Imaginary Time

Imaginary-time evolution is the nonunitary operator semigroup

$$\mathsf S(\tau) = e^{-\tau\mathcal K/\hbar}, \qquad \tau\geq0,$$

generated by the Hamiltonian or by the ensemble generator $\mathcal K$.

For a canonical ensemble, $\mathcal K=H$. For a grand-canonical ensemble with conserved particle number,

$$\mathcal K = H-\mu N, \qquad [H,N]=0.$$

The same semigroup underlies two constructions that should be distinguished. On a long open interval, it filters toward low-energy states. At $\tau=\beta\hbar$, taking the trace constructs thermal equilibrium.

Imaginary time is not physical clock time. It is a spectral and equilibrium coordinate that converts oscillatory phases into energy-dependent damping.

Helpful background. The spectral theorem defines functions of a self-adjoint operator; thermal density operators explains why their traces give equilibrium weights.

The operator $e^{-\tau\mathcal K/\hbar}$ is defined by spectral calculus. A path integral or contour rotation can represent it, but neither is needed for its definition.

The imaginary-time generator and its domain

Unless stated otherwise:

1. $\mathcal K$ is self-adjoint;
2. its spectrum is bounded below;
3. $\tau$ has units of time;
4. $\beta=(k_{\mathrm B}T)^{-1}$ has units of inverse energy;
5. the thermal interval has length $\beta\hbar$.

If

$$\inf\operatorname{spec}\mathcal K = \kappa_{\min},$$

then shifting the generator to

$$\widetilde{\mathcal K} = \mathcal K-\kappa_{\min}I$$

makes it nonnegative. This shift changes an unnormalized propagator by an overall factor but does not change normalized states or normalized Gibbs expectations.

For the thermal construction assume $\beta>0$. The thermal trace requires more than a lower bound. One must also have

$$\operatorname{Tr} e^{-\beta\mathcal K} \lt \infty$$

in the regulated finite-volume problem. Infinite-volume systems are handled through densities, local observables, or a controlled thermodynamic limit rather than a literal trace over the full infinite system.

Real-Time Group and Imaginary-Time Semigroup

Real-time evolution is

$$U(t) = e^{-it\mathcal K/\hbar}, \qquad t\in\mathbb R.$$

Imaginary-time evolution is

$$\mathsf S(\tau) = e^{-\tau\mathcal K/\hbar}, \qquad \tau\geq0.$$

They share the same generator but have different operator geometry.

Property	Real time $U(t)$	Imaginary time $\mathsf S(\tau)$
parameter	$t\in\mathbb R$	$\tau\geq0$
composition	$U(t_1)U(t_2)=U(t_1+t_2)$	$\mathsf S(\tau_1)\mathsf S(\tau_2)=\mathsf S(\tau_1+\tau_2)$
norm behavior	unitary	energy-dependent damping
inverse	$U(-t)$ is bounded	$e^{+\tau\mathcal K/\hbar}$ is generally unbounded
spectral factor	$e^{-it\kappa/\hbar}$	$e^{-\tau\kappa/\hbar}$
typical use	physical dynamics	equilibrium, projection, heat kernels

The imaginary-time family is called a semigroup because only nonnegative $\tau$ is used. Running it backward would amplify arbitrarily high energies and is generally unstable or undefined as a bounded operation.

Semigroup law

Because both factors are functions of the same self-adjoint operator,

$$\begin{aligned} \mathsf S(\tau_1) \mathsf S(\tau_2) &= e^{-\tau_1\mathcal K/\hbar} e^{-\tau_2\mathcal K/\hbar} \\ &= e^{-(\tau_1+\tau_2)\mathcal K/\hbar} \\ &= \mathsf S(\tau_1+\tau_2). \end{aligned}$$

After shifting $\mathcal K$ so that $\mathcal K\geq0$,

$$\left\| \mathsf S(\tau) \right\| \leq 1.$$

Before the shift,

$$\left\| e^{-\tau\mathcal K/\hbar} \right\| = e^{-\tau\kappa_{\min}/\hbar}.$$

Spectral Definition

Let $P(d\kappa)$ be the projection-valued spectral measure of $\mathcal K$. Functional calculus defines

$$\mathsf S(\tau) = \int_{\operatorname{spec}\mathcal K} e^{-\tau\kappa/\hbar} P(d\kappa).$$

