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The Underpinnings of Modern Physics: A Symphony of Symmetries

• • • • • • • • • • • • • • • • Part of a series of articles about Quantum mechanics

iℏ

d

d t

|Ψ⟩ = H^|Ψ⟩ ${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$

Schrödinger equation

• Introduction

• Glossary

• History

Background

• Classical mechanics

• Old quantum theory

• Bra–ket notation

• Hamiltonian

• Interference

Fundamentals

• Complementarity

• Decoherence

• Entanglement

• Energy level

• Measurement

• Nonlocality

• Quantum number

• State

• Superposition

• Symmetry

• Tunnelling

• Uncertainty

• Wave function

• Collapse

Experiments

• Bell's inequality

• CHSH inequality

• Davisson–Germer

• Double-slit

• Elitzur–Vaidman

• Franck–Hertz

• Leggett inequality

• Leggett–Garg inequality

• Mach–Zehnder

• Popper

• Quantum eraser

• Delayed-choice

• Schrödinger's cat

• Stern–Gerlach

• Wheeler's delayed-choice

Formulations

• Overview

• Heisenberg

• Interaction

• Matrix

• Phase-space

• Schrödinger

• Sum-over-histories (path integral)

Equations

• Dirac

• Klein–Gordon

• Pauli

• Rydberg

• Schrödinger

Interpretations

• Bayesian

• Consciousness causes collapse

• Consistent histories

• Copenhagen

• de Broglie–Bohm

• Ensemble

• Hidden-variable

• Many-worlds

• Objective-collapse

• Quantum logic

• Superdeterminism

• Relational

• Transactional

Advanced topics

• Relativistic quantum mechanics

• Quantum field theory

• Quantum information science

• Quantum computing

• Quantum chaos

• EPR paradox

• Density matrix

• Scattering theory

• Quantum statistical mechanics

• Quantum machine learning

Scientists

• Aharonov

• Bell

• Bethe

• Blackett

• Bloch

• Bohm

• Bohr

• Born

• Bose

• de Broglie

• Compton

• Dirac

• Davisson

• Debye

• Ehrenfest

• Einstein

• Everett

• Fock

• Fermi

• Feynman

• Glauber

• Gutzwiller

• Heisenberg

• Hilbert

• Jordan

• Kramers

• Lamb

• Landau

• Laue

• Moseley

• Millikan

• Onnes

• Pauli

• Planck

• Rabi

• Raman

• Rydberg

• Schrödinger

• Simmons

• Sommerfeld

• von Neumann

• Weyl

• Wien

• Wigner

• Zeeman

• Zeilinger

• v • t • e

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Quantum field theory | Feynman diagram | History

