Spontaneous Breaking in Gauge Theories

Introduction and Motivation Spontaneous Symmetry Breaking in Symmetry Breaking—A Natural Gauge Theories Phenomenon?

Local Gauge Symmetry Breaking?

SSB of Global Subgroups Simon Friederich Fachbereich C – Naturwissenschaften und Mathematik Philosophical Universität Wuppertal Implications

30.05.2011 / Wuppertal Spontaneous Outline of the Presentation Symmetry Breaking in Gauge Theories

Introduction and Motivation

Introduction and Motivation Symmetry Breaking—A Natural Phenomenon? Symmetry Breaking—A Natural Phenomenon? Local Gauge Symmetry Breaking?

SSB of Global Local Gauge Symmetry Breaking? Subgroups Philosophical Implications SSB of Global Subgroups

Philosophical Implications Spontaneous Motivation Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Breaking—A Natural I Interpretation of (gauge) and (gauge) Phenomenon?

symmetry breaking: central topic in the philosophy of Local Gauge Symmetry science. Breaking?

I Gauge symmetries: “purely formal” (Healey), no SSB of Global Subgroups

physical instantiations. Philosophical Implications I Received view: SSB of local gauge symmetry crucial for the . Spontaneous Criticisms of the received view Symmetry Breaking in Gauge Theories

Introduction and Motivation I ’t Hooft: SSB of local gauge symmetry “something of Symmetry Breaking—A a misnomer” Natural Phenomenon?

I Earman: should use constraint Hamiltonian Local Gauge Symmetry formalism to clarify whether there is any SSB in the Breaking?

Higgs mechanism SSB of Global Subgroups I Lyre: There is no “Higgs mechanism”. Philosophical Implications I Elitzur (1975): Local gauge symmetry cannot be spontaneously broken (rigorous result).

Let’s take a closer look! Spontaneous SSB and symmetry breaking order Symmetry Breaking in Gauge parameters Theories

Introduction and Motivation

Symmetry Breaking—A I Basic idea of SSB: State not transformable into one Natural Phenomenon?

with the same symmetries as the underlying “laws”. Local Gauge Symmetry I Need infinitely many degrees of freedom. Breaking?

SSB of Global I In quantum theories: Unitary inequivalence of Subgroups

representation associated to different symmetry Philosophical breaking states. Implications

I ⇒ Need “more than one” Hilbert space.

I If hAi= 6 hα(A)i, then hAi is called a “symmetry breaking order parameter” for the symmetry α. Spontaneous SSB as a natural phenomenon Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry I SSB is something that “occurs”. Breaking—A Natural I It applies to (the states of) objects themselves, not Phenomenon? merely our description. Local Gauge Symmetry Breaking? I Liu and Emch (2005) conclude: SSB is a “natural SSB of Global phenomenon”... Subgroups

I ... in contrast to mere “theoretical concepts” such as Philosophical Implications “” or “”.

But what about gauge SSB? Spontaneous Gauge SSB Symmetry Breaking in Gauge Theories

Introduction and Motivation I Gauge SSB not a “natural phenomenon” in the sense Symmetry of asymmetric reality governed by symmetric laws. Breaking—A Natural Phenomenon? I But: Gauge SSB possibly a “natural phenomenon” in Local Gauge a (slightly) weaker sense: Symmetry Breaking? I Transition from unbroken to broken gauge symmetry SSB of Global may indicate a qualitative change in observable Subgroups Philosophical properties. Implications I In that sense gauge SSB can be called a “natural phenomenon” if it signals a phase transition.

I Spontaneous breaking of an observable symmetry always corresponds to a phase transition. Spontaneous BEC as an example Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Breaking—A Example of gauge SSB: Bose-Einstein condensation Natural Phenomenon? (BEC) Local Gauge Symmetry I Broken global gauge symmetry. Breaking? I Distinction between broken and unbroken symmetry SSB of Global Subgroups

lines up with phase transition. Philosophical Implications I Macroscopic observables different in the two phases (e. g. compressibility, viscosity, specific heat). Spontaneous Local gauge symmetry: classical case Symmetry Breaking in Gauge Theories Consider

∗ µ 1 µν Introduction and L = Dµφ D φ − V (φ) − FµνF , (1) Motivation 4 Symmetry Breaking—A with (Abelian) local U(1) gauge symmetry: Natural Phenomenon? 1 Local Gauge φ(x) 7→ eiα(x)φ(x), A (x) 7→ A (x) − ∂ α(x) (2) Symmetry µ µ e µ Breaking? SSB of Global and potential: Subgroups Philosophical 2 ∗ ∗ 2 Implications V (φ) = m0φ φ + λ0(φ φ) , (3)

2 where m0 < 0. Classical ground states: √ φ(x) = eiθ(x)v/ 2 (4)

⇒ SSB of local gauge symmetry. Spontaneous Local gauge symmetry: Quantization Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Generating functional: Breaking—A Natural Z  Z  Phenomenon? 4 µ W [η, J] = N DφDAµ exp i d x(L + ηφ + JµA ) . (5) Local Gauge Symmetry Breaking?

