Spontaneous symmetry breaking in classical systems
| Franco Strocchi (2011), Scholarpedia, 6(10):11195. | doi:10.4249/scholarpedia.11195 | revision #91809 [link to/cite this article] |
The mechanism of spontaneous symmetry breaking in classical systems amounts to the fact that a symmetric dynamics, typically described by a symmetric Hamiltonian or Lagrangian, may allow a non-symmetric behavior of the system.
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Introduction
The mechanism of spontaneous symmetry breaking has become one of the corner stones of modern theoretical physics, with crucial applications to many body theory and to elementary particle physics. It amounts to the fact that a symmetric dynamics, typically described by a symmetric Hamiltonian or Lagrangian, may allow a non-symmetric behavior of the system. Since the mechanism is very subtle, its precise definition requires some care. The literature on the meaning of spontaneous symmetry breaking, especially from the point of view of general philosophy, is rather rich also because it is somewhat intriguing to understand how a symmetry of the dynamics may get lost.
In the standard popular accounts, spontaneous symmetry breaking is identified with the existence of a non-symmetric lowest energy configuration or state, but this explanation without further qualification may be misleading, since it does not in general account for the dramatic loss of symmetry which accompanies spontaneous symmetry breaking. E.g., for a particle on a horizontal plane subject to a constant gravitational field perpendicular to the plane, each point of the plane defines a lowest energy configuration, which is not invariant under translations on the plane, but it would be improper to speak of symmetry breaking, since the behavior of the system does not display a loss of symmetry.
Even less satisfactory is the identification of spontaneous symmetry breaking with the existence of non-symmetric solutions in a theory described by symmetric equations; e.g. in the two-body problem with a central potential the existence of orbits which are not circles have not been (and should not be) regarded as the breaking of rotational symmetry.
As a matter of fact, a non-symmetric behavior of a system described by a symmetric Hamiltonian seems to run into conflict with the general wisdom of classical mechanics.
The "Noether theorem"
The classical "Noether theorem" for Hamiltonian systems with a finite number \(2 s\) of degrees of freedom, described by the canonical variables \(q_k, p_k\ ,\) \(k = 1, \cdots s\ ,\) states that the symmetry of the Hamiltonian under an \(N\)-dimensional Lie group \(G\) implies the existence of \(N\) functions of the canonical variables, \(Q^i\ ,\) \(i = 1, \cdots N\ ,\) which are constants under time evolution (briefly constants of motion) and which are the generators of the symmetry transformations of the canonical variables
\[\delta^i (q_k) = \{q_k, \,Q^i\}, \,\,\,\,\delta^i (p_k) = \{p_k, \,Q^i\}, \]
where \(\delta^i\) denotes the infinitesimal variation under the \(i\)-th subgroup of \(G\ ,\) and \(\{\,, \,\}\) the Poisson bracket
\[ \{ f, g \} = \frac{\partial f}{\partial q_k} \, \frac{ \partial g }{ \partial p_k } - \frac { \partial f }{ \partial p_k } \, \frac{ \partial g }{ \partial q_k } \]
(a sum over repeated indices being always understood).
The time independence of \(Q^i\) implies that
\[ \delta^i ( q_k(t) ) = \{ q_k(t), \, Q^i \} = \{ q_k(t), \, Q^i(t) \} = ( \delta^i q_k )(t) \]
\[ \delta^i ( p_k(t) ) = ( \delta^i p_k )(t), \]
i.e. the symmetry transformation commutes with the time evolution.