Background

• Field theory

• Electromagnetism

• Weak force

• Strong force

• Quantum mechanics

• Special relativity

• General relativity

• Gauge theory

• Yang–Mills theory

Symmetries

• Symmetry in quantum mechanics

• C-symmetry

• P-symmetry

• T-symmetry

• Lorentz symmetry

• Poincaré symmetry

• Gauge symmetry

• Explicit symmetry breaking

• Spontaneous symmetry breaking

• Noether charge

• Topological charge

Tools

• Anomaly

• Background field method

• BRST quantization

• Correlation function

• Crossing

• Effective action

• Effective field theory

• Expectation value

• Feynman diagram

• Lattice field theory

• LSZ reduction formula

• Partition function

• Path Integral Formulation

• Propagator

• Quantization

• Regularization

• Renormalization

• Vacuum state

• Wick's theorem

• Wightman axioms

Equations

• Dirac equation

• Klein–Gordon equation

• Proca equations

• Wheeler–DeWitt equation

• Bargmann–Wigner equations

• Schwinger-Dyson equation

• Renormalization group equation

Standard Model

• Quantum electrodynamics

• Electroweak interaction

• Quantum chromodynamics

• Higgs mechanism

Incomplete theories

• String theory

• Supersymmetry

• Technicolor

• Theory of everything

• Quantum gravity

Scientists

• Adler

• Anderson

• Anselm

• Bargmann

• Becchi

• Belavin

• Bell

• Berezin

• Bethe

• Bjorken

• Bleuer

• Bogoliubov

• Brodsky

• Brout

• Buchholz

• Cachazo

• Callan

• Cardy

• Coleman

• Connes

• Dashen

• DeWitt

• Dirac

• Doplicher

• Dyson

• Englert

• Faddeev

• Fadin

• Fayet

• Fermi

• Feynman

• Fierz

• Fock

• Frampton

• Fritzsch

• Fröhlich

• Fredenhagen

• Furry

• Glashow

• Gell-Mann

• Glimm

• Goldstone

• Gribov

• Gross

• Gupta

• Guralnik

• Haag

• Hagen

• Han

• Heisenberg

• Hepp

• Higgs

• 't Hooft

• Iliopoulos

• Ivanenko

• Jackiw

• Jaffe

• Jona-Lasinio

• Jordan

• Jost

• Källén

• Kendall

• Kinoshita

• Kim

• Klebanov

• Kontsevich

• Kreimer

• Kuraev

• Landau

• Lee

• Lee

• Lehmann

• Leutwyler

• Lipatov

• Łopuszański

• Low

• Lüders

• Maiani

• Majorana

• Maldacena

• Matsubara

• Migdal

• Mills

• Møller

• Naimark

• Nambu

• Neveu

• Nishijima

• Oehme

• Oppenheimer

• Osborn

• Osterwalder

• Parisi

• Pauli

• Peccei

• Peskin

• Plefka

• Polchinski

• Polyakov

• Pomeranchuk

• Popov

• Proca

• Quinn

• Rouet

• Rubakov

• Ruelle

• Sakurai

• Salam

• Schrader

• Schwarz

• Schwinger

• Segal

• Seiberg

• Semenoff

• Shifman

• Shirkov

• Skyrme

• Sommerfield

• Stora

• Stueckelberg

• Sudarshan

• Symanzik

• Takahashi

• Thirring

• Tomonaga

• Tyutin

• Vainshtein

• Veltman

• Veneziano

• Virasoro

• Ward

• Weinberg

• Weisskopf

• Wentzel

• Wess

• Wetterich

• Weyl

• Wick

• Wightman

• Wigner

• Wilczek

• Wilson

• Witten

• Yang

• Yukawa

• Zamolodchikov

• Zamolodchikov

• Zee

• Zimmermann

• Zinn-Justin

• v • t • e

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Symmetries in quantum mechanics are the bedrock upon which modern physics is built. They’re not just pretty mathematical flourishes; they are the fundamental constraints that govern spacetime and the very particles that populate it, from the dizzying heights of quantum field theory down to the granular realities of condensed matter physics. In essence, symmetry in physics, its invariance, and the resulting conservation laws are the sharp edges that carve out the landscape of our understanding. They’re not just tools for solving problems; they predict what can happen, laying down the essential rules of the game. While conservation laws might not always hand you the solution on a silver platter, they provide the crucial framework, the first, undeniable steps towards unraveling complex phenomena. Furthermore, grasping these symmetries reveals deeper truths about the expected states of a system. For instance, the presence of non-commuting symmetry operators hints at degenerate states, while non-degenerate states are, by definition, eigenvectors of these very symmetry operators.

This exhaustive exploration delves into the connection between the classical manifestations of continuous symmetries and their quantum operator counterparts. It charts their relationship to the intricate structures of Lie groups and the fundamental relativistic transformations within the Lorentz group and the Poincaré group.

Notation: The Language of Precision

The notational conventions employed herein are as follows. Boldface is reserved for vectors, four vectors, matrices, and vectorial operators. Quantum states are expressed using the elegant bra–ket notation. Broad hats signify operators, while narrow hats denote unit vectors, including their components when expressed in tensor index notation. The summation convention is applied to repeated tensor indices unless explicitly stated otherwise. The Minkowski metric employs the signature (+−−−).

Symmetry Transformations on the Wavefunction in Non-Relativistic Quantum Mechanics

Continuous Symmetries: The Flow of Invariance

At its core, the relationship between continuous symmetries and conservation laws is elegantly captured by Noether's theorem.

The fundamental nature of quantum operators, such as the energy operator being intrinsically linked to the partial time derivative and the momentum operator to the spatial gradient, becomes starkly clear when we consider an initial state and then subtly alter a single parameter. This can be achieved through displacements in space (lengths), shifts in time (durations), or rotations (angles). Moreover, the invariance of certain physical quantities can be observed by performing these changes in lengths and angles, thereby illustrating the conservation of those very quantities.

For the purposes of this discussion, we shall primarily focus on transformations affecting single-particle wavefunctions, taking the form:

ψ(𝐁, t) = ψ(𝐁′, t′) ${\displaystyle {\widehat {\Omega }}\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t')}$

Here, Ω^ denotes a unitary operator. Unitarity is a non-negotiable requirement for operators representing transformations of space, time, and spin, as the norm of a quantum state—which fundamentally represents the total probability of finding a particle in a particular configuration—must remain invariant under such transformations. The inverse of a unitary operator is its Hermitian conjugate:

Ω^⁻¹ = Ω^²† {\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }}}

These principles can be readily extended to encompass many-particle wavefunctions. Expressed in the standard Dirac notation, these transformations on quantum state vectors appear as:

Ω^|r(t)⟩ = |r′(t′)⟩ ${\displaystyle {\widehat {\Omega }}\left|\mathbf {r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle }$

It follows that an operator Α^ that remains invariant under the action of Ω^ must satisfy the condition:

Α^ψ = Ω^²† Α^ Ω^ ψ ∑ Ω^ Α^ ψ = Α^ Ω^ ψ ${\displaystyle {\widehat {A}}\psi ={\widehat {\Omega }}^{\dagger }{\widehat {A}}{\widehat {\Omega }}\psi \quad \Rightarrow \quad {\widehat {\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat {\Omega }}\psi .}$

Consequently, for any state ψ, this implies:

[Ω^, Α^]ψ = 0 ${\displaystyle [{\widehat {\Omega }},{\widehat {A}}]\psi =0}$

This means Ω^ and Α^ must commute. Furthermore, quantum operators representing observables are mandated to be Hermitian, ensuring that their eigenvalues, which represent the possible measurement outcomes, are always real numbers. This condition is mathematically expressed as Α^ = Α^²†.

An Overview of Lie Group Theory: The Architecture of Symmetry

For a deeper, more exhaustive understanding, one should consult the dedicated articles on Lie group and Generator (mathematics). The following presents the essential concepts of group theory crucial for comprehending quantum theory, with illustrative examples woven throughout. An alternative perspective, utilizing matrix groups, can be found in the works of Hall [1] [2].

Let G represent a Lie group. This is a group that, in its local structure, can be described by a finite number, N, of real continuously varying parameters, denoted ξ¹, ξ², ..., ξ&supN;. More formally, G is a smooth manifold that also possesses group properties, with its group operations being smooth functions.

• The dimension of the group, N, corresponds precisely to the number of parameters required to describe its elements.

• The group elements, denoted by g, are functions of these parameters:

  g = G(ξ¹, ξ², ...) {\displaystyle g=G(\xi _{1},\xi _{2},\dots )}}

  Crucially, setting all parameters to zero returns the identity element of the group:

  I = G(0, 0, ...) {\displaystyle I=G(0,0,\dots )}}

  Group elements frequently manifest as matrices that act on vectors or as transformations that operate on functions.

• The generators of the group are derived by taking the partial derivatives of the group elements with respect to each parameter, and then evaluating these derivatives at the point where all parameters are zero:

  Xⱼ = ∂g / ∂ξⱼ |_ξⱼ=0 {\displaystyle X_{j}=\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}}

  In the language of differential geometry, these generators form the tangent space to the group manifold at the identity element. They are also referred to as infinitesimal group elements or, more technically, as the elements of the Lie algebra associated with G. (This concept is further elaborated in the discussion of the commutator below.)

  In theoretical physics, generators often manifest as operators embodying specific symmetries. These can be represented as matrices or as differential operators. Within the framework of quantum theory, for unitary representations of a group, the generators are conventionally multiplied by a factor of i:

  Xⱼ = i (∂g / ∂ξⱼ) |_ξⱼ=0 {\displaystyle X_{j}=i\left.{\frac {\partial g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}}

  The set of generators forms a vector space, meaning that linear combinations of generators also act as generators.

• These generators adhere to specific commutation relations:

  [Xₐ, X<0xE2><0x82><0x99>] = i fₐ<0xE2><0x82><0x99>c X<0xE1><0xB5><0xA8> {\displaystyle \left[X_{a},X_{b}\right]=if_{abc}X_{c}}}

  Here, fₐ<0xE2><0x82><0x99>c are the structure constants of the group, which are dependent on the chosen basis. Together with the vector space property, these relations define the Lie algebra of the group. The inherent antisymmetry of the commutator ensures that the structure constants are antisymmetric in their first two indices.

• The representations of the group describe the various ways the group G (or its Lie algebra) can act upon a vector space. This vector space could, for instance, be the space of eigenvectors for a Hamiltonian exhibiting G as its symmetry group. We denote these representations with a capital D. Differentiating D to obtain a representation of the Lie algebra, often also denoted by D, yields representations related as follows:

  D[g(ξⱼ)] ≡ D(ξⱼ) = e^{iξⱼD(Xⱼ)} ${\displaystyle D[g(\xi _{j})]\equiv D(\xi _{j})=e^{i\xi _{j}D(X_{j})}}$

  (Note: the summation convention is not applied to the repeated index j here.) Representations are linear operators that map group elements to operators, preserving the group's composition rule:

  D(ξₐ) D(&#x3be><0xE2><0x82><0x99>) = D(ξₐ&#x3be><0xE2><0x82><0x99>) ${\displaystyle D(\xi _{a})D(\xi _{b})=D(\xi _{a}\xi _{b}).}$

  A representation that cannot be decomposed into a direct sum of smaller representations is termed irreducible. It is conventional to label irreducible representations with a superscripted integer, such as D⁽ⁿ⁾, or with multiple integers if necessary, like D⁽ⁿ, ᵐ, ...⁾.

  A subtle point arises in quantum theory: when two vectors differ only by a scalar multiple, they represent the same physical state. In this context, the relevant concept is a projective representation, which satisfies the composition law only up to a scalar factor. These are known as spinorial representations, particularly relevant in the study of quantum mechanical spin.

Momentum and Energy: The Generators of Motion and Time

The space translation operator, ℭ^(╮r), acts on a wavefunction by shifting its spatial coordinates by an infinitesimal displacement ╮r. The explicit form of ℭ^ can be deduced by considering the Taylor expansion of ψ(𝐁 + ╮r, t) around 𝐁. Keeping only the first-order term (and neglecting higher-order terms) allows us to replace the spatial derivatives with the momentum operator, 𝔏^. Similarly, for the time translation operator, we examine the Taylor expansion of ψ(𝐁, t + ╮t) around t, replacing the time derivative with the energy operator, E^.

Name	Translation operator ℭ^(╮r) ψ(𝐁, t) = ψ(𝐁 + ╮r, t)	Time translation/evolution operator ℱ^(╮t) ψ(𝐁, t) = ψ(𝐁, t + ╮t)
Infinitesimal operator	ℭ^(╮r) = I + (i/ℏ) ╮r ⋅ 𝔏^	ℱ^(╮t) = I - (i/ℏ) ╮t E^
Finite operator	limN→∞ (I + (i/ℏ) (╮r/N) ⋅ 𝔏^)N = exp( (i/ℏ) ╮r ⋅ 𝔏^) = ℭ^(╮r)	limN→∞ (I - (i/ℏ) (╮t/N) E^)N = exp( -(i/ℏ) ╮t E^) = ℱ^(╮t)
Generator	Momentum operator 𝔏^ = -iℏ∇	Energy operator E^ = iℏ ∂ / ∂t

The exponential functions arise naturally from these definitions, a consequence of Euler's work, and can be understood both physically and mathematically. A finite translation can be viewed as a composition of many infinitesimally small translations. To obtain the translation operator for a finite increment ╮r, we effectively divide ╮r by N (where N is a large positive integer) and apply the infinitesimal operator N times, then take the limit as N approaches infinity. The same logic applies to time translations.

Crucially, space and time translations commute. This implies that their corresponding operators and generators also commute.