For a discrete eigenbasis,

$$\mathcal K|n\rangle = \kappa_n|n\rangle,$$

this becomes

$$\mathsf S(\tau) = \sum_n e^{-\tau\kappa_n/\hbar} |n\rangle\langle n|.$$

Thus every spectral component acquires a real damping factor. Ratios of components depend only on energy differences:

$$\frac{ e^{-\tau\kappa_n/\hbar} }{ e^{-\tau\kappa_m/\hbar} } = e^{-\tau(\kappa_n-\kappa_m)/\hbar}.$$

This equation is the common mechanism behind low-energy projection, exponential imaginary-time decay, and Boltzmann weighting.

Matrix elements are Laplace transforms

For states $|\phi\rangle$ and $|\psi\rangle$,

$$\langle\phi| \mathsf S(\tau) |\psi\rangle = \int e^{-\tau\kappa/\hbar} d\mu_{\phi,\psi}(\kappa),$$

where

$$d\mu_{\phi,\psi}(\kappa) = \langle\phi| P(d\kappa) |\psi\rangle.$$

Imaginary-time data are therefore Laplace-type transforms of spectral data. Laplace transforms smooth fine spectral structure. Their inversion is much less stable than their forward evaluation.

Imaginary-Time Schrödinger Equation

For an initial state $|\psi_0\rangle$, define

$$|\psi(\tau)\rangle = e^{-\tau\mathcal K/\hbar} |\psi_0\rangle.$$

Differentiation on the domain of $\mathcal K$ gives

$$-\hbar \frac{\partial}{\partial\tau} |\psi(\tau)\rangle = \mathcal K |\psi(\tau)\rangle.$$

This is the imaginary-time Schrödinger equation. It is first order in $\tau$, like the real-time equation, but it generates damping rather than unitary rotation.

The norm is generally not preserved:

$$\frac{d}{d\tau} \langle\psi(\tau)|\psi(\tau)\rangle = -\frac{2}{\hbar} \langle\psi(\tau)| \mathcal K |\psi(\tau)\rangle.$$

The sign of this derivative depends on the chosen energy zero. After shifting $\mathcal K\geq0$, the norm cannot increase.

Heat-Equation Form

For one particle,

$$H = -\frac{\hbar^2}{2m}\nabla^2 +V(\mathbf x),$$

the coordinate wavefunction obeys

$$\frac{\partial\psi}{\partial\tau} = \frac{\hbar}{2m} \nabla^2\psi -\frac{V}{\hbar}\psi.$$

The kinetic term is a diffusion operator with diffusion constant

$$D_{\mathrm E} = \frac{\hbar}{2m}.$$

The potential acts as a position-dependent growth or killing term. This heat-equation structure explains the name heat kernel for coordinate matrix elements of $e^{-\tau H/\hbar}$.

Free-particle heat kernel

For $V=0$ in $d$ spatial dimensions, write $\mathbf r=\mathbf x-\mathbf x'$ for the endpoint separation. Then

$$K_{\mathrm E} (\mathbf r,\tau) = \int \frac{d^d p}{(2\pi\hbar)^d} \times e^{i\mathbf p\cdot\mathbf r/\hbar} e^{-\tau\mathbf p^2/(2m\hbar)}.$$

Evaluating the Gaussian gives

$$K_{\mathrm E} = \left( \frac{m}{2\pi\hbar\tau} \right)^{d/2} \times \exp \left[ -\frac{ m|\mathbf r|^2 }{ 2\hbar\tau } \right].$$

For $\tau>0$, the kernel is a real Gaussian. It satisfies:

$$\int d^d x\, K_{\mathrm E} (\mathbf x,\tau;\mathbf x',0) = 1,$$

and the composition law

$$\int d^d y\, K_{\mathrm E} (\mathbf x,\tau_2;\mathbf y,0) K_{\mathrm E} (\mathbf y,\tau_1;\mathbf x',0) = K_{\mathrm E} (\mathbf x,\tau_1+\tau_2;\mathbf x',0).$$

The Gaussian broadens with $\tau$, even though spectral evolution suppresses high momentum. These are the same statement in conjugate representations. Here the heat equation propagates a wavefunction amplitude, rather than the probability density $|\psi|^2$; $D_{\mathrm E}$ is therefore not a measured particle diffusivity in laboratory time.

