Resolve a thermal circle into Matsubara modes

Continue real time as $t=-i\tau$, with $0\le\tau<\beta$ and $\beta=1/T$ in the module's natural units. Euclidean time ordering combines the Gibbs trace relation with operator statistics. A bosonic two-point function is periodic across the thermal interval, while a fermionic time-ordered Green function is antiperiodic. Equal-time contact discontinuities require their usual limiting prescriptions. The negative sign for fermions belongs to this ordered object and should not be inserted into raw occupation probabilities.

Expand a periodic function using modes $e^{-i\omega_n\tau}$. Requiring each mode to agree after a shift by beta gives integer multiples of $2\pi T$. Antiperiodicity requires a minus sign after the same shift, giving odd multiples of $\pi T$. Bosons therefore have a zero-frequency mode, whereas fermions do not. This difference is important in infrared thermal calculations: the bosonic static sector can be soft even when every nonzero Matsubara frequency is of order temperature.

For a unit-mass harmonic mode with positive frequency $\Omega$, the positive Euclidean coordinate correlator has Fourier coefficient $G_E(i\omega_n)=1/(\omega_n^2+\Omega^2)$. Its equal-time variance is T times the sum over all integers. The factor T comes from Fourier inversion on an interval of length beta. Evaluating the sum by residues or the elementary oscillator trace gives the vacuum half-quantum plus the Bose occupation. This convergent example is useful because no ultraviolet subtraction obscures the thermal normalization.

As temperature tends to zero, the Matsubara grid becomes dense and the sum approaches the vacuum integral, yielding $1/(2\Omega)$. At temperature much greater than the oscillator frequency, the hyperbolic cotangent expands and the leading variance is $T/\Omega^2$, the classical equipartition value. The bosonic zero mode alone gives that leading term. A comparable denominator summed on the fermionic frequency grid produces a hyperbolic tangent instead; that algebraic identity is not by itself the full spinor propagator of an interacting fermion field.

Discrete imaginary frequencies can simplify equilibrium perturbation theory, but an experiment usually asks for a retarded response on the real axis. That response must obey causality, its spectral representation, and appropriate high-frequency behavior. Analytic continuation uses those structures and the precise correlator convention. Replacing a discrete label by a real number without accounting for poles, boundary values, and overall signs can give an advanced function or the wrong spectral weight. The next lesson makes the oscillator continuation explicit.

Numerical Euclidean data are also finite and uncertain. Many different fine spectral features can be consistent with such a dataset, making real-time reconstruction ill conditioned without additional justified information. A narrow transport peak is especially difficult to infer from a handful of Matsubara points. This is an information limitation of the reconstruction, not a failure of the exact equilibrium formalism. State any spectral ansatz, positivity condition, asymptotic constraint, and resolution limit when turning a Euclidean calculation into a proposed real-time measurement prediction.