Commutators

Operators	Generators
[ℭ^(𝐁¹), ℭ^(𝐁²)]ψ(𝐁, t) = 0	[𝔏^ᵢ, 𝔏^ⱼ]ψ(𝐁, t) = 0
[ℱ^(t¹), ℱ^(t²)]ψ(𝐁, t) = 0	[E^, 𝔏^ᵢ]ψ(𝐁, t) = 0

For a time-independent Hamiltonian, energy is conserved over time, and quantum states evolve into stationary states. These states are eigenstates of the Hamiltonian, with eigenvalues representing the energy levels:

ℱ^(t) = exp(-i╮tE / ℏ) ${\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac {i\Delta tE}{\hbar }}\right)}$

And all stationary states take the form:

ψ(𝐁, t + t₀) = ℱ^(t - t₀) ψ(𝐁, t₀) ${\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat {U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}$

Here, t₀ signifies the initial time, conventionally set to zero due to the continuity of the evolution. An alternative notation for the time evolution operator is ℱ(t - t₀) ≡ U(t, t₀).

Angular Momentum: The Generator of Rotations

Orbital Angular Momentum

The rotation operator, R^, transforms a wavefunction by rotating the spatial coordinates of a particle through a specified angle ╮θ about a given axis, defined by a unit vector â:

R^(╮θ, â) ψ(𝐁, t) = ψ(𝐁′, t) ${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t)}$

where 𝐁′ represents the rotated coordinates, and R^(╮θ, â) is the rotation matrix corresponding to this transformation. In the language of group theory, these rotation matrices are group elements, and the parameters ╮θ and â define the axis and angle of rotation, belonging to the three-dimensional special orthogonal group, SO(3).

For rotations about the standard Cartesian axes (êₓ, ê<0xE1><0xB5><0xA7>, ê<0xE2><0x82><0x9B>) through an angle ╮θ, the corresponding generators of rotation are J = (Jₓ, J<0xE1><0xB5><0xA7>, J<0xE2><0x82><0x9B>):

R^ₓ ≡ R^(╮θ, êₓ) = diag(1, cos ╮θ, -sin ╮θ; sin ╮θ, cos ╮θ) ${\displaystyle {\widehat {R}}_{x}\equiv {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}1&0&0&0\\0&\cos \Delta \theta &-\sin \Delta \theta &0\\0&\sin \Delta \theta &\cos \Delta \theta &0\\0&0&0&1\\\end{pmatrix}}\,,}$

Jₓ ≡ J₁ = i (∂R^(╮θ, êₓ) / ∂╮θ) |_╮θ=0 = i diag(0, 0, -1; 0, 1, 0) ${\displaystyle J_{x}\equiv J_{1}=i\left.{\frac {\partial {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})}{\partial \Delta \theta }}\right|_{\Delta \theta =0}=i{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{pmatrix}}\,,}$

(And similarly for J<0xE1><0xB5><0xA7> and J<0xE2><0x82><0x9B>.)

More generally, for rotations about an axis defined by â, the matrix elements of the rotation are given by:

[R^(θ, â)]ᵢⱼ = (δᵢⱼ - aᵢaⱼ) cos θ - εᵢⱼ<0xE2><0x82><0x99> a<0xE2><0x82><0x99> sin θ + aᵢaⱼ ${\displaystyle [{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos \theta -\varepsilon _{ijk}a_{k}\sin \theta +a_{i}a_{j}}$

where δᵢⱼ is the Kronecker delta and εᵢⱼ<0xE2><0x82><0x99> is the Levi-Civita symbol.

Deriving the infinitesimal rotation operator, which corresponds to small ╮θ, involves using the small angle approximations (sin(╮θ) ≈ ╮θ, cos(╮θ) ≈ 1), followed by a Taylor expansion of the wavefunction and substitution of the angular momentum operator components.

Rotation about ê<0xE2><0x82><0x9B>	Rotation about â
Action on wavefunction: R^(╮θ, ê<0xE2><0x82><0x9B>) ψ(x, y, z, t) = ψ(x - ╮θy, ╮θx + y, z, t)	Action on wavefunction: R^(╮θ, â) ψ(&#x1d401>ᵢ, t) = ψ(𝐁ᵢ′, t) = ψ(&#x1d401>ᵢ - εᵢⱼ<0xE2><0x82><0x99> a<0xE2><0x82><0x99> ╮θ rⱼ, t)
Infinitesimal operator: R^(╮θ, ê<0xE2><0x82><0x9B>) = I - (i/ℏ) ╮θ L^<0xE2><0x82><0x9B>	Infinitesimal operator: R^(╮θ, â) = I - (i╮θ/ℏ) â⋅L^
Generator: L^<0xE2><0x82><0x9B> = iℏ ∂ / ∂θ	Generator: L^ = iℏ â ∂ / ∂θ

The finite rotation operator is obtained by exponentiating the infinitesimal generator, analogous to the case of translations:

lim_N→∞ (I - (i/ℏ) (╮θ/N) â⋅L^)^N = exp(-(i/ℏ) ╮θ â⋅L^) = R^ ${\displaystyle \lim _{N\to \infty }\left(1-{\frac {i}{\hbar }}{\frac {\Delta \theta }{N}}{\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\right)^{N}=\exp \left(-{\frac {i}{\hbar }}\Delta \theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}\right)={\widehat {R}}}$

Rotations about the same axis commute, as expected: R(θ¹ + θ², âᵢ) = R(θ¹, âᵢ) R(θ², âᵢ), and [R(θ¹, âᵢ), R(θ², âᵢ)] = 0. However, rotations about different axes do not commute. The general commutation rules are encapsulated by:

[Lᵢ, Lⱼ] = iℏ εᵢⱼ<0xE2><0x82><0x99> L<0xE2><0x82><0x99> ${\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon _{ijk}L_{k}.}$

This is a hallmark of the angular momentum operator, mirroring the intuitive properties of physical rotations. You can test this yourself with an everyday object: rotating it first about one axis, then another, yields a different final orientation than performing the rotations in the reverse order.

