Discrete & Cont.. Symmetries
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Discrete and Continuous Symmetries

Conservation Laws and Corresponding Quantum Numbers

Symmetry in Physics

A symmetry of a physical system is a transformation that leaves the physical laws invariant. Symmetries play a fundamental role in modern physics, especially in quantum mechanics and quantum field theory.

The term symmetry means that a system, state or quantity remains unchanged (or invariant) as a result of a particular transformation, such as a change in space coordinates, or change in time, etc. Symmetries also predict degeneracy between different physical states, which can be connected through the corresponding transformations. Every conservation law is related to a particular invariance (or symmetry) principle. For example, conservation of total energy is due to the invariance of a system under shift in time, called time translation. Conservation of linear or angular momentum is due to invariance under displacement in space (space translation) or rotation in space respectively. These transformations can be abstract also, i.e. they may have no relation to actual space, time, etc. Examples of these are like conservation of charge, lepton number, baryon number, isospin etc. Besides these continuous symmetries, there also exist discrete symmetries like parity and C-parity. In the following, we will describe briefly some aspects of special symmetries in particle physics, which are SU(2) symmetry, SU(3) symmetry, and higher symmetries.

In physics, SU(2) symmetry refers to a specific mathematical framework used to describe how certain physical systems remain unchanged under specific "rotations." The name stands for Special Unitary group of degree 2. While that sounds like a mouthful, it is the backbone of how we understand quantum properties like spin and the weak nuclear force.

Every continuous symmetry of nature leads to a conservation law — this is the essence of Noether’s Theorem.

Classification of Symmetries

Symmetries are broadly classified into:

  • Continuous symmetries
  • Discrete symmetries

Continuous Symmetries

A continuous symmetry involves transformations that can be performed by an arbitrarily small amount.

Examples

  • Space translation
  • Time translation
  • Rotation in space
  • Gauge transformations

Noether’s Theorem

Noether’s theorem states that:

Every continuous symmetry of the action corresponds to a conserved quantity.

Continuous Symmetries and Conservation Laws

Symmetry Transformation Conserved Quantity Quantum Number
Time translation \(t \rightarrow t + \delta t\) Energy Energy eigenvalue
Space translation \(\vec{r} \rightarrow \vec{r} + \delta \vec{r}\) Linear momentum \(\vec{p}\)
Rotational invariance Rotation in space Angular momentum \(J, m\)
Gauge symmetry \(U(1)\) Phase change Electric charge \(Q\)

Internal Continuous Symmetries

Internal symmetries act in internal spaces, not on spacetime coordinates.

Symmetry Group Conserved Quantity Quantum Number
\(U(1)\) Electric charge \(Q\)
\(SU(2)\) Isospin \(I, I_3\)
\(SU(3)\) Color charge Red, Green, Blue

Discrete Symmetries

Discrete symmetries involve transformations that cannot be made continuously. They involve finite changes.

Fundamental Discrete Symmetries

  • Parity (P)
  • Charge Conjugation (C)
  • Time Reversal (T)

Parity Symmetry (P)

Parity corresponds to spatial inversion:According to conservation of parity, if a process is subjected to a reflection in mirror ( also called inversion), so that the sign of all its Cartesian coordinates are changed the resulting process will be indistinguishable from the original one. In other words, the conservation of parity requires that if an event is possible, its reflection in a mirror represents an equally probable event. It was found that in strong and electromagnetic interactions parity is always conserved. Until 1956, it was thought that parity must always be conserved. However, Lee and Yang in 1956 suggested that parity is not conserved in weak interactions. Examples of such cases are b-decay, decay of muons, kaons, etc. Parity causes reflection of the wave function, i.e. changes x co-ordinate to –x, y to –y and z to –z, If operator P is applied second time, the original wave function is obtained, or

\[ \vec{r} \rightarrow -\vec{r} \] \[ P^2 \psi(\vec r) =P\psi(-\vec r) =\psi(\vec r) \]

