T1: SR and Classical Field Theory Flashcards
Define pseudo-orthogonality
Λ^T η Λ = η
i.e. Preserving the metric under lorentz transform
Define the Lorentz group
The set of matrices which obey the pseudo-orthogonality condition.
These are O(1,3) extending orthogonal group to 1-temporal, 3-spacial component vectors.
Define continuously connected to the identity for a matrix.
A matrix can be expressed as:
Λ(ω) = exp(ωM)
at ω = 0, Λ(0) = I
What expression governs the Lorentz generators. How do we find it?
ηM + M^t η = 0
By expanding the continuously connected expression and putting in the pseudo-orthogonality contraint.
What two Lorentz transforms do we get from detΛ =-1
Parity → flip the spacial coords
Time-reversal → flip the time coord
Define the Poincare group
The set of Lorentz, and space-time translations.
x → x’ = Λx - a
Define the generators of the Poincare group (coordinates)
M^σρ - Lorentz
P_µ - space-time
Define scalar field
A field that is invariant under Poincare transformation
φ’(x’) = φ(x)
Define locality for a Lagrangian
The Lagrangian cannot contain terms which couple fields at different spacial points.
What does locality impart on the Lagrangian in field theory
We can pass the spacial integral out the front of the Lagrangian since locality says we can only act at a single point.
State the boundary conditions for classical field theory
δϕ_a = 0 at initial and final t and as |x| tends to ∞
State the EL equation for fields
∂L/∂ϕ + ∂_μ [∂L/∂(∂_μϕ)]
Define ∂μ with upper and lower indices
∂_μ = (∂/∂t, ∇)
∂^μ = (-∂/∂t, ∇)
State Noeter’s theorem
To each continuous symmetry of action there is a corresponding conservation law/time-independent quantity.
Define the continuous transformation of scalar field ϕ and define all terms
ϕ’ = ϕ + αΔϕ + ho
Where Δ denotes the generator of the continuous transformation