Imaginary-Time Heisenberg Operators

State evolution uses $e^{-\tau\mathcal K/\hbar}$. An imaginary-time Heisenberg operator is defined by the similarity transform

$$A(\tau) = e^{\tau\mathcal K/\hbar} A e^{-\tau\mathcal K/\hbar}.$$

This similarity transform is immediate for finite matrices. For unbounded generators it is understood on a suitable common domain, or inside thermal correlations where the products exist; $e^{+\tau\mathcal K/\hbar}$ need not act on every vector. It obeys

$$\hbar \frac{\partial A(\tau)}{\partial\tau} = \left[ \mathcal K,A(\tau) \right].$$

The positive exponential on the left of $A$ does not mean that physical states evolve with $e^{+\tau\mathcal K/\hbar}$. It appears because the operator carries the evolution removed from the state, just as in the ordinary Heisenberg picture.

Grand-canonical one-particle mode

Let

$$\mathcal K = \sum_{\mathbf k} \xi_{\mathbf k} c_{\mathbf k}^\dagger c_{\mathbf k}, \qquad \xi_{\mathbf k} = \epsilon_{\mathbf k}-\mu.$$

Using

$$\left[ \mathcal K,c_{\mathbf k} \right] = -\xi_{\mathbf k}c_{\mathbf k},$$

one obtains

$$c_{\mathbf k}(\tau) = e^{-\xi_{\mathbf k}\tau/\hbar} c_{\mathbf k}.$$

Similarly,

$$c_{\mathbf k}^\dagger(\tau) = e^{+\xi_{\mathbf k}\tau/\hbar} c_{\mathbf k}^\dagger.$$

The notation $c^\dagger(\tau)$ means the imaginary-time evolution of $c^\dagger$, not the adjoint of $c(\tau)$. In fact,

$$[c(\tau)]^\dagger=c^\dagger(-\tau).$$

Imaginary-time evolution is not a unitary similarity transformation. An operator can grow along part of the imaginary-time interval. Thermal correlation functions remain controlled because the trace supplies Boltzmann weights and KMS cyclicity relates the two ends of the interval.

Open and Closed Imaginary-Time Boundaries

The same semigroup answers different questions depending on its boundary condition.

The same exponential answers two different questions. Open propagation suppresses an excited component relative to the ground component; normalization and nonzero initial ground-state overlap are essential. A closed trace sums over all states and gives a thermal mixture at finite temperature. The diagram uses a nondegenerate two-state Hamiltonian with gap $\Delta>0$.

Construction	Operator object	Boundary data	Physical role
state propagation	$e^{-\tau H/\hbar}\lvert\psi_0\rangle$	initial state fixed	low-energy filtering
kernel	$\langle x_f\rvert e^{-\tau H/\hbar}\lvert x_i\rangle$	two endpoints fixed	heat kernel and Euclidean amplitude
matrix element	$\langle\phi\rvert e^{-\tau H/\hbar}\lvert\psi\rangle$	bra and ket fixed	spectral Laplace transform
thermal trace	$\operatorname{Tr}e^{-\beta\mathcal K}$	endpoints summed and identified	partition function
thermal correlator	$\operatorname{Tr}(e^{-\beta\mathcal K}\mathcal T_\tau\cdots)$	insertions on a circle	equilibrium fluctuations

Open propagation and a closed trace should not be interchanged. The limit $\tau\to\infty$ in an open projector is not the same operation as lowering the temperature in a normalized thermal trace, especially when degeneracies or the thermodynamic limit matter.