Therefore, \( P^2 = 1 \) or \( P = 1 \) If P = +1, then wave function is said to have even parity. And if P = –1, then wave function is said to have odd parity. Thus, depending upon the behaviour of particle wave functions, \[ \psi (–x, –y, –z) = \psi (x, y, z) \] they carry either positive or negative parity quantum number. Mesons (bosons) can carry positive or negative parity. Baryons are assigned positive parity, whereas their antiparticles (antibaryons) carry negative parity. Similarly, antileptons carry opposite parity to leptons. Photon and gluons have negative parity. Since weak interactions can violate parity, no fixed parity can be assigned to W- and Z-bosons. Parity is a multiplicative quantum number, i.e. parity of a composite state is given by product of parities of its constitutents.

Mesons (bosons) can carry positive or negative parity. Baryons are assigned positive parity, whereas their antiparticles (antibaryons) carry negative parity. Similarly, antileptons carry opposite parity to leptons. Photon and gluons have negative parity. Since weak interactions can violate parity, no fixed parity can be assigned to W- and Z-bosons. Parity is a multiplicative quantum number, i.e. parity of a composite state is given by product of parities of its constitutents

  • Strong and electromagnetic interactions conserve parity
  • Weak interaction violates parity

Associated Quantum Number: Parity \(P = \pm 1\)

Charge Conjugation (C)

Charge conjugation changes particles into their antiparticles.

  • Charges reverse sign
  • Photon has \(C = -1\)
  • Neutral pion has \(C = +1\)

The charge conjugation operator is normally denoted by C. Like parity, charge conjugation symmetry is applicable to strong and electromagnetic interactions only. For weak interactions it fails. A combination of C-parity with isospin is called G-parity which is conserved only in the strong interactions. Like P-parity, C-parity is also multiplicative in nature. Combining the two parities, CP can also be defined for particles and their processes.

Associated Quantum Number: C-parity

Time Reversal Symmetry (T)

Time reversal corresponds to reversing the direction of time:

\[ t \rightarrow -t \]
  • Momentum and angular momentum change sign
  • Most interactions conserve T

If a motion picture film was made of an event of say a particle formation or decay, the event would appear to be equally possible regardless of whether the film is run forward or backward

Associated Quantum Number: T-parity

CPT Theorem

A fundamental theorem of quantum field theory states:

According to postulates of advanced quantum mechanics (or quantum field theory), all interactions should be invariant under the combination of \( C, P \& T \) i.e. CPT operation,

\begin{equation} \psi_{particle}(\vec x,t)\rightarrow \psi_{antparticle}(- \vec x,-t) \end{equation}

This is also known as CPT theorem.

All physical laws are invariant under the combined operation of CPT.
  • CPT symmetry is always conserved
  • Violation of CPT would indicate new physics

Conservation Laws from Discrete Symmetries

Symmetry Conserved Quantity Status
Parity (P) Parity quantum number Violated in weak interaction
Charge Conjugation (C) C-parity Violated in weak interaction
CP CP quantum number Small violation observed
CPT Fundamental invariance Always conserved

Summary Table

Type of Symmetry Example Conserved Quantity Quantum Number
Continuous Time translation Energy \(E\)
Continuous Rotation Angular momentum \(J\)
Discrete Parity Parity \(\pm 1\)
Discrete Charge conjugation C-parity \(\pm 1\)

Symmetri Properties and Conservation Laws – MCQ Quiz

According to Noether’s theorem, conservation of energy is associated with:




Conservation of linear momentum arises due to:




Conservation of angular momentum is related to:




Parity transformation involves:




Which interaction violates parity symmetry?




Charge conjugation (C) transformation changes:




The combined CPT symmetry is:




Conservation of electric charge is associated with:




Lepton number conservation means:




Baryon number conservation implies:




Time reversal symmetry changes:




CP violation has been observed in:




If the laws of physics are invariant under spatial translation, then:




Symmetry breaking in weak interaction leads to:




Noether’s theorem connects: