noncompact phase field misses.

Fermionic Coherent Fields

For one fermionic mode, an unnormalized coherent state can be defined by

$$|\eta\rangle = e^{-\eta a^\dagger}|0\rangle,$$

where $\eta$ is Grassmann odd. With a consistent convention,

$$a|\eta\rangle = \eta|\eta\rangle, \qquad \langle\bar\eta|\eta'\rangle = e^{\bar\eta\eta'}.$$

The overlap produces the first-order bilinear

$$S_{\mathrm F} = \int dt\, \left[ i\hbar\bar\eta\dot\eta - H_{\mathrm N}(\bar\eta,\eta) \right].$$

For spatial modes, this becomes schematically

$$S_{\mathrm F} = \int dt \int d^d x\, \left[ i\hbar\bar\psi\partial_t\psi - \mathcal H_{\mathrm N} \right].$$

This is a graded construction, not an ordinary classical phase plane. $\bar\eta$ and $\eta$ are algebraically independent, nilpotent variables; they are not numerical complex conjugates. Berezin integration extracts coefficients rather than averaging over a positive measure.

In an untwisted ordinary thermal trace over the full fermionic Fock space, coherent fields close antiperiodically. The minus sign comes from the coherent-state trace construction, not from replacing the physical trace by a supertrace. A number projection modifies the closure by a phase twist. A Gaussian in complex Grassmann pairs produces a determinant; a real/Majorana Gaussian instead produces a Pfaffian. Those signs and measure-ordering rules are derived in Coherent-State Path Integrals Preview.

The fermionic kinetic term is often called a Berry term by analogy. The analogy is useful at the level of first-order overlap geometry, but it should not be used to assign an ordinary area or probability distribution to Grassmann space.

The Spin Coherent-State Sphere

Fix a spin-$s$ irreducible representation. Spin coherent states are rotations of an extremal-weight state and resolve the identity as

$$I = \frac{2s+1}{4\pi} \int_{S^2}d\Omega\, |\mathbf n;s\rangle \langle\mathbf n;s|.$$

The parameter manifold is the sphere of directions $\mathbf n$, not the complex plane. For $s>0$, use the highest-weight state, parameterize $\mathbf n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, and adopt the convention

$$\mathcal A = i\hbar \langle\mathbf n;s|d|\mathbf n;s\rangle.$$

A gauge regular near the north pole has

$$\mathcal A_{\mathrm N} = \hbar s (\cos\theta-1) d\phi.$$

A gauge regular near the south pole has

$$\mathcal A_{\mathrm S} = \hbar s (\cos\theta+1) d\phi.$$

On their overlap,

$$\mathcal A_{\mathrm S} - \mathcal A_{\mathrm N} = 2\hbar s\,d\phi.$$

Both give the same curvature,

$$\Omega_s = -\hbar s \sin\theta\, d\theta\wedge d\phi.$$

The sign follows from the highest-weight and rotation conventions. Using the lowest-weight orbit at the same direction labels, or reversing the orientation of those labels, reverses the sign. A mere phase-gauge change cannot reverse the curvature. For $s=0$ the family is one state, so the sphere labels are redundant and its curvature vanishes.

The curvature flux is

$$\frac{1}{2\pi\hbar} \int_{S^2}\Omega_s = -2s.$$

Because $2s$ is an integer, the phase acquired when switching patches around a closed winding is unity:

$$\exp\left( \frac{i}{\hbar} \oint (\mathcal A_{\mathrm S} -\mathcal A_{\mathrm N}) \right) = e^{i4\pi s m} = 1.$$

The familiar spin quantization condition is thus visible as the integrality needed to glue local coherent-state actions into one quantum amplitude.

Spin Action and Precession

In the northern patch, the real-time action is

$$S[\mathbf n] = \int dt\, \left[ \hbar s (\cos\theta-1) \dot\phi - H_\sigma(\mathbf n) \right].$$

For

$$H = \omega S_z,$$

the coherent-state expectation is

$$H_Q = \hbar\omega s\cos\theta.$$

The Euler–Lagrange equations give

$$\dot\theta = 0, \qquad \dot\phi = \omega.$$

This agrees with exact rotation about the $z$ axis. The Berry term is essential: without it, the action would have no symplectic structure and no spin precession equation.

For a lattice of spins, the total Berry term is initially a sum over sites. Coarse graining can combine those local phases into continuum Wess–Zumino or theta-like structures. Whether terms cancel, add, or interfere depends on the microscopic spin representation, lattice, ordering pattern, and spacetime topology. From Berry Phase to Topological Terms develops that continuation.