Thermal Trace

At temperature $T$,

$$\tau_\beta = \beta\hbar.$$

The grand partition function is

$$\mathcal Z = \operatorname{Tr} e^{-\beta\mathcal K} = \operatorname{Tr} e^{-\tau_\beta\mathcal K/\hbar}.$$

In an eigenbasis of $\mathcal K$,

$$\mathcal Z = \sum_n e^{-\beta\kappa_n}.$$

The normalized equilibrium state is

$$\rho_\beta = \frac{ e^{-\beta\mathcal K} }{ \mathcal Z }.$$

The trace closes the interval because

$$\operatorname{Tr} e^{-\beta\mathcal K} = \sum_a \langle a| e^{-\beta\mathcal K} |a\rangle.$$

The state at the final endpoint is identified with the state at the initial endpoint and then summed. In a coordinate path integral this becomes a closed path; in a coherent-state path integral it leads to periodic bosonic or antiperiodic fermionic fields.

Matsubara frequencies arise from the boundary conditions of ordered correlations. A fixed-$N$ trace supports the usual cyclic argument for operators preserving that sector; number-changing fields here use the full grand-canonical Fock-space trace.

Ground-State Projection Preview

Suppose

$$H|n\rangle = E_n|n\rangle, \qquad E_0\leq E_1\leq\cdots,$$

and

$$|\psi_0\rangle = \sum_n c_n|n\rangle.$$

Then

$$e^{-\tau H/\hbar} \lvert\psi_0\rangle = e^{-\tau E_0/\hbar} \lvert\widetilde\psi(\tau)\rangle,$$

where

$$\lvert\widetilde\psi(\tau)\rangle = c_0\lvert0\rangle + \sum_{n>0} c_n e^{-\tau(E_n-E_0)/\hbar} \lvert n\rangle.$$

If $c_0\neq0$ and the normalizable ground state is unique with a positive gap, the normalized state approaches it exponentially.

The suppression scale for the first excited component is

$$e^{-\tau\Delta/\hbar}, \qquad \Delta=E_1-E_0.$$

Several qualifications are essential:

• if $c_0=0$, evolution cannot create ground-state overlap;
• a preserved symmetry can confine the state to a sector whose lowest state is excited globally;
• a degenerate ground space is projected as a subspace, with coefficients inherited from the initial state;
• in a gapless spectrum convergence need not be exponential; without a normalizable lowest eigenstate there need not be any limiting normalized vector;
• normalization is required for a stable state interpretation;
• backward imaginary-time evolution is ill conditioned.

For example, if two states span a degenerate ground space, an open projection of $(\lvert0,a\rangle+\lvert0,b\rangle)/\sqrt2$ preserves that pure superposition. A normalized Gibbs trace as $T\to0$ instead gives an equal mixture on the two-dimensional ground space, provided the rest is gapped. The open and closed constructions agree on the lowest energy without producing the same state.

Imaginary-time projection develops the numerical algorithm and convergence checks.

Worked Example: Two-Level Filtering

Let

$$H = E_0\lvert0\rangle\langle0\rvert + E_1\lvert1\rangle\langle1\rvert.$$

Write the positive level spacing as $\Delta=E_1-E_0>0$, and choose

$$|\psi_0\rangle = \cos\theta\,|0\rangle +\sin\theta\,|1\rangle.$$

After imaginary time $\tau$,

$$|\psi(\tau)\rangle = e^{-E_0\tau/\hbar} \big[ \cos\theta\,|0\rangle + \sin\theta\, e^{-\Delta\tau/\hbar} |1\rangle \big].$$

The ground-state probability in the normalized state is

$$P_0(\tau) = \frac{ \cos^2\theta }{ \cos^2\theta + \sin^2\theta\, e^{-2\Delta\tau/\hbar} }.$$

The excited-state probability is

$$P_1(\tau) = \frac{ \sin^2\theta\, e^{-2\Delta\tau/\hbar} }{ \cos^2\theta + \sin^2\theta\, e^{-2\Delta\tau/\hbar} }.$$

Thus

$$\langle H\rangle_\tau = E_0+\Delta P_1(\tau).$$

If $\cos\theta=0$, then $P_0(\tau)=0$ for every $\tau$. Imaginary time suppresses unwanted components; it does not manufacture a missing overlap.

