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Finite Density and Conserved Charges

Finite density is not one operation. An ordinary grand-canonical state uses compatible conserved charges; an integrable GGE requires a complete commuting charge family; noncommuting constraints define a different maximum-entropy problem; a chemical potential may be represented as a thermal holonomy; response and onset depend on limit order; and computational access is bounded by sign, overlap, convergence, and validation. This chapter identifies which problem is actually being asked before a method or interpretation is chosen.

Finite-temperature field theory with conserved chemical potentials and its Euclidean conventions are developed in Laine and Vuorinen 2016, ch. 3.

Enter this chapter

Section titled “Enter this chapter”

You should be able to identify a conserved charge and state the KMS condition. Repair either gap with Continuous Symmetries, Generators, and Charges and Thermal Density Operators and the KMS Condition.

Choose a route

Section titled “Choose a route”
GoalRouteDemonstrable result
Free finite-density QFTGrand-canonical states → distributions → responseDerive particle/antiparticle occupations, density, susceptibility, and stability threshold
Integrable stationary dataCharge conditions → integrable GGEMap a complete charge family to a thermodynamic macrostate and expose truncation bias
Non-Abelian equilibriumCharge conditions → noncommuting constraintsConstruct the exponential state and its residual symmetry without assigning simultaneous sharp charges
Imaginary chemical potentialDistributions → holonomy → accessTranslate trace, derivative shift, twist, large-gauge period, and analytic domain
Density onsetResponse → Silver BlazeLocate the lowest E/QE/Q threshold and test homogeneous, paired, and inhomogeneous instabilities

Guide to the eight pages

Section titled “Guide to the eight pages”
  1. Conserved Charges and Grand-Canonical States gives the admissibility, stability, gauge, anomaly, and superselection tests for H−μaQaH-\mu_aQ_a.
  2. Chemical Potentials and Finite-Density Ensembles derives free distributions, density, susceptibility, KMS shifts, and the canonical convention table.
  3. Generalized Gibbs Ensembles for Integrable Charges maps commuting local and quasilocal charges to a TBA macrostate and tests completeness.
  4. Noncommuting Charges and Generalized Gibbs States derives the non-Abelian maximum-entropy state, Kubo–Mori response, and residual symmetry.
  5. Chemical Potentials as Temporal Gauge Fields and Holonomies connects derivative shifts, thermal twists, imaginary chemical potential, and large gauge transformations.
  6. Finite-Density Correlators and Charge Susceptibilities treats charged detailed balance, contact terms, Ward identities, diffusion, and noncommuting static/dynamic limits.
  7. Silver Blaze Behavior, Thresholds, and Density Instabilities derives zero-temperature independence below the smallest E/QE/Q and classifies onset mechanisms.
  8. Euclidean and Real-Time Access at Finite Density compares expansion, continuation, reweighting, canonical, complexified, real-time, and EFT routes by failure tests and claim ceilings.

Synthesis and misconceptions

Section titled “Synthesis and misconceptions”

The invariant equilibrium object is K=H−μaQaK=H-\mu_aQ_a together with the physical time generator, charge normalization, and state domain. The same chemical potential can appear as a trace weight, a charged KMS factor, an Euclidean derivative shift, or a holonomy; the canonical table provides round trips among them.

Four distinctions organize the chapter:

  • mutually commuting integrable charges are not noncommuting non-Abelian constraints;
  • a stable homogeneous Hessian does not exclude a paired or finite-momentum instability;
  • explicit μ\mu dependence of an integrand does not defeat Silver Blaze when a contour shift remains legal; and
  • a method that fits available data has not necessarily solved sign, overlap, or identifiability.

Review the chapter

Section titled “Review the chapter”

Derivation. Starting from Ω=−Tlog⁡Ξ\Omega=-T\log\Xi, derive particle and antiparticle density and the susceptibility of a free gas. State the bosonic stability bound.

Translation. Move μ\mu from Dτ=∂τ−qμD_\tau=\partial_\tau-q\mu to a boundary twist and recover the same shifted frequencies. A correct answer declares the Fourier sign and charge normalization.

Comparison. Contrast an integrable GGE with ρ∝e−λ⋅S\rho\propto e^{-\boldsymbol\lambda\cdot\mathbf S}. A correct answer names charge completeness in the first case and incompatibility/reference-frame dependence in the second.

Failure diagnosis. A finite-density method reproduces its own forward observables but misses Silver Blaze in a solvable benchmark. A correct answer rejects extrapolation and investigates contour correctness, overlap, regulator, and order of limits.

Synthesis. Given a negative eigenvalue of χab(k)\chi_{ab}(\mathbf k), state what it establishes. A correct answer identifies instability of the assumed branch in that channel and momentum, but does not yet identify the globally preferred phase or its dynamics.

Continue

Section titled “Continue”

Thermal Perturbation Theory and Renormalization organizes finite-T,μT,\mu loops and ultraviolet control. Response, Transport, and Inference develops contact terms and Kubo limits. QCD Matter and Multistage Collision Inference owns current dense-QCD evidence boundaries; this chapter does not turn generic method availability into a QCD result.

References

Section titled “References”