SSB of Global Subgroups I The integral in Eq. (5) is “badly divergent”. Philosophical Implications I No perturbation expansion possible.

I Either: Quantize non-perturbatively on a lattice.

I Or: Fix the gauge. Spontaneous Elitzur’s theorem Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Without gauge fixing: Breaking—A Natural Phenomenon?

Local gauge symmetry cannot be spontaneously broken. Local Gauge Symmetry (Elitzur, 1975) Breaking?

SSB of Global Crux of proof: Can’t implement nonzero order parameter Subgroups Philosophical by fixing it on a finite volume boundary. Implications

Reason: Gauge transformations depend on infinitely many parameters. Spontaneous Gauge fixing Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Breaking—A Natural Roughly: Replace the S in the functional integral Phenomenon? W [η, J] by an “effective action” Local Gauge Symmetry Breaking?

SSB of Global Subgroups Seff = S + Sgf + Sghost , (6) Philosophical Implications where Sgf breaks local gauge invariance. Spontaneous Post-gauge fixing global symmetries Symmetry Breaking in Gauge Theories

Introduction and Motivation Sgf may either break local gauge invariance completely Symmetry or not. Breaking—A Natural I If so: Only one configuration per gauge orbit Phenomenon? compatible with gauge fixing condition. Local Gauge Symmetry Breaking? I If not: Sgf still invariant under remnant global gauge SSB of Global symmetry. Subgroups Examples: Philosophical Implications I Unitary gauge: θ(x) = 0

I Coulomb gauge, Landau gauge, temporal gauge,... Remnant global gauge symmetries may break spontaneously! Spontaneous Global gauge SSB and phase transitions I Symmetry Breaking in Gauge Theories

Introduction and Motivation

Symmetry Breaking—A Natural Question: Are these global gauge SSBs alwas Phenomenon? Local Gauge associated with phase transitions? Symmetry Breaking?

SSB of Global Subgroups

Philosophical Implications Answer: No! Spontaneous Global gauge SSB and phase transitions II Symmetry Breaking in Gauge Theories

Introduction and Whether remnant global gauge symmetry is broken or Motivation Symmetry unbroken depends on the gauge. Breaking—A Natural Phenomenon? Example: Different unbroken/broken transitions in Landau Local Gauge Symmetry gauge and Coulomb gauge for an SU(2)-Higgs model Breaking? (Caudy and Greensite, 2008) SSB of Global Subgroups

Philosophical No rigid connection between phase transitions and global Implications gauge SSBs.

⇒ Global gauge SSB is “ambiguous”, not (in general) a “natural phenomenon’. Spontaneous Kosso on epistemology of SSB Symmetry Breaking in Gauge Theories

Kosso: Gauge symmetry breakings belong to the class of Introduction and Motivation

cases where “the relevant laws of nature are exactly Symmetry Breaking—A symmetric, but the phenomena expressing these laws are Natural not.” (Kosso 2000) Phenomenon? Local Gauge Symmetry Breaking?

SSB of Global Subgroups Criticism: Philosophical Implications I broken gauge symmetry: No asymmetry in observable phenomena (see Bose-Einstein condensation).

I in gauge theories: breaking of global subgroups is ambiguous. Spontaneous Morrison on “vacuum hypotheses” Symmetry Breaking in Gauge Theories

Morrison: “it would be folly to accept a robust physical Introduction and Motivation

interpretation of the SSB story.” (Morrison 2003) Symmetry Reasonable conclusion, but based on confused Breaking—A Natural reasoning (“theoretical story about the nature of the Phenomenon? Local Gauge vacuum”): Symmetry Breaking?

SSB of Global Criticism: Subgroups I Without gauge fixing: is Philosophical Implications zero (Elitzur).

I With gauge fixing: vacuum expectation value depends on the gauge and...

I ...especially, whether or not it is zero depends on the gauge. Spontaneous Weinberg on “reality of gauge symmetries” Symmetry Breaking in Gauge Theories

Introduction and Motivation Weinberg: “if a gauge symmetry becomes unbroken for Symmetry sufficiently high temperature, it becomes difficult to doubt Breaking—A Natural its reality.” (Weinberg 1978) Phenomenon? Local Gauge Symmetry Whatever “reality” is supposed to mean: Breaking? SSB of Global I No evidence against standard view of gauge Subgroups symmetries as purely formal! Philosophical Implications I Gauge symmetry “restoration” depends on the gauge, is ambiguous. ⇒ No indication for “reality” of gauge symmetries in whatever sense.