space time symmetry and conservation laws, 40 mcqs with explained answers foe competitive exam
Here are 40 multiple-choice questions (MCQs) on Space-Time Symmetry and Conservation Laws with explained answers for competitive exam preparation:
## Basic Concepts
1. Noether's theorem states that every symmetry of the Lagrangian leads to a:
- A) Force law
- B) Conservation law
- C) Equation of motion
- D) Potential energy
**Answer: B) Conservation law**
**Explanation:** Noether's theorem establishes a direct connection between continuous symmetries of the action and conserved quantities in physical systems.
2. Translational symmetry in space leads to conservation of:
- A) Energy
- B) Momentum
- C) Angular momentum
- D) Charge
**Answer: B) Momentum**
**Explanation:** Uniformity of space (translational invariance) implies linear momentum is conserved.
3. Time translation symmetry (time invariance) conserves:
- A) Momentum
- B) Angular momentum
- C) Energy
- D) Charge
**Answer: C) Energy**
**Explanation:** Homogeneity of time leads to conservation of total energy.
4. Rotational symmetry in space conserves:
- A) Linear momentum
- B) Energy
- C) Angular momentum
- D) Charge
**Answer: C) Angular momentum**
**Explanation:** Isotropy of space (rotational invariance) implies angular momentum conservation.
## Advanced Symmetries
5. Which internal symmetry conserves electric charge?
- A) Time reversal
- B) U(1) gauge symmetry
- C) Space inversion
- D) Lorentz boost
**Answer: B) U(1) gauge symmetry**
**Explanation:** Local U(1) phase invariance leads to charge conservation via Noether's theorem.
6. CPT theorem states that all local quantum field theories respect:
- A) CP symmetry only
- B) CT symmetry only
- C) CPT symmetry
- D) None of these
**Answer: C) CPT symmetry**
**Explanation:** CPT (charge conjugation, parity, time reversal) is a fundamental symmetry of all Lorentz-invariant local QFTs.
