Discrete and Continuous Symmetries

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.

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:

\[
\vec{r} \rightarrow -\vec{r}
\]

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 \(C = +1\)

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

Associated Quantum Number: T-parity

CPT Theorem

A fundamental theorem of quantum field theory states:

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	\(\pm 1\)