Curie's Principle and spontaneous symmetry breaking

Abstract

In 1894 Pierre Curie announced what has come to be known as Curie's Principle: the asymmetry of effects must be found in their causes. In the same publication Curie discussed a key feature of what later came to be known as spontaneous symmetry breaking: the phenomena generally do not exhibit the symmetries of the laws that govern them. Philosophers have long been interested in the meaning and status of Curie's Principle. Only comparatively recently have they begun to delve into the mysteries of spontaneous symmetry breaking. The present paper aims to advance the discussion of both of these twin topics by tracing their interaction in classical physics, ordinary quantum mechanics and quantum field theory. The features of spontaneous symmetry that are peculiar to quantum field theory have received scant attention in the philosophical literature. These features are highlighted here, along with an explanation of why Curie's Principle, though valid in quantum field theory, is nearly vacuous in that context.

Notes

Correspondence: Department of History and Philosophy of Science, 1017 C.L., University of Pittsburgh, Pittsburgh, PA 15260, USA. E‐mail: [email protected].

‘Lorsque certains effects re´ve`lent une certaine dissyme´trie, cette dissyme´trie doit se retrouver dans les causes qui lui ont donne´ naissance.’ An English translation of Curie's paper can be found in Rosen (1982, pp. 17–25).

For cases where the equations of motion/field equations are derivable from an action principle the point becomes that there is a symmetry of the Lagrangian that is not a symmetry of the physically significant state. This will be discussed in detail below.

I take this qualifier from Roberts (1992, p. 404).

‘C'est la dissyme´trie qui cre´e le phe´nome`ne.’

See Roberts (1992, p. 404). Weinberg (1996) quips that ‘we do not have far to look for examples of spontaneous symmetry breaking (p. 163)’, e.g. most of the objects we encounter in daily life have a definite orientation in space although the laws governing the motions and interactions of the atoms that compose these objects are rotationally symmetric. But Weinberg immediately goes on to discuss how spontaneous symmetry breaking in quantum field theory differs from such homely examples, and the key difference concerns the behavior of the quantum vacuum state.

Attempts by philosophers of science to come to grips with the implications of spontaneous symmetry breaking can be found in Morrison (1995, 2000, Sec. 4.4), Cao (1997, Ch. 10), Kosso (2000), Liu (2003), Balashov (2002) and Part III (‘Symmetry breaking’) of Brading and Castellani (2003). Historical surveys are to be found in Radicati (1987) and Brown and Cao (1991).

I leave open the question of whether the present proposal captures Curie's original intentions. Ismael (1997), who gives a similar reading of Curie's Principle, offers textual evidence about Curie's intentions.

For a precise statement and proof of these assertions, see Olver (1993, Sec. 4.2). A symmetry of the equations of motion need not be a variational symmetry.

For a readable treatment of the Noether theorems and their applications in physics see Barbashov and Nesterenko (1983).

The quantum treatment of phase transitions belongs to quantum statistical mechanics and is beyond the scope of this paper. One of the features that emerges from this treatment is the necessity to invoke unitarily inequivalent representations of the canonical commutations relations which, as will be seen below, is a key to spontaneous symmetry breaking in quantum field theory. For an accessible introduction to some of the issues involved in quantum phase transitions see Ruetsche (2003).

Needless to say, the idea that collapse of the state vector is an objective physical process is highly controversial. See Albert (1998) for an overview of different ways of treating the measurement problem in QM.

See Ismael (1997) for a start on this project.

With U _m(s) := exp(iq _m s) and V _n(t) := exp(ip _n t), the Weyl form of the canonical commutation relations p _n q _m − q _m p _n = −iδ_nm, etc., is given by V _n(t)U _m(s) = U _m(s)V _n(t) exp(istδ_nm) and U _m(s)U _n(t) − U _n(t)U _m(s) = 0 = V _m(s)V _n(t) − V _n(t)V _n(s). This form of the commutation relations avoids problems about domains of definition for unbounded operators. When the ranges of m and n are finite the Stone–von Neumann theorem says that all irreducible representations of these relations by continuous unitary groups on Hilbert space are unitarily equivalent. For an infinite number of degrees of freedom, in particular for field theories, the theorem no longer applies.

There are many good reviews of the physics of spontaneous symmetry breaking. Among the ones I found most helpful are Coleman (1985, Ch. 5) and Guralnik et al. (1968).

