An intuitive construction of modular flow

Abstract

The theory of modular flow has proved extremely useful for applying thermodynamic reasoning to out-of-equilibrium states in quantum field theory. However, the standard proofs of the fundamental theorems of modular flow use machinery from Fourier analysis on Banach spaces, and as such are not especially transparent to an audience of physicists. In this article, I present a construction of modular flow that differs from existing treatments. The main pedagogical contribution is that I start with thermal physics via the KMS condition, and derive the modular operator as the only operator that could generate a thermal time-evolution map, rather than starting with the modular operator as the fundamental object of the theory. The main technical contribution is a new proof of the fundamental theorem stating that modular flow is a symmetry. The new proof circumvents the delicate issues of Fourier analysis that appear in previous treatments, but is still mathematically rigorous.

Figures (9)

• Figure 1: A flowchart of the standard topologies on operator algebras. Arrows point from stronger topologies to weaker topologies. So, for example, every sequence that converges in norm also converges ultrastrongly.
• Figure 2: If $T$ is bounded and invertible, then the function $T^z$ is norm analytic in the entire complex plane. If it is bounded but not invertible, then $T^z$ is norm analytic in the right half-plane, and strongly continuous on the imaginary axis.
• Figure 3: If $T$ is an unbounded positive operator, and the vector $\ket{\psi}$ is in the domain of $T^w,$ then it is also in the domain of $T^z$ for each $z$ in the vertical strip between $w$ and the imaginary axis. In the example sketched here, $w$ has positive real part and the imaginary axis is the left boundary of the strip. The function $z \mapsto T^z \ket{\psi}$ is holomorphic in the interior of this strip and continuous on the boundary.
• Figure 4: A sketch of the statement of theorem \ref{['thm:vector-continuation']}. To check that a vector $\ket{\psi}$ is in the domain of $T^{w},$ it is sufficient to check that all functions of the form $\langle \xi | T^{it} | \psi\rangle$ admit an analytic continuation from the imaginary axis to the vertical strip bounded by $w$. The overlap $\langle \xi | T^w |\psi \rangle$ is obtained by evaluating this analytic function at $w$. The main difference from figure \ref{['fig:vector-strip']} is that it is easier to study the analyticity of complex-valued functions like $\langle \xi | T^z | \psi \rangle$ than the analyticity of vector-valued functions like $T^z \ket{\psi}.$ Note that once we know $\ket{\psi}$ is in the domain of $T^w$, it follows from figure \ref{['fig:vector-strip']} that the vector-valued function also admits an analytic continuation.
• Figure 5: A sketch of the statement of theorem \ref{['thm:operator-continuation']}. Given an unbounded, invertible, positive operator $T$, a bounded operator $a,$ and a complex number $w$ such that $T^w a T^{-w}$ is densely defined and bounded on its domain, the function on the strip between the imaginary axis and $w$ given by $z \mapsto \overline{T^z a T^{-z}}$ is norm analytic in the interior of the strip and strongly continuous on its boundary. On the imaginary axis, it takes the values $T^{it} a T^{-it}.$

Theorems & Definitions (56)

• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5: Properties of adjoints
• Definition 2.6
• Definition 2.7
• Theorem 2.8: Spectral theorem
• Theorem 2.9
• Definition 2.10