Spin Angular Momentum: An Intrinsic Quantum Property

While orbital angular momentum has a classical analogue, spin is a purely quantum mechanical property, an intrinsic angular momentum possessed by particles. The spin vector operator is denoted S^ = (Sₓ^, S<0xE1><0xB5><0xA7>^, S<0xE2><0x82><0x9B>^). The eigenvalues of its components represent the possible outcomes of measuring the spin projection along a specific axis.

Rotations in space, acting on multicomponent wavefunctions (spinors), are represented by:

Spin rotation operator (finite): S^(θ, â) = exp(-(i/ℏ) θ â⋅S^) ${\displaystyle {\widehat {S}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {S} }}\right)}$

A key distinction from orbital angular momentum lies in the spin quantum number, s. While orbital angular momentum quantum numbers ℓ are integers, spin quantum numbers can be half-integers. For the simplest non-trivial case, s = 1/2, the spin operator is given by:

S^ = (ℏ/2) σ ${\displaystyle {\widehat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }}}$

where σ represents the Pauli matrices:

σ₁ = σₓ = diag(0, 1; 1, 0) ${\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad }$

σ₂ = σ<0xE1><0xB5><0xA7> = diag(0, -i; i, 0) ${\displaystyle \sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad }$

σ₃ = σ<0xE2><0x82><0x9B> = diag(1, 0; 0, -1) ${\displaystyle \sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

Evaluating the exponential for a given spin quantum number s yields a (2s + 1)-dimensional spin matrix. This matrix governs the transformation of a spinor—a column vector of 2s + 1 components—under rotations.

Total Angular Momentum

The total angular momentum operator is the sum of the orbital and spin angular momentum operators:

J^ = L^ + S^ ${\displaystyle {\widehat {\mathbf {J} }}={\widehat {\mathbf {L} }}+{\widehat {\mathbf {S} }}}$

This quantity is of paramount importance in the study of multi-particle systems, particularly in nuclear physics and the quantum chemistry of complex atoms and molecules. The transformation properties of the total angular momentum operator are analogous to those of orbital and spin angular momentum.

Conserved Quantities in the Quantum Harmonic Oscillator: Hidden Symmetries

The dynamical symmetry group of the n-dimensional quantum harmonic oscillator is the special unitary group SU(n). For instance, the number of infinitesimal generators for the Lie algebras of SU(2) and SU(3) are three and eight, respectively. This directly translates to precisely three and eight independent conserved quantities (beyond the Hamiltonian itself) within these systems.

The two-dimensional isotropic quantum harmonic oscillator exhibits the expected conserved quantities—the Hamiltonian and angular momentum—but also possesses additional, "hidden," conserved quantities related to energy level differences and another form of angular momentum. This hints at deeper symmetries at play, not immediately apparent from the basic structure.

The Lorentz Group in Relativistic Quantum Mechanics: Spacetime's Dance

A thorough exploration of the Lorentz group, encompassing boosts and rotations within spacetime, can be found in sources such as T. Ohlsson (2011) [4] and E. Abers (2004) [5].

Lorentz transformations are parameterized by rapidity φ for boosts along a specific direction (defined by a unit vector în) and by an angle θ for rotations about an axis (defined by a unit vector â). These parameters – three for rotations and three for boosts – collectively form the six parameters defining the Lorentz group, a six-dimensional entity.

Pure Rotations in Spacetime

The rotation matrices and generators previously discussed form the spatial component of a four-dimensional matrix representation of pure-rotation Lorentz transformations. The three generators J = (J₁, J₂, J₃) for these rotations are:

R^ₓ ≡ R^(╮θ, êₓ) = diag(1, 1, cos ╮θ, -sin ╮θ; sin ╮θ, cos ╮θ) ${\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \Delta \theta &-\sin \Delta \theta \\0&0&\sin \Delta \theta &\cos \Delta \theta \\\end{pmatrix}}\,,}$

Jₓ = J₁ = i diag(0, 0, 0, -1; 0, 1, 0) ${\displaystyle J_{x}=J_{1}=i{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{pmatrix}}\,,}$

(with analogous matrices for J<0xE1><0xB5><0xA7> and J<0xE2><0x82><0x9B>). These rotation matrices act on any four vector A = (A₀, A₁, A₂, A₃), rotating its spatial components while leaving the time-like component unchanged.

Pure Boosts in Spacetime

A boost, representing a change in velocity, is characterized by velocity c tanh φ. For boosts along the x, y, or z directions (defined by Cartesian basis vectors), the corresponding boost transformation matrices B^ₓ, B^<0xE1><0xB5><0xA7>, B^<0xE2><0x82><0x9B> and their generators K = (K₁, K₂, K₃) are:

B^ₓ ≡ B^(φ, êₓ) = diag(cosh φ, sinh φ, 1, 1; sinh φ, cosh φ, 0, 0) ${\displaystyle {\widehat {B}}_{x}\equiv {\widehat {B}}(\varphi ,{\hat {\mathbf {e} }}_{x})={\begin{pmatrix}\cosh \varphi &\sinh \varphi &0&0\\\sinh \varphi &\cosh \varphi &0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}\,,}$

Kₓ = K₁ = i diag(0, 1, 0, 0; 1, 0, 0, 0) ${\displaystyle K_{x}=K_{1}=i{\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{pmatrix}}\,,}$

(with analogous matrices for K<0xE1><0xB5><0xA7> and K<0xE2><0x82><0x9B>). These boost matrices transform a four vector A by mixing its time-like and space-like components.

Combining Boosts and Rotations: The Lorentz Transformation

Products of rotations yield another rotation, forming a subgroup. However, combining boosts with boosts, or boosts with rotations, does not generally result in a pure boost or a pure rotation. Any Lorentz transformation can, in fact, be decomposed into a sequence of a pure rotation and a pure boost. For a comprehensive treatment, consult B.R. Durney (2011) [6] and H.L. Berk et al. [7].