For continuous classical systems described by a set of fields \( \phi_a(\mathbf x,t) \ ,\) \( a = 1, \cdots n\ ,\) \(\mathbf x \in \mathbf R^s \ ,\) \(s\) being the space dimensions, Noether theorem (Noether 1918) states that the invariance of the Lagrangian under an \(N\)-dimensional Lie group \(G\) implies the existence of \(N\) conserved currents
\( J_\mu^i(\mathbf x, t) \ ,\) \( \mu = 0, 1, 2, \cdots s, \) \( i = 1, \cdots N, \)
\[ \partial^\mu J^i_\mu(x) = \partial_t J_0^i(x, t) + \mbox{div} \mathbf J^i(x, t)= 0 , \]
and of \(N\) time independent "charges" \(Q^i\ ,\)
\[Q^i(t) = \int d^s x \, J_0^i(\mathbf x, t) = Q^i(0) = Q^i.\]
Furthermore, as functionals of the canonical variables \( \phi_a, \pi_a\ ,\) (\(\pi_a\) denotes the field canonically conjugated to \(\phi_a\)), the charges \(Q^i\) are the generators of the symmetry transformations of the canonical variables, with the Poisson brackets defined by functional derivatives in place of partial derivatives,
see e.g. (Schmid 1987). Hence, the time evolution of the system does not display a loss of symmetry.
Symmetry breaking
To understand how a possible loss of symmetry may arise, we recall that a symmetry \(g\) of a classical physical system is an invertible transformation of the set coordinates \(\gamma\) which identify its configurations \(S_\gamma\ ,\) \( g: \gamma \rightarrow g \, \gamma\ ,\) such that it commutes with the time evolution \(a_t\ ,\) \(t \in \mathbf R \) \[ a_t g S_\gamma = a_t S_{g \gamma} = S_{(g \gamma)(t)} = g a_t S_\gamma , \] (for simplicity, we do not consider the more general case in which the dynamics transforms covariantly under the symmetry). For classical canonical systems, such a commutation is equivalent to the invariance of the Hamiltonian under the action of \(g\ .\)
As we learned from the foundations of special relativity and from the principles of quantum mechanics, a good theoretical concept must always be confronted with its operational counterpart. Therefore, the symmetric behavior of the system has an operational meaning, i.e. it can be detected, if both the configuration \(S_\gamma\) and its transformed one \(S_{g \gamma}\) are accessible by the same physical observer.
For this purpose, it is convenient to introduce the following structure, namely the partition of the set of configurations into classes or phases \(\Gamma\ ,\) each being characterized by the property that any two configurations \(S_{\gamma_1}, \,S_{\gamma_2} \in \Gamma \) may be obtained one from the other by physically realizable operations, so that they are both accessible by the same physical observer. For example, this is not the case if the two configurations are separated by an infinite potential barrier, since infinite energy would be required for going from one to the other.
Thus, different phases describe physically disjoint realizations of the system.
A symmetry \(g\) is physically realized or unbroken in a phase \(\Gamma\) if \(g\) leaves \(\Gamma \) stable (i.e. it maps configurations of \(\Gamma\) into configurations of \(\Gamma)\ ,\) and it is broken in \(\Gamma\) otherwise.
This definition of unbroken symmetry accounts for the fact that not only the dynamics is symmetric, but, more crucially, the symmetric behavior of the system can be checked by confronting the time evolution of any given configuration \(S_{\gamma} \in \Gamma\) and of its transformed one \(S_{g \gamma}\ ,\) since both belong to \(\Gamma\) and are therefore accessible by the same observer.
It is worthwhile to apply the above definition of phases to the popular mechanical examples of the particle in a (finite) one-dimensional double well, corresponding to the potential energy \(V(q) = \lambda (q^2 - a^2)^2\ ,\) as well as that of the particle in a Mexican hat potential (in three space dimensions), \( V(q) = \lambda (\mathbf q^2 - a^2)^2 \)
(\(a>0\) is assumed in both cases).
In the first case, unless one puts an artificial bound on the kinetic energy, all the configurations fall in only one phase, since there is no physical obstruction for going from one configuration to another. In the second case, the degenerate ground states (defined by the points at the bottom of the Mexican hat) have all the same energy and one cannot envisage any physical obstruction which prevents from going from one ground state to the other; hence, there is only one phase. Then, according to the above definition, no symmetry can be broken and such mechanical examples are not good examples of spontaneous symmetry breaking. On the other hand, if one lets the local maximum of the above one-dimensional double well potential to go to infinity, e.g. by adding to the potential the function \(v(q) = q^{-2} - (a/2)^{-2}\ ,\) for \(|q| \leq a/2\ ,\) \(v(q) = 0\ ,\) for \(|q| > a/2 \ ,\) \(a > 0\ ,\) then two phases arise, \(\Gamma_1 = \{q > 0, \, p \in \mathbf R \}\ ,\) \(\Gamma_2 = \{ q < 0, \,p \in \mathbf R \}\ ,\) and the discrete symmetry \(g: q \rightarrow - q\) is broken in each phase.
