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Symmetry is among the most fundamental and powerful concepts in nature, whose existence is usually taken as given, without explanation. We explore whether symmetry can be derived from more fundamental principles from the perspective of quantum information. Starting with a two-qubit system, we show there are only two minimally entangling logic gates: the Identity and the SWAP. We further demonstrate that, when viewed as an entanglement operator in the spin-space, the $S$-matrix in the two-body scattering of fermions in the $s$-wave channel is uniquely determined by unitarity and rotational invariance to be a linear combination of the Identity and the SWAP. Realizing a minimally entangling $S$-matrix would give rise to global symmetries, as exemplified in Wigners spin-flavor symmetry and Schrodingers conformal invariance in low energy Quantum Chromodynamics. For $N_q$ species of qubit, the Identity gate is associated with an $[SU(2)]^{N_q}$ symmetry, which is enlarged to $SU(2N_q)$ when there is a species-universal coupling constant.
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Entanglement suppression in the strong interaction $S$-matrix is shown to be correlated with approximate spin-flavor symmetries that are observed in low-energy baryon interactions, the Wigner $SU(4)$ symmetry for two flavors and an $SU(16)$ symmetry for three flavors. We conjecture that dynamical entanglement suppression is a property of the strong interactions in the infrared, giving rise to these emergent symmetries and providing powerful constraints on the nature of nuclear and hypernuclear forces in dense matter.
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We extend our study of deriving the local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory by considering self-interactions of the scalar field and inclusion of the electromagnetic interaction. By analyzing renormalizability and the scale separation conditions of three-, four- and five-point vertex functions of the scalar field, we fix the two couplings of the scalar field self-interactions of the leading order Lagrangian. Next we add the electromagnetic interaction and derive conditions relating the magnetic moment of the charged vector boson to its charge and the masses of the charged and neutral massive vector bosons to each other and the two independent couplings of the theory. We obtain the bosonic part of the Lagrangian of the electroweak Standard Model as a unique solution to the conditions imposed by the self-consistency conditions of the considered effective field theory.
We revisit the problem of deriving local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory. Previous derivations were based on the condition of tree-order unitarity. However, the modern point of view considers the Standard Model as the leading order approximation to an effective field theory. As tree-order unitarity is in any case violated by higher-order terms in an effective field theory, it is instructive to investigate a formalism which can be also applied to analyze higher-order interactions. In the current work we consider an effective field theory of massive vector bosons interacting with a massive scalar field. We impose the conditions of generating the right number of constraints for systems with spin-one particles and perturbative renormalizability as well as the separation of scales at one-loop order. We find that the above conditions impose severe restrictions on the coupling constants of the interaction terms. Except for the strengths of the self-interactions of the scalar field, that can not be determined at this order from the analysis of three- and four-point functions, we recover the gauge-invariant Lagrangian with spontaneous symmetry breaking taken in the unitary gauge as the leading order approximation to an effective field theory. We also outline the additional work that is required to finish this program.
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