CRC Press eBooks

Symmetries

Bruno Juliá Díaz, Arnau Rios Huguet, Héctor Briongos Merino, Javier Rozalén Sarmiento, Artur Polls Martí

Published Sep 17, 2025• 0 citations

Abstract

Symmetry considerations lie at the very heart of quantum many-body theory. In particular, symmetries dictate and restrict the properties of many-body wavefunctions. A many-body system with https://www.w3.org/1998/Math/MathML" display="inline"> N particles can be described at zero temperature by a many-body wavefunction https://www.w3.org/1998/Math/MathML" display="inline"> Ψ x → 1 , ⋯ , x → A , where https://www.w3.org/1998/Math/MathML" display="inline"> x → i represents the coordinate of particle https://www.w3.org/1998/Math/MathML" display="inline"> i . Note that this coordinate can be discrete, e.g. a lattice site in a spin system; continuous, if, for instance, it describes the position of a particle; or a combination of both, when we work with particles of definite position and spin, https://www.w3.org/1998/Math/MathML" display="inline"> x → i = x → i , σ i .

DOI: 10.1201/9781003191339-1

Related papers

Mathematical introduction

In-Gee Kim

The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .

Mar 14, 20181 citation

Introduction

Jiannis K. Pachos

Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).

Cambridge University Press eBooksApr 12, 20120 citations

Angular Momentum and Spin

Martin Bojowald A

If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .

CRC Press eBooksMay 8, 20260 citations

Symmetry in Many-Body Physics

Vanderlei Salvador Bagnato, R. G. Nazmitdinov, Vyacheslav I. Yukalov

The harmony of nature is expressed through the implementation of symmetry providing optimal structures for complex systems from snowflakes to graphene lattices [...]

SymmetryDec 27, 20221 citation

Quantum Information

Martin Bojowald A

There is more information in a quantum state than in the classical initial values https://www.w3.org/1998/Math/MathML" display="inline"> ( x 0 , p 0 ) that determine the complete 1-dimensional motion of a point particle for a given force. The initial wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , 0 ) requires an infinite number of parameters, which can be expressed in observational terms as the two basic expectation values https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ^ ⟩ and https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ p ^ ⟩ together with the two variances https://www.w3.org/1998/Math/MathML" display="inline"> ( σ x 2 , σ p 2 ) and the covariance 5.1 https://www.w3.org/1998/Math/MathML" display="block"> C x p = 1 2 ⟨ x ^ p ^ + p ^ x ^ ⟩ − ⟨ x ^ ⟩ ⟨ p ^ ⟩ ,

CRC Press eBooksMay 8, 20260 citations

2 On the Wave Functions of Many-Electron Systems

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

From the very ﬁrst applications of quantum mechanics to many-electron problems it appeared necessary to state the symmetry of the Schro¨dinger wave function corresponding to a certain energy level of an atomic system.

CRC Press eBooksMay 21, 20042 citations