The generators of boosts (K) and rotations (J) have representations, denoted D(K) and D(J) respectively, which obey specific commutation relations:

Generators	Representations
Pure rotation	[Jₐ, J<0xE2><0x82><0x99>] = i εₐ<0xE2><0x82><0x99>c J<0xE1><0xB5><0xA8>
Pure boost	[Kₐ, K<0xE2><0x82><0x99>] = -i εₐ<0xE2><0x82><0x99>c J<0xE1><0xB5><0xA8>
Lorentz transformation	[Jₐ, K<0xE2><0x82><0x99>] = i εₐ<0xE2><0x82><0x99>c K<0xE1><0xB5><0xA8>

Notice how the boost generators mix with the rotation generators, unlike the pure rotations which simply yield another rotation. Exponentiating these generators yields the boost and rotation operators that combine to form the general Lorentz transformation. This transformation dictates how spacetime coordinates change between different inertial frames. Similarly, exponentiating the representations of these generators describes how a particle's spinor field transforms under these operations.

The transformation laws are summarized as:

Transformations	Representations
Pure boost: B^(φ, în) = exp(-(i/ℏ) φ în⋅K)	D[B^(φ, în)] = exp(-(i/ℏ) φ în⋅D(K))
Pure rotation: R^(θ, â) = exp(-(i/ℏ) θ â⋅J)	D[R^(θ, â)] = exp(-(i/ℏ) θ â⋅D(J))
Lorentz transformation: Λ(φ, în, θ, â) = exp(-i/ℏ (φ în⋅K + θ â⋅J))	D[Λ(θ, â, φ, în)] = exp(-i/ℏ (φ în⋅D(K) + θ â⋅D(J)))

The generators K and J can be combined into a single antisymmetric four-dimensional matrix M, where M⁰ᵃ = -Mᵃ⁰ = Kₐ and Mᵃᵇ = εᵃᵇᶜ J<0xE1><0xB5><0xA8>. Similarly, the parameters φ and θ are consolidated into an antisymmetric four-dimensional matrix ω, with ω⁰ᵃ = -ωᵃ⁰ = φnₐ and ωᵃᵇ = θ εᵃᵇᶜ a<0xE1><0xB5><0xA8>. The general Lorentz transformation then becomes:

Λ(φ, în, θ, â) = exp(-i/2 ω_αβ M^αβ) = exp(-i/2 (φ în⋅K + θ â⋅J)) ${\displaystyle \Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{2}}\omega _{\alpha \beta }M^{\alpha \beta }\right)=\exp \left[-{\frac {i}{2}}\left(\varphi {\hat {\mathbf {n} }}\cdot \mathbf {K} +\theta {\hat {\mathbf {a} }}\cdot \mathbf {J} \right)\right]}$

This matrix Λ acts on any four-vector A = (A₀, A₁, A₂, A₃), transforming it between different reference frames.

Transformations of Spinor Wavefunctions in Relativistic Quantum Mechanics

In the realm of relativistic quantum mechanics, wavefunctions transcend their simple scalar nature. They become multi-component spinor fields, with the number of components dictated by the particle's spin. Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, single-particle quantum states ψ_σ transform according to a specific representation D of the Lorentz group: [8] [9]

ψ_σ(r, t) → D(Λ) ψ_σ(Λ⁻¹(r, t)) ${\displaystyle \psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))}$

Here, D(Λ) is a finite-dimensional representation, essentially a (2s + 1)×(2s + 1) square matrix, and ψ is treated as a column vector containing components for each of the (2s + 1) possible values of the spin index σ.

Real Irreducible Representations and Spin

The irreducible representations (irreps) D(K) and D(J) of the generators K and J are fundamental building blocks for understanding spin representations of the Lorentz group. By defining new operators:

A = (J + iK) / 2, B = (J - iK) / 2 ${\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,}$

we observe that A and B are complex conjugates of each other and satisfy commutation relations analogous to those of angular momentum:

[Aᵢ, Aⱼ] = εᵢⱼ<0xE2><0x82><0x99> A<0xE2><0x82><0x99>, [Bᵢ, Bⱼ] = εᵢⱼ<0xE2><0x82><0x99> B<0xE2><0x82><0x99>, [Aᵢ, Bⱼ] = 0 ${\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}$

These operators A and B, along with their components, form algebras similar to angular momentum. The irreducible representations are then labeled by integers or half-integers, a and b, with corresponding magnetic quantum numbers m and n. The matrices representing these operators take a specific form involving the angular momentum matrices J⁽ᵐ⁾ and J⁽ⁿ⁾.

Applying this framework to particles with spin s:

• Left-handed (2s + 1)-component spinors transform under the real irreps D(s, 0).
• Right-handed (2s + 1)-component spinors transform under the real irreps D(0, s).
• The direct sum D(m, n) ⊕ D(n, m) (where m + n = s) describes the transformation of 2(2s + 1)-component spinors. These are also real irreps, but they decompose into complex conjugate representations.

In these contexts, D refers to any of the representations D(J), D(K), or the full Lorentz transformation D(Λ).

Relativistic Wave Equations: The Dirac and Weyl Spinors

The Dirac equation and Weyl equation are cornerstones of relativistic quantum mechanics. The Weyl spinors, solutions to the Weyl equation, transform under the simplest irreducible spin representations of the Lorentz group, D(1/2, 0) for left-handed and D(0, 1/2) for right-handed components. Dirac spinors, satisfying the Dirac equation, transform under the direct sum of these two representations: D(1/2, 0) ⊕ D(0, 1/2).