Normalized Imaginary-Time Flow

For projection calculations, define the normalized state

$$|\widehat\psi(\tau)\rangle = \frac{ |\psi(\tau)\rangle }{ \sqrt{ \langle\psi(\tau)|\psi(\tau)\rangle } }.$$

For a time-independent $H$, differentiating the propagated state and its normalization gives the nonlinear equation below. If $H$ is self-adjoint and bounded below, this differentiation is valid at every $\tau>0$ for any nonzero initial Hilbert-space vector: spectral damping puts the propagated state in the domain of every power of $H$. At $\tau=0$, take the initial state in $D(H)$ to justify the state derivative.

$$\hbar \frac{\partial}{\partial\tau} |\widehat\psi\rangle = - \left( H-\langle H\rangle_\tau \right) |\widehat\psi\rangle.$$

For a time-independent self-adjoint Hamiltonian with finite second energy moment,

$$\frac{d}{d\tau} \langle H\rangle_\tau = -\frac{2}{\hbar} \left[ \langle H^2\rangle_\tau -\langle H\rangle_\tau^2 \right].$$

Therefore

$$\frac{d}{d\tau} \langle H\rangle_\tau = -\frac{2}{\hbar} \operatorname{Var}_\tau(H) \leq0.$$

The energy derivative vanishes exactly when the state has support within one energy eigenspace. Monotone decrease by itself does not prove convergence to a normalizable ground state. This is a variational descent property of normalized imaginary-time flow, not a law of physical dissipation.

Imaginary-Time Correlators and Energy Gaps

For an operator $A$, define

$$A(\tau) = e^{\tau H/\hbar} A e^{-\tau H/\hbar}.$$

At zero temperature, assume a nondegenerate ground state and consider

$$C_A(\tau) = \langle0| A(\tau) A^\dagger(0) |0\rangle.$$

Inserting energy eigenstates gives

$$C_A(\tau) = \sum_n \left| \langle n|A^\dagger|0\rangle \right|^2 e^{-(E_n-E_0)\tau/\hbar}.$$

If $\langle0|A|0\rangle\ne0$, the $n=0$ term is a constant and the lowest visible energy is zero. To extract an excitation gap, first center $A$ by subtracting that expectation. The operator then selects which excitations appear. If $\Delta_A$ is the smallest positive energy with nonzero matrix element, then at large $\tau$

$$C_A(\tau) \sim Z_A e^{-\Delta_A\tau/\hbar},$$

provided that an isolated leading exponential exists.

Effective gap

When $C_A(\tau)>0$, define

$$\Delta_{\mathrm{eff}}(\tau) = -\hbar \frac{d}{d\tau} \ln C_A(\tau).$$

For a positive sum of discrete exponentials, this is a weighted average of excitation gaps. Its derivative is

$$\frac{ d\Delta_{\mathrm{eff}} }{ d\tau } = -\frac1\hbar \operatorname{Var}_{w(\tau)}(\Delta_n) \leq0.$$

The effective gap approaches the smallest gap carrying nonzero spectral weight. A plateau supports a single-exponential fit only after finite-$\tau$, finite-size, statistical, and excited-state uncertainties are checked.

At finite temperature, the trace includes transitions among thermally occupied states and propagation around the compact interval. Correlators can contain forward and wrap-around terms. Thermal Green Functions develops the finite-temperature Lehmann and Matsubara formulas.

From Imaginary to Real Time

Writing

$$z = t-i\tau$$

gives formally

$$e^{-iz\mathcal K/\hbar} = e^{-it\mathcal K/\hbar} e^{-\tau\mathcal K/\hbar}.$$

For a lower-bounded self-adjoint generator, suitable matrix elements define analytic functions in a half-plane after an energy shift. This supplies the mathematical origin of the substitution $t=-i\tau$ in favorable cases.

Three logically distinct operations should not be conflated:

1. Spectral definition: define $e^{-\tau\mathcal K/\hbar}$ directly by functional calculus.
2. Contour rotation: deform a real-time integral into a Euclidean contour while avoiding singularities and controlling boundary terms.
3. Analytic reconstruction: infer real-frequency information from imaginary-time or Matsubara data.

The first is well defined under standard spectral assumptions. The second requires analyticity of the integrand and a valid contour deformation. The third can be severely ill conditioned when data are finite or noisy.