It is clear from these examples that in classical Hamiltonian systems with a finite number of degrees of freedom it is difficult to have spontaneous symmetry breaking: discrete symmetries can be broken in very artificial cases while continuous symmetries can never be broken.
The situation changes substantially if one considers infinitely extended systems. The new situation is well illustrated by the familiar prototypic system described by the (hyperbolic) field equation \[\tag{1} \square \phi(x,t) = U'(\phi(x, t)) \]
with \(\phi\) an \(n\)-component field (\(\phi: \mathbf R^s \rightarrow \mathbf R^n\)), and \(U(\phi)\) a potential satisfying suitable regularity conditions (Parenti, Strocchi and Velo 1975) in \(s = 1\) space dimensions \(U\) is required to be an entire function, in \(s = 2\) space dimensions if \(U(\phi) = \sum_{a \in \mathbf N^n}^{\infty} C_a \phi^a\ ,\) \(a\) being a multi index \(\phi^a = \phi_1^{a_1}\cdots \phi_n^{a_n}\ ,\) \(a_i \in \mathbf N\ ,\) then the following condition is required\[\sum_{a \in \mathbf N^n}^{\infty} |C_a| |a|^{|a|/2} |\phi|^{|a|} < \infty\] and finally in \(s = 3\) space dimensions \(U\) is required to be a twice differentiable real function satisfying\[\mbox{sup}_{\phi}\,(1 + |\phi|^2)^{-1}\,| U''(\phi)| < \infty.\] As we shall see, the relevant point is the emergence of a natural non-trivial structure of phases, i.e. of physically disjoint realizations of the system described by the above field equation.
The crucial physical property is the inevitable localization in space of any physical operation. Thus, for infinitely extended systems a condition for the physical realizability of two initial conditions (by the same observer) is that they have the same behavior at space infinity, since no physically realizable operation can change the boundary condition of the phase or ``universe`` in which the observer is living. Moreover, since the time evolution is one of the realizable operations, if an initial data is realizable by one observer, so is also its time translated one.
The above considerations are technically formalized by the following notion: a sector \(S\) is a set of initial data such that i) any pair of them \( u =( \phi, \,\psi)\ ,\) \(\psi = \partial_t \phi \ ,\) \(u' = ( \phi', \,\psi')\) differ by elements of \((H^1(\mathbf R^s), L^2(\mathbf R^s))\ ,\) i.e. \(\phi' - \phi\) and \(\partial_j (\phi' - \phi)\ ,\) \(j = 1, \cdots s\) are square integrable and \(\psi' - \psi \in L^2(\mathbf R^s)\ ,\) (a motivation for the choice of such a functional space is that it leads to a Hilbert space structure stable under time evolution, see below), ii) if \(u\) belongs to \(S\ ,\) also its time translated one \(u_\tau = u(\tau)\) belongs to \(S\ .\) Clearly, the sector \(S_u\ ,\) to which \(u\) belongs, is a subset of the Hilbert space \(H_u = u + (H^1(\mathbf R^s), L^2(\mathbf R^s))\ .\) A distinguished class of sectors are those of the form \(S_u = H_u\ ;\) they are isomorphic to Hilbert spaces and are called Hilbert sectors.
A necessary and sufficient condition for the stability of a sector \(H_u\ ,\) \(u = (\phi, \psi)\ ,\) under space translation is that \(\partial_j \phi \in L^2(\mathbf R^s)\ ,\) \(j = 1, \cdots s\) (Strocchi 2008).