The Poincaré Group: The Full Symmetry of Spacetime

The Poincaré group encompasses all the symmetries discussed so far: space translations, time translations, rotations, and boosts. It is a ten-dimensional group, with generators for each of these transformations.

In special relativity, spacetime is described by a four-position vector X = (ct, -r), and similarly, energy and momentum combine into a four-momentum vector P = (E/c, -p). The generators of spacetime translations are encapsulated in the four-momentum operator:

P^ = (E^/c, -p^) = iℏ (1/c ∂/∂t, ∇) ${\displaystyle {\widehat {\mathbf {P} }}=\left({\frac {\widehat {E}}{c}},-{\widehat {\mathbf {p} }}\right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,,}$

The transformation under spacetime translations is given by:

X^(╮X) = exp(-i/ℏ ╮X⋅P^) = exp(-i/ℏ (╮t E^ + ╮r⋅p^)) {\displaystyle {\widehat {X}}(\Delta \mathbf {X} )=\exp \left(-{\frac {i}{\hbar }}\Delta \mathbf {X} \cdot {\widehat {\mathbf {P} }}\right)=\exp \left[-{\frac {i}{\hbar }}\left(\Delta t{\widehat {E}}+\Delta \mathbf {r} \cdot {\widehat {\mathbf {p} }}\right)\right]\,.

The Poincaré algebra defines the fundamental commutation relations between the generators of spacetime translations (P) and Lorentz transformations (M): [10] [11]

• [P_μ, P_ν] = 0 ${\displaystyle [P_{\mu },P_{\nu }]=0\,}$
• (1/i) [M_μν, P_ρ] = η_μρ P_ν - η_νρ P_μ ${\displaystyle {\frac {1}{i}}[M_{\mu \nu },P_{\rho }]=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho }P_{\mu }\,}$
• (1/i) [M_μν, M_ρσ] = η_μρ M_νσ - η_μσ M_νQ - η_νQ M_μσ + η_νσ M_μQ ${\displaystyle {\frac {1}{i}}[M_{\mu \nu },M_{\rho \sigma }]=\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho }\,}$

These equations, where η is the Minkowski metric, encapsulate the fundamental properties of spacetime as currently understood. They have a classical analogue where commutators are replaced by Poisson brackets.

The Pauli–Lubanski pseudovector, W_μ = (1/2) ε_μνᅹσ J^νᅹ P_σ, is a Casimir operator that characterizes the intrinsic spin of a particle, independent of its momentum. It commutes with the generators of spacetime translations and exhibits specific commutation relations with the Lorentz generators.

Symmetries in Quantum Field Theory and Particle Physics: Unveiling the Fundamental Forces

Unitary Groups: The Pillars of Quantum Transformations

Group theory provides an abstract yet powerful framework for analyzing symmetries. In quantum theory, unitary groups are paramount because unitary operators preserve inner products, and thus probabilities. Let 𝐁^ be a unitary operator, meaning its inverse is its Hermitian adjoint, 𝐁^⁻¹ = 𝐁^²†. If this operator commutes with the Hamiltonian, [𝐁^, H^] = 0, then the observable corresponding to 𝐁^ is conserved, and the Hamiltonian is invariant under the transformation represented by 𝐁^. Physicists seek unitary transformations to represent the symmetries of nature.

Important subgroups of U(N) are the unimodular unitary groups, known as special unitary groups, SU(N), which have a determinant of 1.

U(1): The Simplest Symmetry

The U(1) group, comprising complex numbers of modulus 1, is the simplest unitary group. Its elements are of the form U = e^-iθ, where θ is the group parameter. This is an Abelian group, as all one-dimensional matrices commute. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If a quantum number 'a' is associated with this symmetry (like baryon or lepton number in electromagnetic interactions), the transformation becomes U = e^-iaθ.

U(2) and SU(2): The Algebra of Spin

A general element of U(2) can be parameterized by complex numbers a and b:

U = [[a, b], [-b*, a*]] ${\displaystyle U={\begin{pmatrix}a&b\\-b^{\star }&a^{\star }\\\end{pmatrix}}}$

For SU(2), the determinant is constrained to 1: |a|² + |b|² = 1. The Pauli matrices serve as the generators for SU(2), satisfying commutation relations similar to angular momentum, albeit with a factor of 2: [σₐ, &#x3c3><0xE2><0x82><0x99>] = 2iℏ εₐ<0xE2><0x82><0x99>c σ<0xE1><0xB5><0xA8>. A group element of SU(2) is expressed as U(θ, j) = e^iθσⱼ/2, where θ is the angle of rotation and j specifies the axis. The two-dimensional isotropic quantum harmonic oscillator exhibits SU(2) symmetry.

U(3) and SU(3): The Foundation of Strong Interactions

The eight Gell-Mann matrices, denoted λₙ, are fundamental to quantum chromodynamics and originally arose in the theory of flavor SU(3). They are the generators of SU(3), and its group elements can be written as:

U(θ, j) = exp(-i/2 Σ_n=1⁸ θₙ λₙ) ${\displaystyle U(\theta ,{\hat {\mathbf {e} }}_{j})=\exp \left(-{\frac {i}{2}}\sum _{n=1}^{8}\theta _{n}\lambda _{n}\right)}$

where θₙ are eight independent parameters. The structure constants fₐ<0xE2><0x82><0x99>c, derived from the commutation relation [λₐ, &#x3bb><0xE2><0x82><0x99>] = 2i fₐ<0xE2><0x82><0x99>c λ<0xE1><0xB5><0xA8>, are totally antisymmetric. In the color charge basis (|r⟩, |g⟩, |b⟩), the Gell-Mann matrices act on these states, with λ₃ and λ₈ diagonal, while others mix the color states. The eight gluons, mediating the strong force, are eigenstates of the adjoint representation of SU(3).