Time Slicing and Product Formulas

For

$$\mathcal K = T+V$$

with noncommuting pieces, one cannot generally write

$$e^{-\tau(T+V)/\hbar} = e^{-\tau T/\hbar} e^{-\tau V/\hbar}.$$

The Lie–Trotter product formula instead gives, under appropriate operator-domain assumptions,

$$e^{-\tau(T+V)/\hbar} = \operatorname*{s-lim}_{M\to\infty} \left[ e^{-\tau T/(M\hbar)} e^{-\tau V/(M\hbar)} \right]^M.$$

A symmetric finite-step factorization is

$$e^{-\Delta\tau(T+V)/\hbar} = e^{-\Delta\tau V/(2\hbar)} e^{-\Delta\tau T/\hbar} \times e^{-\Delta\tau V/(2\hbar)} +O(\Delta\tau^3),$$

for bounded operators or suitably controlled unbounded operators. Repeating this step over a fixed total interval produces a global error of order $\Delta\tau^2$ in the usual regular setting.

These factorizations lead to transfer matrices, worldline representations, path integrals, and projector algorithms. Operator domains and convergence mode matter in rigorous applications.

Transfer-Matrix Viewpoint

Divide the thermal interval into $M$ slices:

$$\Delta\tau = \frac{\beta\hbar}{M}.$$

Define a short-time transfer operator

$$\mathsf T = e^{-\Delta\tau\mathcal K/\hbar}.$$

Then

$$e^{-\beta\mathcal K} = \mathsf T^M,$$

and

$$\mathcal Z = \operatorname{Tr} \mathsf T^M.$$

This makes a quantum system resemble a statistical system with one additional discrete direction. The resemblance is exact at the operator level, but the resulting classical weights need not be local, real, or nonnegative.

For lattice Hamiltonians, checkerboard or Suzuki–Trotter decompositions can turn noncommuting local terms into layers of commuting gates. The continuum limit in imaginary-time spacing must be tested rather than assumed.

Checking an imaginary-time calculation

For projection, vary the total imaginary time and time step independently. A stable energy alone can hide a missing symmetry sector or small overlap; also check the energy variance and compare states or correlators where possible. The projection benchmark illustrates these tests.

For a thermal calculation, temperature fixes the interval $\beta\hbar$: increasing the number of slices refines discretization rather than lowering temperature. Check finite-size effects, contact terms in the estimator, and sampling correlations separately. A sign or phase problem remains possible even though the time coordinate is imaginary.

For spectral reconstruction, precise Euclidean data are only a starting point. Their covariance, sum rules, and synthetic-data resolution tests constrain the compatible spectra; see analytic continuation.

Common pitfalls

A backward heat equation is unstable. Recovering high-energy components multiplies them by $e^{+\tau E/\hbar}$. Small errors can therefore become large even when the forward propagation is well behaved.

An energy shift changes amplitudes, not normalized predictions. Shifting $H$ multiplies an unnormalized state or partition function by a scalar. It leaves normalized state ratios and expectations of unchanged observables invariant.

A fit may see the wrong gap. Selection rules can remove the lowest excitation, a nonzero expectation produces a constant, and a continuum can add a power-law prefactor. Identify the operator channel before interpreting a decay rate.

Related constructions

• Euclidean path integrals derives the coordinate kernel and its time-sliced representation.
• Thermal path integrals includes the closed trace and identical-particle exchange sectors.
• Matsubara formalism turns compact imaginary time into discrete frequency sums.
• Euclidean QFT extends the construction to fields and reflection positivity.

Exercises

1. Semigroup norm

Let $\mathcal K$ be self-adjoint with

$$\inf\operatorname{spec}\mathcal K = \kappa_{\min}.$$

Use the spectral theorem to show

$$\left\| e^{-\tau\mathcal K/\hbar} \right\| = e^{-\tau\kappa_{\min}/\hbar}.$$

What changes after shifting $\mathcal K$ by $-\kappa_{\min}I$?