A particularly clear treatment is found in Aitchison (1982, Ch. 6). The discussion assumes that the dimension of space is two or greater; for the reasons why this assumption is needed see Coleman (1973).

In different approaches to QFT the vacuum is expected to satisfy different conditions. In Fock representations the vacuum state is the zero‐particle state. In most approaches it is postulated, and sometimes proved, that the vacuum is the unique (up to phase) state that is Poincare´ invariant.

This result is peculiar to the zero mass Klein–Gordon field. Streater (1966) shows that when m > 0 spontaneous symmetry breaking cannot occur for the Klein–Gordon field.

Commentators will sometimes say that a spontaneously broken symmetry is represented in a ‘non‐Wigner’ mode. This terminology invites confusion. Wigner (1959) was concerned with ray mappings of a separable Hilbert space ℋthat preserve transition probabilities and he showed that the action of such a mapping is given by a vector mapping that is either linear unitary or antilinear antiunitary. So the notion that a spontaneously broken symmetry is represented in a ‘non‐Wigner’ mode has led some commentators to conclude that a spontaneously broken symmetry does not conserve probability (see, for example, Fonda & Ghirardi, 1970, p. 446). But as will be seen below, ‘symmetry’ in spontaneous symmetry breaking does not invoke the ray or vector maps at the basis of Wigner's theorem.

For an account of what Noether (1918) did in her classic paper on variational symmetries and her purpose in stating two theorems see Brading and Brown (2003).

Using similar heuristic reasoning, T. D. Lee (1973) discusses the spontaneous breaking of P, CP and T. For an application of the (assumed) spontaneous breakdown of CP symmetry to cosmology see Zel'Dovich et al. (1975).

My ‘Rough guide to spontaneous symmetry breaking’ (Earman, 2003b) gave a brief and (alas!) all‐too‐rough introduction to the use of the algebraic formalism for QFT to illuminate features of symmetry breaking.

Haag's theorem shows, for example, that irreducible representations of a free scalar field and a self‐interacting scalar field are unitarily inequivalent.

See Appendix for more details. The mathematics of algebraic QFT is developed in Bratteli and Robinson (1987, 1996).

For a construction of the Weyl algebra for the Klein–Gordon field see Wald (1994).

Here || • || is the norm of the C*‐algebra. Note that || ϑ(A)|| = ||A|| for all A ∈ 𝒜.

And in the case where there are an uncountable infinity of degenerate vacuum states, the Hilbert space that accommodates them all would be nonseparable.

π_ω1+ω2 (𝒜)′ denotes the commutant of π_ω1+ω2 (𝒜), i.e. the set of all bounded operators on ℋ_ω1+ω2 that commute with each element of π_ω1+ω2 (𝒜). The equivalence of the definitions of disjointness is proved in Bratteli and Robinson (1996).

A state ω on a C*‐algebra 𝒜 is said to be a factor iff the von Neumann algebra π_ω (𝒜)″ (where ″ indicates the double commutant) has a trivial center, i.e. π_ω(𝒜)″ ∩ π_ω(𝒜)′ consists of multiples of the identity.

One might also require of a vacuum representation that it be unitarily equivalent to a Fock representation with |0⟩ being the no‐particle state. However, this is not essential for the present point.

The reader is invited to compare the present discussion with the ‘Panel discussion’ that followed Simon Saunders' question about the possibility of a dynamical representation of spontaneous symmetry breaking (see Cao, 1999, pp. 382–383).

This model is taken from Goldstone (1961).

For a nice antidote to this hype see Martin (2001).

For an account of the discovery of this mechanism see Higgs (1997).

For details about constrained Hamiltonian systems and techniques for quantizing them see Henneaux and Teitelboim (1992). For an introduction designed to be accessible to philosophers of science see Earman (2003a).

In classical mechanics the Legendre transformation takes one from the Lagrangian variables (q, q˙) to the Hamiltonian variables (q, p), where the canonical momenta p are given by ∂ℒ/∂q˙. For unconstrained Hamiltonian systems p will be m q˙.

There is another approach to quantizing constrained Hamiltonian systems called Dirac constraint quantization. It promotes the classical Hamiltonian constraints to operators on a Hilbert space and identifies the physical sector as consisting of vectors that are annihilated by the operator constraints. It is mathematically possible that reduced phase space and Dirac constraint quantization yield physically inequivalent results. As far as I am aware this possibility is not realized in the types of examples at issue.

Assuming, of course, that the actual universe is spatially infinite. The latest evidence from cosmological observations is that this assumption is correct.