Euclidean Spin Berry Term

For any real-time first-order action,

$$S = \int\mathcal A - \int dt\,H,$$

Wick rotation gives

$$S_{\mathrm E} = \int d\tau\,H - i\int\mathcal A.$$

In the northern spin patch,

$$S_{\mathrm E}[\mathbf n] = \int d\tau\, \Big[ H_\sigma(\mathbf n) + i\hbar s (1-\cos\theta) \partial_\tau\phi \Big].$$

The Euclidean Berry term is imaginary. Therefore a perfectly Hermitian spin Hamiltonian can lead to a complex Euclidean weight in this representation. That phase can encode interference among topological sectors; it is not an algebraic error to be deleted merely to obtain a positive sampling weight.

Berry Does Not Mean Adiabatic Here

The one-form $i\hbar\langle\xi|d\xi\rangle$ is mathematically the same kind of connection used in the adiabatic Berry phase. In a coherent-state path integral, however, it arises from neighboring basis overlaps and requires no adiabatic approximation. Every path in the regulated sum carries it.

Adiabatic Berry phase concerns an instantaneous eigenstate transported slowly through an external parameter space. Coherent-state Berry phase concerns the geometry of the chosen overcomplete state manifold. The two constructions can coincide in special problems, but their assumptions and parameter spaces must not be conflated. Berry Phase explains the adiabatic theorem setting.

Operator Symbols Are Part of the Action

Several functions can represent one operator. For normalized bosonic coherent states, the covariant or $Q$ symbol is

$$H_Q(\bar z,z) = \langle z|H|z\rangle.$$

If $H$ is normal ordered, its off-diagonal normal symbol is fixed by

$$\frac{ \langle z'|H|z\rangle }{ \langle z'|z\rangle } = H_{\mathrm N}(\bar z',z).$$

An upper or $P$ symbol instead satisfies a representation of the form

$$H = \int d\mu(z)\, H_P(\bar z,z) |z\rangle\langle z|.$$

Weyl symbols use symmetric operator ordering and a different phase-space kernel. These symbols generally differ. In a specified classical scaling their differences can be organized in powers of $\hbar$ or inverse spin; without that scaling they need not be numerically small. Such corrections can be subleading in a classical trajectory while remaining order one in the phase $S/\hbar$.

The safe rule is:

1. derive the adjacent-slice matrix element;
2. identify the symbol it contains;
3. keep that symbol and discretization paired;
4. benchmark against an exactly soluble operator problem.

For example, replacing $(a^\dagger a)^2$ by $(\bar z z)^2$ skips the commutator needed to normal order the operator. The detailed correction and finite-temperature benchmark are derived in Coherent-State Path Integrals Preview.

Semiclassical Corrections and Continuum Hazards

A stationary coherent-state path is only the first term of an asymptotic calculation. A phase-accurate propagator may also require:

• the correct mixed-boundary classical solution;
• a fluctuation determinant and its branch;
• a Maslov-like or caustic prescription;
• symbol-dependent order-$\hbar$ terms;
• a Solari–Kochetov-type correction for relevant conventions;
• a contour choice for complex saddles.

There is no universal extra phase that can be appended independently of the coherent family, symbol, slicing, and measure. Coherent-State Semiclassics Preview develops that calculation.

Even before a saddle approximation, a naive continuum coherent-state action can fail for nonlinear Hamiltonians if the time lattice and symbol are removed inconsistently. A smooth-looking expression alone does not establish exact operator equivalence. Retain a finite-slice definition or another mathematically controlled regulator until exact benchmarks agree.

Bridge to Quantum Field Theory

Coherent-state path integrals lead to several field-theory structures, but not to one universal action.

• Bosonic Fock modes: complex labels $z_r,\bar z_r$ produce $i\hbar\bar z_r\dot z_r$ and lead to nonrelativistic bosonic fields.
• Fermionic Fock modes: Grassmann labels $\eta_r,\bar\eta_r$ produce $i\hbar\bar\eta_r\dot\eta_r$ and lead to fermion functional integrals.
• Fixed spins: directions $\mathbf n_r\in S^2$ produce local spin Berry one-forms and lead to sigma models or magnetic effective theories.
• Constrained or group-orbit families: quotient-space coordinates produce local connections and can lead to gauge, collective, or topological field variables.

Four distinctions remain essential.

First-order versus relativistic kinetics

A nonrelativistic bosonic coherent field naturally has $i\hbar\psi^*\partial_t\psi$. A relativistic scalar field commonly has a second-order kinetic term such as $|\partial_t\phi|^2$. The latter packages independent canonical data and, after quantization, particle and antiparticle modes. It is not obtained by merely renaming the nonrelativistic coherent label.

Integration field versus order parameter

The exact path integral integrates over all coherent labels allowed by the regulator. A saddle $\psi_*$ can approximate an order parameter, but the integration variable itself is not an expectation value. At finite volume, symmetry restoration, source limits, and fluctuations must still be handled.

