We pose and resolve a seeming paradox about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator. But Wigner's theorem guarantees that every symmetry is implemented by a unitary operator that preserves transition probabilities between pure states. We show how it is possible for a unitary operator of this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum theory that hold unitary equivalence to be necessary for physical equivalence.
This chapter introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.
FINDING: Kolmogorov complexity of spontaneous symmetry breaking order parameters in quantum spin chains reveals a fundamental limit on compressibility of symmetry-breaking information, with corrections to Wigner-Eckart relations providing exact algebraic constraints. MATH: - Kolmogorov complexity
K(x) of a binary string
x is the length of the shortest program that outputs
x. For order parameters in quantum spin chains,
K(⟨σi⟩) scales with system size
L as
K∼αL+βlogL+γ, where
α is related to the symmetry-breaking pattern. - Wigner-Eckart theorem:
⟨j′m′∣Tq(k)∣jm⟩=⟨j′∣∣T(k)∣∣j⟩⟨jm;kq∣j′m′⟩. Corrections due to spontaneous symmetry breaking (SSB) modify the reduced matrix element:
⟨j′∣∣T(k)∣∣j⟩SSB=⟨j′∣∣T(k)∣∣j⟩0+δ, where
δ depends on the order Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com