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where ' (x) = g( x, 1). From a theoretical point of view, the dimensions D h , Dr are more fundamental than the exponents a, {3, etc. The former tell us how thermodynamic variables behave under a scale transformation; the latter are combinations dictated purely by experimental convenience. The critical exponents can all be expressed in terms of Dh and Dr as follows:

a= 2 - -

FIGURE 7.57 It is helpful to leave older trendlines in place, as they can help illustrate the general strength or weakness of a stock historically.

1 (16.59)

p= -

1/ = 2

The first four relations can be obtained from (16.58) by using (16.3)-(16.6) and the definitions (16.21)-(16.24). The last two came from (16.56) and the Fisher scaling law. We have derived the scaling laws via various routes, using different ways to realize the scaling hypothesis. None of these are satisfactory, for they rely on ad hoc assumptions. However, they do suggest plausible reasons for the experimental success of the scaling laws. Motivated by these discussions, we set the goal of theory as a derivation of scale invariance directly from more basic premises, and calculation of the dimensions of various thermodynamic quantities. This goal will be attained with the help of the renormalization-group method, as will be discussed in 18.

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We must comment on that most remarkable fact, which underlies Landau's original conception of the order parameter, that macroscopic systems generally have a lesser degree of symmetry at low temperatures than at high temperatures. The symmetry manifested at high temperatures is usually a property of the system's microscopic Hamiltonian. As such, it cannot cease to exist, even when it appears to be violated. The question is, where does it go For example, the microscopic Hamiltonian of a ferromagnet is rotationally invariant. Lowering the temperature of the system surely does not change that. What is changed, however, is the mode in which the symmetry is expressed. One might find it natural to assume (in the best tradition of Aristotelian logic) that in the most perfect expression of the symmetry, the ground state should be invariant under the symmetry operation. Nature, alas, is not perfect. In most instances, the system possesses many equivalent ground states that transform into one another

under the symmetry operation. But, since the system can actually exist in only one of these states, the symmetry appears to be broken. This phenomenon, that the ground state of the system does not possess the symmetry of the Hamiltonian, is called "spontaneous symmetry breaking." We have encountered the simplest manifestation of spontaneous symmetry breaking in the two-dimensional Ising model. The total energy is invariant under a simultaneous sign change of all the spins. Yet, the lowest-energy configuration is that in which all spins are aligned. In this case the symmetry is expressed through the fact that the spins could also have aligned themselves along the opposite direction. A more subtle example is the nonvanishing of the superfluid order parameter, as discussed in 12. When a continuous symmetry is spontaneously broken in a quantum mechanical system, interesting consequences follow. In this case, there is a noncountable infinity of ground states (with energy taken to be zero) orthogonal to one another. The system must choose one of these. As a consequence of the degeneracy, there emerges a type of excited state in which the local ground state changes very gradually over space, so as to form a "wave" of very long wavelength. Such a state is orthogonal to anyone ground state, with an energy approaching zero when the wavelength approaches infinity. This is called a "Goldstone excitation." The underlying symmetry is said to be realized in the "Goldstone mode." Familiar examples of Goldstone excitations include phonons in a crystal lattice, which expresses the broken translational invariance, and spin waves in a ferromagnetic, which realizes the basic rotational invariance. A new twist occurs when the system is coupled to a zero-mass vector field such as the photon field. In this case the Goldstone mode gives way to the" Higgs

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