Solution

For the bounded continuous function used here, spectral calculus gives

$$\|f(\mathcal K)\| = \sup_{\kappa\in\operatorname{spec}\mathcal K} |f(\kappa)|.$$

With

$$f(\kappa) = e^{-\tau\kappa/\hbar}$$

and $\tau\geq0$, the largest value occurs at the lower spectral edge. Hence

$$\left\| e^{-\tau\mathcal K/\hbar} \right\| = e^{-\tau\kappa_{\min}/\hbar}.$$

For

$$\widetilde{\mathcal K} = \mathcal K-\kappa_{\min}I,$$

the lower edge is zero, so

$$\left\| e^{-\tau\widetilde{\mathcal K}/\hbar} \right\| = 1.$$

The shift multiplies the unnormalized propagator by $e^{+\tau\kappa_{\min}/\hbar}$ but leaves normalized state ratios unchanged. For a general bounded Borel function the norm uses the spectral-measure essential supremum; continuity justifies the ordinary spectral supremum above.

2. Energy descent

Suppose a normalized state obeys

$$\hbar \partial_\tau|\psi\rangle = - \left( H-\langle H\rangle \right) |\psi\rangle.$$

Derive

$$\partial_\tau\langle H\rangle = -\frac{2}{\hbar} \operatorname{Var}(H).$$

Solution

For time-independent $H$,

$$\partial_\tau\langle H\rangle = \langle\partial_\tau\psi|H|\psi\rangle + \langle\psi|H|\partial_\tau\psi\rangle.$$

Insert the evolution equation and its adjoint:

$$\begin{aligned} \partial_\tau\langle H\rangle &= -\frac1\hbar \langle (H-\langle H\rangle)H \rangle -\frac1\hbar \langle H(H-\langle H\rangle) \rangle \\ &= -\frac{2}{\hbar} \left( \langle H^2\rangle -\langle H\rangle^2 \right). \end{aligned}$$

The variance is nonnegative, so normalized imaginary-time flow cannot increase the energy expectation.

3. Projection time for a two-level state

For the two-level example, find a condition on $\tau$ that guarantees

$$P_1(\tau) \leq \varepsilon,$$

where $0\lt\varepsilon\lt1$ and both initial amplitudes are nonzero.

Solution

Write

$$r = \tan^2\theta\, e^{-2\Delta\tau/\hbar}.$$

Then

$$P_1 = \frac{r}{1+r}.$$

The condition $P_1\leq\varepsilon$ is equivalent to

$$r \leq \frac{\varepsilon}{1-\varepsilon}.$$

Therefore it is sufficient that

$$\tau \geq \frac{\hbar}{2\Delta} \ln \left[ \tan^2\theta \frac{1-\varepsilon}{\varepsilon} \right].$$

If the logarithm is negative, the requested tolerance already holds at $\tau=0$, so the minimal nonnegative projection time is zero.

4. Free heat kernel

Show that the $d$-dimensional free kernel satisfies

$$\partial_\tau K_{\mathrm E} = \frac{\hbar}{2m} \nabla^2 K_{\mathrm E}$$

and approaches a delta distribution as $\tau\to0^+$.

Solution

Write

$$K_{\mathrm E}(\mathbf r,\tau) = \left( \frac{m}{2\pi\hbar\tau} \right)^{d/2} \exp \left( -\frac{m\mathbf r^2}{2\hbar\tau} \right).$$

Direct differentiation gives

$$\frac{ \partial_\tau K_{\mathrm E} }{ K_{\mathrm E} } = -\frac{d}{2\tau} + \frac{ m\mathbf r^2 }{ 2\hbar\tau^2 }.$$

The Laplacian gives

$$\frac{ \nabla^2 K_{\mathrm E} }{ K_{\mathrm E} } = -\frac{ dm }{ \hbar\tau } + \frac{ m^2\mathbf r^2 }{ \hbar^2\tau^2 }.$$

Multiplying by $\hbar/(2m)$ reproduces the time derivative. The Gaussian is normalized for every $\tau>0$, and its width is of order

$$\sqrt{ \frac{\hbar\tau}{m} }.$$

As $\tau\to0^+$, the width vanishes while the integral remains one. Hence the kernel converges to $\delta^{(d)}(\mathbf r)$ in the distributional sense.

5. Energy shifts in a thermal trace

Let

$$H' = H+C I.$$

Show how $Z$, $\rho_\beta$, and a normalized thermal expectation change.