Local geometry versus global sectors

The bosonic plane admits a global one-form, whereas the spin sphere requires patches. In a field theory, spatially varying coherent labels can have vortices, instantons, skyrmions, or other sectors that no single smooth chart covers. Global information can survive even when the local equations look ordinary.

Regulated modes versus continuum fields

Replacing a finite sum by $\int d^d x$ introduces an ultraviolet limit. Couplings, composite operators, and the measure can require matching or renormalization. Coherent-state geometry does not remove those obligations. Path Integrals from QM to Field Path Integrals and From Euclidean Time to Euclidean QFT develop the wider field-theory boundary.

Connections

• Path Integrals for Many-Body Systems compares coherent fields with worldline and local-basis representations.
• Coherent States develops oscillator eigenstates, displacement, statistics, and exact motion.
• Coherent States in Phase Space develops Wigner geometry and phase-space localization.
• Normal Ordering in Many-Body QM develops reference-dependent normal ordering and induced lower-body terms.
• Spin Coherent States develops the fixed-spin coherent family on $S^2$.
• Berry Connection and Berry Curvature develop gauge and curvature concepts independently of path integrals.
• Heisenberg Model supplies the canonical lattice setting for spin coherent fields.
• Off-Diagonal Long-Range Order distinguishes condensate order from an arbitrary coherent integration label.
• Nonrelativistic Field Theory from Many-Body QM develops operator fields, currents, propagators, and contact matching.

Exercises

1. Berry Form of a Bosonic Coherent State

Starting from

$$\langle z'|z\rangle = \exp\left[ -\frac{|z'|^2}{2} -\frac{|z|^2}{2} +\bar z'z \right],$$

derive $\langle z|d z\rangle$, $\mathcal A_{\mathrm B}$, and $\Omega_{\mathrm B}$.

Solution

Set $z'=z$ in the bra and vary the ket. To first order,

$$\langle z|z+dz\rangle = \exp\left[ \frac12 (\bar z\,dz-z\,d\bar z) \right] + \mathcal O(|dz|^2).$$

Therefore

$$\langle z|d z\rangle = \frac12 (\bar z\,dz-z\,d\bar z).$$

Multiplying by $i\hbar$ gives

$$\mathcal A_{\mathrm B} = \frac{i\hbar}{2} (\bar z\,dz-z\,d\bar z).$$

Taking an exterior derivative,

$$\begin{aligned} d\mathcal A_{\mathrm B} &= \frac{i\hbar}{2} \left( d\bar z\wedge dz - dz\wedge d\bar z \right) \\ &= i\hbar\, d\bar z\wedge dz. \end{aligned}$$

2. Total Derivative and Open Endpoints

Show that the holomorphic and symmetric bosonic kinetic terms differ by a total derivative. Explain why the difference vanishes for a smooth periodic path but cannot be dropped in an open propagator.

Solution

Use

$$\frac{d}{dt}(\bar z z) = \dot{\bar z}z + \bar z\dot z.$$

Then

$$i\hbar\bar z\dot z - \frac{i\hbar}{2} (\bar z\dot z-\dot{\bar z}z) = \frac{i\hbar}{2} \frac{d}{dt}(\bar z z).$$

For a smooth periodic path, $\bar z(T)z(T)=\bar z(0)z(0)$, so the integral of the derivative is zero. In an open kernel it equals

$$\frac{i\hbar}{2} \left[ \bar z(T)z(T) - \bar z(0)z(0) \right].$$

That contribution combines with coherent-state normalization and the finite-slice endpoint term. Dropping it changes the kernel.

3. Number–Phase Hamilton Equations

For $z=\sqrt n\,e^{-i\theta}$, derive the Berry form and the equations of motion generated by $H(n,\theta)$.

Solution

Direct differentiation gives

$$\bar z\,dz-z\,d\bar z = -2in\,d\theta.$$

Hence

$$\mathcal A_{\mathrm B} = \hbar n\,d\theta.$$

The action is

$$S = \int dt\, (\hbar n\dot\theta-H).$$

Variation with respect to $n$ gives

$$\hbar\dot\theta = \frac{\partial H}{\partial n}.$$

Variation with respect to $\theta$ gives

$$\hbar\dot n = - \frac{\partial H}{\partial\theta}.$$

Thus $\theta$ and $\hbar n$ form a canonical pair on the patch $n>0$.

4. Harmonic-Oscillator Kernel

Use the mixed endpoint equations for $H_{\mathrm N}=\hbar\omega\bar z z$ to recover the normalized coherent-state kernel.

