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Inverse temperature is $\beta =1/(k_{B}T)$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature. A canonical ensemble weights an energy $E$ by its Boltzmann factor $e^{-\beta E}$. Calling couplings dimensionless means absorbing this factor $\beta$ into their energy coefficients. The alternative convention $k_B=1$ makes inverse temperature $1/T$; physical and dimensionless couplings must not be mixed.

In a thermal Euclidean worldline path integral with $\hslash =1$, $\beta$ is also the circumference of the imaginary-time circle.

The trace of the Euclidean evolution kernel identifies its initial and final positions. Thus the Euclidean worldline path integral is over periodic paths $x(\tau +\beta )=x(\tau )$, with $\tau \in [0,\beta ]$:The circle has circumference $\beta$, not an arbitrary unit length. The Euclidean action follows by Wick rotation of the unit-mass oscillator action. Equivalently, integration by parts with periodic boundary conditions gives $S_E=\frac{1}{2}⟨x,L_{\omega }x⟩$ for $L_{\omega }=-{\partial }_{\tau }^2+{\omega }^2$. The path integral measure is formal until a regularization and its normalization are specified; the next part supplies a finite-dimensional version.