The realization and relevance of the above definitions is provided by the following theorem (Parenti, Strocchi and Velo 1977, Strocchi 2008)
A constant initial data \( u_0 = (\phi_0, \,0)\ ,\) corresponding to an absolute minimum of the potential \(U\ ,\) defines an Hilbert sector, i.e. a set of configurations \(H_{u_0} = u_0 + (H^1(\mathbf R^s), L^2(\mathbf R^s))\ ,\) with the following properties: i) \(H_{u_0}\) is stable under time evolution and under spaces translations, ii) all the solutions corresponding to initial data \( u = (\phi, \psi) \in H_{u_0}\) have the same asymptotic limit \(\phi_0\) for \(|x| \rightarrow \infty\ ,\) for any time. Furthermore the energy of the configurations \( u \in H_{u_0} \) is bounded from below.
The property of energy bounded from below guarantees a sort of stability, since otherwise small perturbations may lead to solutions with lower and lower energy, i.e. to a collapse. Thus, different (absolute) minima of the potential define disjoint Hilbert sectors, which may be interpreted as disjoint stable realizations of the system (like the different thermodynamical phases of a complex system in the infinite volume limit), since two configurations belonging to different Hilbert sectors cannot be both physically accessible by the same observer.
It is worthwhile to remark that also initial data corresponding to relative minima of the potential define Hilbert sectors, but the energy is not bounded from below; for the general characterization of the initial data which define Hilbert sectors see (Parenti, Strocchi and Velo 1977).
A symmetry \(g\) of the system described by the above field equation is an invertible mapping of the initial data which commutes with the time evolution. The symmetry is broken in the realization of the system described by the Hilbert sector \(H_{u_0}\) if it does not leave \(H_{u_0}\) stable.
In the following, for simplicity, we shall consider groups \(G\) of internal symmetries, i.e. of the form \(g(\phi) = A \phi\ ,\) with \(A\) an invertible matrix satisfying \(A^T \, A = \lambda \mathrm I\ ,\) \(\lambda \in \mathbf R\ ,\) \(A^T\) the transpose of \(A\ ,\) and \(U(A \phi ) = \lambda U(\phi)\) (Parenti, Velo and Strocchi 1977).
Clearly, an internal symmetry of the Lagrangian maps Hilbert sectors into Hilbert sectors and if \(G\) is the group of internal symmetries of the Lagrangian each Hilbert sector \(H_{u_0}\) determines the subgroup \(G_{u_0}\) of \(G\ ,\) called the stability group of \(H_{u_0}\ ,\) which leaves the sector invariant and therefore describes the unbroken subgroup of \(G\ .\)
A time independent functional \(Q\) generates a transformation of the solutions \(u\ ,\) \(u_k = \phi_k\ ,\) for \(k = 1, \cdots n\ ,\) \(u_k = \psi_{k - n}\ ,\) for \(k = n + 1, \cdots 2 n\ ,\) through the Poisson brackets \[ \{u_k(x,t), Q\}=\sum_{a=1}^n \int d^3 \mathbf{y} \left[\frac{\delta u_k(x,t)}{\delta \phi_a(y,t)} \frac{\delta Q} {\delta \psi_a(y,t)} - \frac{\delta u_k(x, t)}{\delta \psi_a(y,t)} \frac{\delta Q}{\delta \phi_a(y,t)}\right],\] where \(\delta F/ \delta \phi_a(y,t)\) denotes the functional derivative of \(F\) with respect to \(\phi_a(y,t)\ ,\) see (Schmid 1987, Marsden and Ratiu 1994).