Solution

The shifted partition function is

$$\begin{aligned} Z' &= \operatorname{Tr} e^{-\beta(H+CI)} \\ &= e^{-\beta C}Z. \end{aligned}$$

The shifted density operator is

$$\rho_\beta' = \frac{ e^{-\beta C}e^{-\beta H} }{ e^{-\beta C}Z } = \rho_\beta.$$

Therefore every normalized expectation

$$\operatorname{Tr} (\rho_\beta A)$$

is unchanged. The free energy shifts by $C$, as expected for a change of energy zero.

6. Effective-gap monotonicity

Let

$$C(\tau) = \sum_n w_n e^{-\Delta_n\tau/\hbar}, \qquad w_n>0.$$

Show that

$$\Delta_{\mathrm{eff}} = -\hbar\partial_\tau\ln C$$

is a weighted mean of the $\Delta_n$ and decreases monotonically with $\tau$.

Solution

Define normalized $\tau$-dependent weights

$$p_n(\tau) = \frac{ w_n e^{-\Delta_n\tau/\hbar} }{ C(\tau) }.$$

Then

$$\sum_n p_n=1$$

and

$$\Delta_{\mathrm{eff}} = \sum_n p_n\Delta_n.$$

Differentiating the weights gives

$$\partial_\tau p_n = -\frac1\hbar p_n \left( \Delta_n-\Delta_{\mathrm{eff}} \right).$$

Therefore

$$\partial_\tau\Delta_{\mathrm{eff}} = \sum_n \Delta_n \partial_\tau p_n.$$

Substitution yields

$$\partial_\tau\Delta_{\mathrm{eff}} = -\frac1\hbar \Bigg[ \sum_n p_n\Delta_n^2 - \left( \sum_n p_n\Delta_n \right)^2 \Bigg].$$

Equivalently,

$$\partial_\tau\Delta_{\mathrm{eff}} = -\frac1\hbar \operatorname{Var}_{p(\tau)}(\Delta) \leq0.$$

Equality holds only when the contributing gaps are all equal. If there is a smallest gap with nonzero weight, it dominates at large $\tau$. For an infinite sum, the differentiations also require convergence of the first two weighted gap moments.

7. Grand-canonical operator evolution

For one fermionic mode,

$$H = \epsilon c^\dagger c, \qquad N = c^\dagger c,$$

derive $c(\tau)$ using $\mathcal K=H-\mu N$. What incorrect factor would result from evolving with $H$ alone?

Solution

The ensemble generator is

$$\mathcal K = (\epsilon-\mu)c^\dagger c = \xi c^\dagger c.$$

Since

$$[c^\dagger c,c] = -c,$$

the imaginary-time Heisenberg equation gives

$$\hbar\partial_\tau c(\tau) = -\xi c(\tau).$$

Hence

$$c(\tau) = e^{-(\epsilon-\mu)\tau/\hbar}c.$$

Using $H$ alone would give $e^{-\epsilon\tau/\hbar}$ and would miss the chemical-potential shift that aligns the propagator with grand-canonical thermal weights.

References

• D. M. Ceperley, “Path Integrals in the Theory of Condensed Helium”, Reviews of Modern Physics 67, 279–355 (1995) – path-integral Monte Carlo and finite-temperature many-body applications.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015) – modern operator and finite-temperature many-body treatment.
• E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press (1989) – semigroups, Schrödinger heat kernels, and spectral estimates.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003 reprint of the 1971 edition) – equilibrium propagators, imaginary time, and spectral representations.
• R. P. Feynman, “Atomic Theory of the $\lambda$ Transition in Helium”, Physical Review 91, 1291–1301 (1953) – early path-integral treatment of quantum statistical mechanics.
• T. Matsubara, “A New Approach to Quantum-Statistical Mechanics”, Progress of Theoretical Physics 14, 351–378 (1955) – imaginary-time many-body formalism.
• B. Simon, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea (2005) – Feynman–Kac methods and functional integration for nonrelativistic quantum mechanics.
• H. F. Trotter, “On the Product of Semi-Groups of Operators”, Proceedings of the American Mathematical Society 10, 545–551 (1959) – product formula underlying time slicing.