Matter and Antimatter: A Fundamental Duality

Relativistic wave equations predict a profound symmetry: every particle has an antiparticle. This is mathematically encoded in the spinor fields. Charge conjugation, denoted C, swaps particles and antiparticles. Physical laws invariant under this transformation exhibit C symmetry.

Discrete Spacetime Symmetries: Parity and Time Reversal

• Parity (P): This operation mirrors spatial coordinates, effectively reflecting space into its mirror image. Physical laws invariant under parity exhibit P symmetry.
• Time Reversal (T): This flips the direction of time, making time run from future to past. A curious aspect of time is its unidirectional nature: particles moving forward in time are mathematically equivalent to antiparticles moving backward. Physical laws invariant under time reversal exhibit T symmetry.

The interplay of these symmetries is captured by the CPT theorem.

Gauge Theory: The Language of Forces

In quantum electrodynamics, the local symmetry group is the Abelian U(1), associated with the electromagnetic interaction mediated by photons. The electromagnetic tensor possesses gauge symmetry through the electromagnetic four-potential. In quantum chromodynamics, the non-Abelian SU(3) group governs the strong interaction, mediated by eight gluons, each carrying color charge. These gluons themselves carry color charge, leading to complex self-interactions.

The strong interaction is characterized by color charge operators Fⱼ = (1/2)λⱼ. As color charge is conserved, these operators commute with the Hamiltonian: [Fⱼ, H^] = 0.

Isospin: A Nuclear Symmetry

Isospin is a conserved quantity in strong interactions, acting as a symmetry relating protons and neutrons.

The Weak and Electromagnetic Interactions: A Unified Front

The weak and electromagnetic forces are unified within the electroweak theory, described by the SU(2) × U(1) gauge group. This unification elegantly explains phenomena like parity violation in weak interactions.

Duality Transformation: Monopoles and Symmetry

The concept of duality transformation suggests a theoretical interchangeability between electric and magnetic charges, hinting at a deeper symmetry, though magnetic monopoles remain hypothetical.

Electroweak Symmetry: Breaking the Unification

Electroweak symmetry is spontaneously broken at low energies, giving mass to the W and Z bosons while the photon remains massless. This electroweak symmetry breaking is a crucial aspect of the Standard Model.

Supersymmetry: A Proposed Extension

Supersymmetry postulates a fundamental relationship between fermions and bosons, suggesting that every known particle has a supersymmetric partner with different spin. This theory, while theoretically appealing for its potential to solve various puzzles in particle physics (like the hierarchy problem and providing dark matter candidates such as the gravitino), has yet to be experimentally confirmed. If it exists, it is believed to be a broken symmetry at observable energies.

Exchange Symmetry: The Indistinguishable Nature of Particles

The principle of exchange symmetry stems from a fundamental postulate of quantum statistics: observable quantities remain unchanged when two identical particles are exchanged. This implies that the wave function ψ must either be symmetric or antisymmetric under such an exchange. The spin-statistics theorem connects this property to the particle's spin: integer-spin particles (bosons) have symmetric wave functions, obeying Bose–Einstein statistics, while half-integer spin particles (fermions) have antisymmetric wave functions, obeying Fermi–Dirac statistics.

The antisymmetric nature of fermionic wave functions leads directly to the Pauli exclusion principle, forbidding two identical fermions from occupying the same quantum state. This principle underpins the stability and rigidity of matter, giving rise to degeneracy pressure. The exchange of two particles is mathematically equivalent to a 360-degree rotation of one particle relative to the other. For bosons, this rotation leaves the wave function unchanged. For fermions, a 360-degree rotation results in a sign flip of the wave function.

See Also

• Casimir operator
• Pauli–Lubanski pseudovector
• Projective representation
• Renormalization group
• Representation of a Lie group
• Representation theory of the Lorentz group
• Representation theory of the Poincaré group
• Spin-statistics theorem
• Symmetric group
• Symmetries in general relativity

Footnotes

• ^ Sometimes the tuple abbreviations:

  (A)_m'n',mn = [(Aₓ)_m'n',mn, (A<0xE1><0xB5><0xA7>)_m'n',mn, (A<0xE2><0x82><0x9B>)_m'n',mn] ${\displaystyle \left(\mathbf {A} \right)_{m'n',mn}\equiv \left[\left(A_{x}\right)_{m'n',mn},\left(A_{y}\right)_{m'n',mn},\left(A_{z}\right)_{m'n',mn}\right]}$

  (B)_m'n',mn = [(Bₓ)_m'n',mn, (B<0xE1><0xB5><0xA7>)_m'n',mn, (B<0xE2><0x82><0x9B>)_m'n',mn] ${\displaystyle \left(\mathbf {B} \right)_{m'n',mn}\equiv \left[\left(B_{x}\right)_{m'n',mn},\left(B_{y}\right)_{m'n',mn},\left(B_{z}\right)_{m'n',mn}\right]}$

  (J⁽ᵐ⁾)_m'm = [(Jₓ⁽ᵐ⁾)_m'm, (J<0xE1><0xB5><0xA7>⁽ᵐ⁾)_m'm, (J<0xE2><0x82><0x9B>⁽ᵐ⁾)_m'm] ${\displaystyle \left(\mathbf {J} ^{(m)}\right)_{m'm}\equiv \left[\left(J_{x}^{(m)}\right)_{m'm},\left(J_{y}^{(m)}\right)_{m'm},\left(J_{z}^{(m)}\right)_{m'm}\right]}$

  are employed.