7. Parity violation was first discovered in:
- A) Electromagnetic interactions
- B) Strong interactions
- C) Weak interactions
- D) Gravitational interactions
**Answer: C) Weak interactions**
**Explanation:** Wu's cobalt-60 experiment (1957) demonstrated parity violation in beta decay (weak interaction).
8. Which conservation law arises from Lorentz invariance?
- A) Energy-momentum 4-vector conservation
- B) Angular momentum only
- C) Charge conservation
- D) Baryon number
**Answer: A) Energy-momentum 4-vector conservation**
**Explanation:** Poincaré group (Lorentz + translations) symmetries yield the energy-momentum tensor conservation.
## Specific Conservation Laws
9. Baryon number conservation arises from:
- A) U(1)_B gauge symmetry
- B) Approximate SU(3) flavor symmetry
- C) Global U(1)_B symmetry
- D) CPT theorem
**Answer: C) Global U(1)_B symmetry**
**Explanation:** Baryon number is conserved due to approximate global U(1)_B symmetry of the Standard Model.
10. Lepton number conservation is related to:
- A) SU(2)_L gauge symmetry
- B) Global U(1)_L symmetry
- C) Strong isospin symmetry
- D) Color symmetry
**Answer: B) Global U(1)_L symmetry**
**Explanation:** Separate lepton numbers for each generation are approximately conserved via global U(1)_L.
## Space-Time Symmetries
11. Which transformation leaves the Minkowski metric invariant?
- A) Galilean transformations
- B) Lorentz transformations
- C) Conformal transformations
- D) Diffeomorphisms
**Answer: B) Lorentz transformations**
**Explanation:** Lorentz group preserves the spacetime interval ds² = -dt² + dx² + dy² + dz².
12. Time reversal symmetry (T) invariance conserves:
- A) Energy
- B) Momentum
- C) Probability current
- D) Charge
**Answer: C) Probability current**
**Explanation:** T symmetry reverses time derivatives, conserving probability current direction.
13. Which of these is NOT a spacetime symmetry?
- A) Translations
- B) Rotations
- C) Scale transformations
- D) Lorentz boosts
**Answer: C) Scale transformations**
**Explanation:** Scale invariance is a conformal symmetry, not part of the Poincaré group.
## Violations and Exceptions
14. CP violation was first observed in:
- A) K⁰ meson decays
- B) B meson decays
- C) Neutrino oscillations
- D) Muon decay
**Answer: A) K⁰ meson decays**
**Explanation:** Cronin-Fitch experiment (1964) discovered CP violation in neutral kaon system.
15. Which interaction approximately conserves strangeness?
- A) Electromagnetic
- B) Weak
- C) Strong
- D) Gravitational
**Answer: C) Strong**
**Explanation:** Strong interaction conserves flavor quantum numbers including strangeness.
## Gauge Symmetries
16. The Standard Model has how many gauge symmetries?
- A) SU(3) × SU(2) × U(1)
- B) SU(3) × U(1) × U(1)
- C) SU(2) × U(1)
- D) U(1) only
**Answer: A) SU(3) × SU(2) × U(1)**
**Explanation:** SU(3)_c × SU(2)_L × U(1)_Y describes strong, weak, and electromagnetic forces.
17. Spontaneous symmetry breaking leads to:
- A) Exact conservation laws
- B) Goldstone bosons
- C) Gauge bosons gaining mass
- D) Both B and C
**Answer: D) Both B and C**
**Explanation:** Higgs mechanism breaks electroweak symmetry, giving W/Z bosons mass and producing Goldstone modes.
## Angular Momentum and Spin
18. Total angular momentum conservation arises from:
- A) Translational invariance
- B) Rotational invariance
- C) Time invariance
- D) Gauge invariance
**Answer: B) Rotational invariance**
**Explanation:** Spatial rotational symmetry conserves total angular momentum J = L + S.
19. Spin-statistics theorem relates:
- A) Spin to charge
- B) Spin to statistics (bosons/fermions)
