Quantum Statistics: the Partition Function as a Trace

For the reader from (mathematics).

The upgrade here is from a sum over a labelled index set to a trace on $\operatorname{End}(\mathcal{H})$ . A trace is basis-independent and cyclic; those two properties are doing real work. The Fock-space examples turn the geometric and finite series of Lecture 1 into the infinite products $\prod(1-q^{n})^{-1}$ and $\prod(1+q^{n})$ - the same series a number theorist meets as generating functions for integer partitions.

1 From a sum to a trace

Quantum mechanically the energies $E_{i}$ are eigenvalues of a Hamiltonian operator $H$ on a Hilbert space $\mathcal{H}$ . The Boltzmann factor is no longer a number attached to a configuration; it is the operator $\mathrm{e}^{-\beta H}$ , and the partition function is its trace:

\boxed{\,Z=\operatorname{Tr}_{\mathcal{H}}\,\mathrm{e}^{-\beta H}=\sum_{i}% \mathrm{e}^{-\beta E_{i}}.\,}

(1)

Evaluating the trace in the energy eigenbasis recovers Lecture 1. But the trace is basis-independent: we may compute it in any basis, and that freedom is exactly what makes $Z$ a robust object rather than an artifact of a chosen labelling.

Thread (The L1 $\to$ L2 distinction - worth dwelling on).

Students routinely conflate “statistical mechanics” and “quantum statistics.” The difference is precisely the promotion of the Boltzmann factor from a scalar $\mathrm{e}^{-\beta E_{i}}$ (L1) to an operator $\mathrm{e}^{-\beta H}$ (L2). Three consequences follow, none cosmetic:

1.

With non-commuting observables there is no preferred basis, so basis-independence - the trace - is what makes $Z$ well-defined.
2.

Indistinguishability of identical quanta forces the Bose vs. Fermi alternatives (§2).
3.

$\mathrm{e}^{-\beta H}$ is imaginary-time evolution, $\mathrm{e}^{-\beta H}=\mathrm{e}^{-\mathrm{i}Ht}\big{|}_{t=-\mathrm{i}\beta}$ , the hinge that swings us into the path integral of Lecture 3.

The (unnormalized) state of the system is the density operator $\rho=\mathrm{e}^{-\beta H}/Z$ , and expectation values are $\langle\mathcal{O}\rangle=\operatorname{Tr}(\rho\,\mathcal{O})$ . Cyclicity of the trace, $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ , is the algebraic seed of the Kubo–Martin–Schwinger condition we will meet in L3 and of modular invariance in L4.

2 Fock space and the two universal products

The trace (1) may be computed in any basis; for a collection of non-interacting quantum modes the convenient one is the occupation-number basis of a Fock space. We build it one mode at a time, and in that basis the trace is a sum of diagonal matrix elements $\langle n\rvert\mathrm{e}^{-\beta H}\lvert n\rangle$ -each basis ket sandwiched around the Boltzmann operator.

A single bosonic mode.

Write the mode with ladder operators $[a,a^{\dagger}]=1$ . The number operator $\hat{n}=a^{\dagger}a$ has the tower of eigenstates $\lvert n\rangle$ , $n=0,1,2,\dots$ , generated from the vacuum by $a^{\dagger}\lvert n\rangle=\sqrt{n+1}\,\lvert n+1\rangle$ (and lowered by $a\lvert n\rangle=\sqrt{n}\,\lvert n-1\rangle$ ). With $H=\hbar\omega(\hat{n}+\tfrac{1}{2})$ these are energy eigenstates, so each sandwich $\langle n\rvert\mathrm{e}^{-\beta H}\lvert n\rangle$ returns the eigenvalue $\mathrm{e}^{-\beta E_{n}}$ and the trace is a bare sum of Boltzmann numbers:

\operatorname{Tr}_{\mathcal{H}}\,\mathrm{e}^{-\beta H}=\sum_{n\geq 0}\langle n% \rvert\mathrm{e}^{-\beta H}\lvert n\rangle=\sum_{n\geq 0}\mathrm{e}^{-\beta% \hbar\omega(n+1/2)}=\frac{\mathrm{e}^{-\beta\hbar\omega/2}}{1-\mathrm{e}^{-% \beta\hbar\omega}}.

(2)

This is (7) of Lecture 1, now read as a trace. A bosonic mode admits any occupation $n\geq 0$ : the tower is infinite.

A single fermionic mode.

Now the operators anticommute, $\{c,c^{\dagger}\}=1$ and $c^{2}=0$ . That $c^{2}=0$ is the Pauli principle, and it truncates the tower to just two states, $\lvert 0\rangle$ and $\lvert 1\rangle=c^{\dagger}\lvert 0\rangle$ : the Fock space is two-dimensional. With $H=\epsilon\,c^{\dagger}c$ the trace is a two-term sum,

\operatorname{Tr}_{\mathcal{H}}\,\mathrm{e}^{-\beta H}=\langle 0\rvert\mathrm{% e}^{-\beta H}\lvert 0\rangle+\langle 1\rvert\mathrm{e}^{-\beta H}\lvert 1% \rangle=1+\mathrm{e}^{-\beta\epsilon},

(3)

recovering (8). Bose versus Fermi is nothing but the length of the tower the trace runs over-an infinite geometric series against two terms.

For the reader from (mathematics).

Fock space is built from the single-particle space $V=\bigoplus_{k}\mathbb{C}$ (one dimension per mode) by a functor: bosonic Fock space is the symmetric algebra $\mathrm{Sym}(V)=\bigoplus_{n}\mathrm{Sym}^{n}V$ , fermionic Fock space is the exterior algebra $\bigwedge V=\bigoplus_{n}\bigwedge^{n}V$ , and occupation number is exactly the symmetric/exterior grading. This is not a passing analogy: in Lecture 5 $\mathrm{Sym}(V)$ becomes the Heisenberg vertex algebra and $\bigwedge V$ a Clifford (super) vertex algebra, generated by exactly the modes assembled here.

Many modes: the trace factorizes.

Assemble all the modes. The Hilbert space is the tensor product $\mathcal{H}=\bigotimes_{k}\mathcal{H}_{k}$ , with occupation-number basis $\lvert\{n_{k}\}\rangle=\bigotimes_{k}\lvert n_{k}\rangle$ ; the Hamiltonian is additive, $H=\sum_{k}H_{k}$ ; and because the $H_{k}$ act on separate factors, $\mathrm{e}^{-\beta H}=\bigotimes_{k}\mathrm{e}^{-\beta H_{k}}$ . A diagonal matrix element therefore splits into a product over modes, $\langle\{n_{k}\}\rvert\mathrm{e}^{-\beta H}\lvert\{n_{k}\}\rangle=\prod_{k}% \langle n_{k}\rvert\mathrm{e}^{-\beta H_{k}}\lvert n_{k}\rangle$ , and summing over every configuration $\{n_{k}\}$ turns a sum of products into a product of sums:

	$\displaystyle Z=\operatorname{Tr}\,\mathrm{e}^{-\beta H}$	$\displaystyle=\sum_{\{n_{k}\}}\;\prod_{k}\langle n_{k}\rvert\mathrm{e}^{-\beta H% _{k}}\lvert n_{k}\rangle$		(4)
		$\displaystyle=\prod_{k}\Big{(}\sum_{n_{k}}\langle n_{k}\rvert\mathrm{e}^{-% \beta H_{k}}\lvert n_{k}\rangle\Big{)}=\prod_{k}Z_{k}.$		(4)

Exchanging “sum over all configurations” for “product over modes” is the operator version of the factorization (9): independent modes multiply, and each factor $Z_{k}$ is a single-mode trace of the kind just computed.

Now specialize to a tower of modes with energies $\hbar\omega_{n}=n\,\hbar\omega$ , $n\geq 1$ , and write $q=\mathrm{e}^{-\beta\hbar\omega}$ . (Note: this is the thermal $q$ , not yet the torus $q$ .) Mode $n$ then carries the weight $q^{n}$ per quantum. Measuring energies from each mode’s ground state, its bosonic factor (2) is $\sum_{m\geq 0}q^{nm}=(1-q^{n})^{-1}$ and its fermionic factor (3) is $1+q^{n}$ . The product (4) over the whole tower is then

	Bosons (each $n$ occupied $0,1,2,\dots$ times):	$\displaystyle Z_{\mathrm{b}}=\prod_{n\geq 1}\frac{1}{1-q^{n}},$		(5)
	Fermions (each $n$ occupied $0$ or $1$ time):	$\displaystyle Z_{\mathrm{f}}=\prod_{n\geq 1}\bigl{(}1+q^{n}\bigr{)}.$		(6)

These two products-one for each statistics-are the protagonists of the rest of the series.

Thread (The boson/fermion thread, now infinite products).

Equation (5) is, up to the factor $q^{-1/24}$ (the zero-point energies of Example 1.2 summed over the tower and regularized in L3), the character $\eta(\tau)^{-1}$ we will meet in L4; (6) is a ratio of theta and eta functions in disguise. Keep both in view: the boson will become the Heisenberg vertex algebra and the fermion a Clifford (super) vertex algebra in L5.

3 The “partition function” pun is literal

Expand (5) as a power series in $q$ :

\prod_{n\geq 1}\frac{1}{1-q^{n}}=\sum_{N\geq 0}p(N)\,q^{N}=1+q+2q^{2}+3q^{3}+5% q^{4}+7q^{5}+\cdots,

(7)

where $p(N)$ is the number of ways of writing $N$ as a sum of positive integers-the number of integer partitions of $N$ . Euler’s generating function and the physicist’s partition function of a bosonic tower are the same series. (The coincidence of names is a genuine coincidence, but a happy one.) Likewise the fermionic product (6) counts partitions of $N$ into distinct parts:

\prod_{n\geq 1}\bigl{(}1+q^{n}\bigr{)}=1+q+q^{2}+2q^{3}+2q^{4}+3q^{5}+\cdots.

(8)

Computing a thermodynamic quantity and counting combinatorial objects are, here, one act.

4 Refining the trace: chemical potential and fugacity

Suppose the system also conserves a particle number $N$ (an operator commuting with $H$ ). Holding $\langle N\rangle$ fixed on average introduces a second Lagrange multiplier, exactly as $\beta$ was introduced for energy in Lecture 1. Call it the chemical potential $\mu$ . The weight becomes $\mathrm{e}^{-\beta(H-\mu N)}$ , and the partition function is the grand canonical

\mathcal{Z}=\operatorname{Tr}\,\mathrm{e}^{-\beta(H-\mu N)}=\operatorname{Tr}% \,z^{N}\mathrm{e}^{-\beta H},\qquad z\vcentcolon=\mathrm{e}^{\beta\mu},

(9)

with $z$ the fugacity. Physically $\mu$ is the energetic cost of adding one particle; formally $z^{N}$ grades the trace by the conserved charge, and $\partial\log\mathcal{Z}/\partial(\beta\mu)=\langle N\rangle$ .

Thread (One conjugate potential per conserved charge).

This is the general move, stated once: for every commuting conserved charge $Q_{a}$ we may insert a conjugate potential and refine the trace, $Z=\operatorname{Tr}\,\exp(-\sum_{a}\beta_{a}Q_{a})$ (the generalized Gibbs ensemble). The sign-graded trace $\operatorname{Tr}\,(-1)^{F}\mathrm{e}^{-\beta H}$ is the same move with the fermion-number charge-the seed of the Witten index. Each such knob reappears, renamed, downstream: $y^{J_{0}}$ on the torus (L4), the Cartan variables of a character (L5), and the $y$ of the elliptic genus.

For the curious (the trace as a twisted character).

The trace itself is the canonical map $\operatorname{End}(\mathcal{H})\to\mathbb{C}$ for a dualizable object $\mathcal{H}$ -the categorical fact the Finale will make precise. The graded trace $\operatorname{Tr}\,g\,\mathrm{e}^{-\beta H}$ for a symmetry operator $g$ (here $g=z^{N}$ or $g=(-1)^{F}$ ) is a twisted version of that same map. When $g$ generates a finite or compact group, summing or integrating over $g$ projects onto invariants-the orbifold/gauging construction. We will not need orbifolds, but it is worth knowing that “insert an operator in the trace” is the universal hook on which charges, boundary conditions, and (in L4) spin structures all hang.

Rosetta notes

•

Trace $\leftrightarrow$ sum over states in any basis, now basis-independent.
•

$\mathrm{e}^{-\beta H}\leftrightarrow$ evolution by imaginary time $\beta$ - the cliffhanger for L3.
•

Graded trace $\operatorname{Tr}\,z^{N}\mathrm{e}^{-\beta H}\leftrightarrow$ inserting a symmetry operator $\leftrightarrow$ choosing a boundary condition $\leftrightarrow$ a spin structure (L3–L4).

Quantum Statistics: the Partition Function as a Trace

For the reader from (mathematics).

1 From a sum to a trace

Thread (The L1 → L2 distinction - worth dwelling on).

2 Fock space and the two universal products

A single bosonic mode.

A single fermionic mode.

For the reader from (mathematics).

Many modes: the trace factorizes.

Thread (The boson/fermion thread, now infinite products).

3 The “partition function” pun is literal

4 Refining the trace: chemical potential and fugacity

Thread (One conjugate potential per conserved charge).

For the curious (the trace as a twisted character).

Rosetta notes

Thread (The L1 $\to$ L2 distinction - worth dwelling on).