Solution

The equations and boundary data are

$$\begin{aligned} \dot z &= -i\omega z, &z(0)&=z_i, \\ \dot{\bar z} &= i\omega\bar z, &\bar z(T)&=\bar z_f. \end{aligned}$$

Therefore

$$z(T) = z_i e^{-i\omega T}.$$

The bulk action vanishes on shell, while the Bargmann endpoint term gives

$$\frac{i\mathcal S_{\mathrm{cl}}}{\hbar} = \bar z_fz_i e^{-i\omega T}.$$

Normalized coherent states supply $e^{-|z_f|^2/2-|z_i|^2/2}$. Thus

$$K = \exp\Bigg[ -\frac{|z_f|^2}{2} -\frac{|z_i|^2}{2} + \bar z_fz_i e^{-i\omega T} \Bigg].$$

5. Spin Patches and Quantization

Using $\mathcal A_{\mathrm S}-\mathcal A_{\mathrm N}=2\hbar s\,d\phi$, show that changing patches around a loop of winding $m$ leaves the amplitude invariant precisely when $2s$ is an integer.

Solution

The action changes by

$$\Delta S = \oint (\mathcal A_{\mathrm S} - \mathcal A_{\mathrm N}).$$

For $\Delta\phi=2\pi m$,

$$\Delta S = 4\pi\hbar s m.$$

The path-integral phase changes by

$$e^{i\Delta S/\hbar} = e^{i4\pi sm} = e^{i2\pi(2s)m}.$$

This equals one for every integer winding $m$ exactly when $2s\in\mathbb Z$.

6. Spin Precession from the Berry Term

For $H_Q=\hbar\omega s\cos\theta$, derive the equations of motion from the northern-patch action.

Solution

The Lagrangian is

$$L = \hbar s(\cos\theta-1)\dot\phi - \hbar\omega s\cos\theta.$$

The $\theta$ equation is

$$-\hbar s\sin\theta\,\dot\phi + \hbar\omega s\sin\theta = 0.$$

Away from the coordinate poles,

$$\dot\phi = \omega.$$

The $\phi$ equation gives

$$-\hbar s\sin\theta\,\dot\theta = 0,$$

so $\dot\theta=0$. At a pole, $\phi$ is undefined and the physical direction is stationary; the vector equation $\dot{\mathbf n}=\omega\,\mathbf e_z\times\mathbf n$ is nonsingular there.

7. Why the Euclidean Spin Weight Is Complex

Show how a real Berry one-form becomes an imaginary term in the Euclidean action. Does a complex weight imply that the Hamiltonian is non-Hermitian?

Solution

Write the real-time action as

$$S = \int\mathcal A - \int dt\,H.$$

With $t=-i\tau$,

$$\frac{iS}{\hbar} = \frac{i}{\hbar} \int\mathcal A - \frac{1}{\hbar} \int d\tau\,H.$$

Therefore

$$S_{\mathrm E} = \int d\tau\,H - i\int\mathcal A.$$

For the northern spin gauge, this gives

$$S_{\mathrm E,B} = i\hbar s \int d\tau\, (1-\cos\theta) \partial_\tau\phi.$$

The complex weight records geometric phase interference. It does not imply a non-Hermitian Hamiltonian.

8. Repair a Naive Coherent-State Calculation

A proposed derivation for $H=U(a^\dagger a)^2$ replaces the Hamiltonian by $U(\bar z z)^2$, fixes both $z$ and $\bar z$ at both endpoints, and then takes the continuum limit before checking a finite-slice expression. Identify the three independent problems.

Solution

First, the operator is not already a single normal-ordered monomial:

$$(a^\dagger a)^2 = (a^\dagger)^2a^2 + a^\dagger a.$$

Its normal symbol is therefore

$$(\bar z z)^2 + \bar z z,$$

not merely $(\bar z z)^2$.

Second, first-order coherent-state equations require mixed endpoint data, such as $z(0)$ and $\bar z(T)$. Fixing both variables at both ends generally overdetermines the saddle.

Third, removing the time lattice before deriving the adjacent-slice symbol and overlap discards the regulator that defines the continuum expression. A correct calculation must restore the finite-slice kernel and verify it against the exact finite-dimensional or spectral answer.

References

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• E. A. Kochetov, “SU(2) Path Integral”, Journal of Mathematical Physics 36, 4667–4679 (1995) – boundary terms and stationary phase on the spin sphere.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998 reissue; originally Addison-Wesley, 1988) – bosonic and fermionic coherent-state functional integrals in many-body physics.
• H. G. Solari, “Semiclassical Treatment of Spin System by Means of Coherent States”, Journal of Mathematical Physics 28, 1097–1102 (1987) – discretization-sensitive spin coherent-state semiclassics.
• M. Stone, K.-S. Park, and A. Garg, “The Semiclassical Propagator for Spin Coherent States”, Journal of Mathematical Physics 41, 8025–8049 (2000) – fluctuation determinants and the Solari–Kochetov phase.
• J. H. Wilson and V. Galitski, “Breakdown of the Coherent State Path Integral: Two Simple Examples”, Physical Review Letters 106, 110401 (2011) – failures of naive continuum prescriptions for nonlinear Hamiltonians.