- C) Angular momentum to parity
- D) Orbital to spin angular momentum
**Answer: B) Spin to statistics (bosons/fermions)**
**Explanation:** Particles with integer spin are bosons; half-integer spin particles are fermions.
## Classical vs Quantum
20. In classical mechanics, which symmetry gives Hamilton's equations?
- A) Time translation
- B) Canonical transformations
- C) Phase space translations
- D) Galilean invariance
**Answer: B) Canonical transformations**
**Explanation:** Hamilton's principle and canonical transformations preserve the Lagrangian structure.
## Continuation (21-40)
21. Which theorem guarantees conservation of probability in QM?
- A) Ehrenfest theorem
- B) Noether's theorem
- C) Unitary evolution
- D) CPT theorem
**Answer: C) Unitary evolution**
**Explanation:** Time evolution operator U(t) is unitary: U†U = 1, preserving probability.
22. Isospin symmetry SU(2) conserves:
- A) Total charge
- B) Third component of isospin
- C) Baryon number
- D) All of these
**Answer: D) All of these**
**Explanation:** SU(2)_I acts on up/down quark doublet conserving I₃ and total charge.
23. Lorentz invariance implies:
- A) E² = p²c² + m²c⁴
- B) E = pc
- C) E = mc²
- D) p = mv
**Answer: A) E² = p²c² + m²c⁴**
**Explanation:** Invariant mass relation from spacetime interval invariance.
24. Which symmetry is broken by the Higgs vacuum expectation value?
- A) SU(3)_c
- B) SU(2)_L × U(1)_Y
- C) U(1)_EM
- D) Translational symmetry
**Answer: B) SU(2)_L × U(1)_Y**
**Explanation:** Electroweak symmetry breaking generates masses for weak bosons.
25. Angular momentum algebra satisfies:
- A) Heisenberg uncertainty
- B) SU(2) Lie algebra
- C) Abelian group
- D) Non-commutative geometry
**Answer: B) SU(2) Lie algebra**
**Explanation:** [J_i, J_j] = iℏ ε_ijk J_k defines angular momentum algebra.
26. Charge conservation in QED arises from:
- A) Global U(1)
- B) Local U(1) gauge symmetry
- C) Dirac equation
- D) Maxwell equations
**Answer: B) Local U(1) gauge symmetry**
**Explanation:** Gauge invariance requires ∂μjμ = 0 (continuity equation).
27. P violation means nature distinguishes:
- A) Left from right
- B) Up from down
- C) Clockwise from anticlockwise
- D) Matter from antimatter
**Answer: A) Left from right**
**Explanation:** Parity transformation inverts spatial coordinates x → -x.
28. The conserved current for time translation symmetry is:
- A) Momentum density
- B) Energy-momentum tensor T^{0μ}
- C) Angular momentum tensor
- D) Charge current
**Answer: B) Energy-momentum tensor T^{0μ}**
**Explanation:** Noether current for time translations gives energy density and momentum flux.
29. Scale invariance conserves:
- A) Energy
- B) Dilatation current
- C) Momentum
- D) Charge
**Answer: B) Dilatation current**
**Explanation:** Conformal symmetry includes dilatation (scale) transformations.
30. Which process violates baryon number?
- A) Proton decay (predicted by GUTs)
- B) Beta decay
- C) Electron capture
- D) Pair production
**Answer: A) Proton decay (predicted by GUTs)**
**Explanation:** Grand Unified Theories predict ΔB ≠ 0 processes at high energies.
31. Poincaré group generators include:
- A) P^μ (translations), M^{μν} (Lorentz)
- B) Only rotations
- C) Only boosts
- D) Scale generators
**Answer: A) P^μ (translations), M^{μν} (Lorentz)**
**Explanation:** Poincaré algebra: translations and Lorentz transformations.
32. Weak interaction violates:
- A) P and CP
- B) Only P
- C) Only CP
- D) C, P, T individually
**Answer: A) P and CP**
**Explanation:** Weak interaction maximally violates parity; CP violation is small.
33. The Noether charge Q satisfies:
- A) dQ/dt = 0 (conserved)
- B) [Q,H] = 0
- C) Both A and B
- D) Q|ψ⟩ = λ|ψ⟩
**Answer: C) Both A and B**
**Explanation:** Conserved charges commute with Hamiltonian and are time-independent.
34. Flavor SU(3) symmetry is broken by:
- A) quark masses
- B) electromagnetic interactions
- C) weak currents
- D) All of these
**Answer: D) All of these**
**Explanation:** QCD has approximate flavor symmetry broken by multiple effects.
35. Which conserved quantity corresponds to phase rotation ψ → e^{iα}ψ?
- A) Momentum
- B) Probability
- C) Particle number
- D) Energy
**Answer: C) Particle number**
**Explanation:** Global U(1) phase symmetry conserves particle number N.
36. In general relativity, conservation laws arise from:
- A) Killing vectors (isometries)
- B) Christoffel symbols
- C) Ricci tensor
- D) Weyl tensor
**Answer: A) Killing vectors (isometries)**
**Explanation:** Spacetime symmetries (Killing vectors) generate conserved quantities.
37. The axial current conservation is broken by:
- A) Anomalies (triangle diagram)
- B) Mass terms
- C) Both A and B
- D) Gauge interactions
**Answer: C) Both A and B**
**Explanation:** QCD axial anomaly and quark masses break U(1)_A symmetry.
38. Boost generators in Poincaré group generate:
- A) Rotations
- B) Translations
- C) Lorentz transformations mixing space and time
- D) Scale changes
**Answer: C) Lorentz transformations mixing space and time**
**Explanation:** K^{μ} generate boosts (velocity changes).
39. Which theorem relates symmetry breaking to mass generation?
- A) Goldstone theorem
- B) Noether's theorem
- C) Ward identity
- D) Coleman-Mandula theorem
**Answer: A) Goldstone theorem**
**Explanation:** Continuous symmetry breaking produces massless Goldstone bosons.
40. The stress-energy tensor T^{μν} is conserved due to:
- A) Translational invariance
- B) Diff invariance
- C) Both A and B
- D) Gauge invariance
**Answer: C) Both A and B**
**Explanation:** ∂_μT^{μν} = 0 from spacetime translation invariance and diffeomorphism invariance.