Then, one has the following generalization of Noether theorem which takes into account the possibility of symmetry breaking
Let \(G\) be an \(N\)-parameter Lie group of internal symmetries of (the Lagrangian of) eq. (1), then there exist \(N\) currents \(J_\mu^i(u(x, t))\ ,\) \(i = 1, \cdots N\ ,\) which obey the continuity equation \(\partial^\mu J_\mu^i = 0 \) (local conservation law). Given a Hilbert sector \(H_{u_0}\) and a one-parameter subgroup \(G^i\) of \(G\ ,\) the Noether charge\[ Q^i(u) = \int d^3 x \, J_0^i(u(x, t))\] exists and it is independent of time for all solutions \(u(x, t) \in H_{u_0}\ ,\) equivalently it defines a linear operator \(\tilde{Q}^i: H_{u_0} \rightarrow H_{u_0}\) acting as the generator of the corresponding transformation\[(\tilde{Q}^i u)_k = \{ u_k, \,Q^i\} = (\delta^i u)_k,\] if and only if \(G^i\) is a subgroup of the stability or unbroken group \(G_{u_0}\) of \(H_{u_0}\ .\) In this case, \(G^i\) is represented by unitary operators in \(H_{u_0}\ .\)
Example. Non-linear scalar field in three space dimensions
The model is defined by the following field equation\[\square \phi = - 2 \lambda \phi (\phi^2 - \mu^2)^2, \,\,\,\,\phi: \mathbf R^3 \rightarrow \mathbf R^n. \] It displays some similarity with the mechanical model of a particle in \(\mathbf R^n\) with a Mexican hat potential \(U(q) = \lambda (q^2 - \mu^2)^2\ .\) However, as explained above, the difference is substantial: in the field case, each point \(q\) has become infinite dimensional and the absolute minima \(\phi_0\) identify disjoint Hilbert sectors. Whereas in the finite dimensional case there is no physical obstruction or barrier which prevents the motion from one minimum to the other, in the field case there is no physically realizable operation which leads from one sector to the other, since this would require to change the asymptotic limit of the configurations and this is not possible by physically realizable operations, which are necessarily localized.
More generally, infinitely extended systems described by hyperbolic equations which admit solutions with non-trivial behaviors at space infinity, have disjoint realizations described by sectors stable under time evolution. Thus, the mechanism of spontaneous symmetry breaking discussed above applies.
Another wide class of infinitely extended systems, which display a structure of phases and spontaneous symmetry breaking is given by complex systems in the thermodynamical limit. A few comments are in order about the realization of spontaneous symmetry breaking in complex systems, some examples of which are used as prototypic examples for explaining the phenomenon of spontaneous symmetry breaking, which, strictly speaking, requires the infinite volume limit. Even if one is always dealing with finitely extended systems, the use of the thermodynamical limit has proved to be very useful for the theoretical description of systems with a very large number of degrees of freedom and with a very large size. The theory of phase transitions is one of the most successful achievements of such a strategy. In the thermodynamical limit different ground states define disjoint thermodynamical phases; they are selected by the boundary conditions imposed in taking the infinite volume limit, as discussed by Ruelle in his general treatment of Statistical Mechanics (Ruelle 2004). Thus, the general philosophy discussed above applies, since different phases describe disjoint realizations of the system, and a non-symmetric ground state implies the breaking of the symmetry in the corresponding phase.
References
- H. Goldstein, Classical Mechanics, 2nd. ed., Addison-Wesley 1980
- J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer 1994
- E. Noether, Invariante Variations Probleme, Nachr. d. Kgl. Ges. d. Wiss. Göttingen (1918), p.235-257; English translation in N.A. Tavel, Milestones in mathematical physics, Transport Theory and Statistical Mechanics, 1, 183-207 (1971)
- C. Parenti, F. Strocchi and G. Velo, Solutions of Classical Field Equations with Local Finite Kinetic Energy, Phys. Lett. 59 B, 157-158 (1975)
- C. Parenti, F. Strocchi and G. Velo, Hilbert space sectors for solutions of non-linear relativistic field equations, Comm. Math. Phys. 53, 65-96 (1977)
- D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Cambridge University Press 2004
- R. Schmid, Infinite Dimensional Hamiltonian Systems, Bibliopolis 1987
- F. Strocchi, Symmetry Breaking, 2nd ed., Springer 2008; there one can find a more extensive discussion, the proofs of the propositions stated in this note and further references.
- A.S. Wightman, Constructive Field Theory, in Fundamental Interactions in Physics and Astrophysics, G. Iverson et al. eds., Plenum 1972
