ФИАН / Nefedyev_Ktp

∂µL = ∂Aσ

∂µAσ + ∂(∂ν Aσ)

∂µ∂ν Aσ = ∂ν

(∂µAσ)∂(∂ν Aσ) ,

∂L

∂L

∂L

∂ν gµν L

∂ν T µν = 0,

T µν = −L gµν

+ (∂µAσ)

∂L

∂(∂ν Aσ)

P µ = Z d3xT µ0(x, t).

P µ

t1

t2

t2

dt

∂T µ0

∂T µi

= hP µ(t2) − P µ(t1)i

t2

dt IΣ→∞ dπiT µi.

0 = ZV d4x∂ν T µν = Z

d3x Zt1

+

+ Zt1

∂t

∂xi

P µ =

.

∂σψµνσ

ψµνσ = −ψµσν

Mµν

∂λ(xµT νλ − xν T µλ) = T µν − T νµ = 0.

t1

t2

Mµν (t2)−Mµν (t1) = 0

P µ

d3x 4Fµν F µν − F 0σF 0σ = Z

d3x [−L + F0iF0i] =

P

0 ≡ W = Z

1

=

Z

d3x 2(H2 − E2) + E2

= 2 Z

d3x(E2

+ H2) = Z

w(x)d3x,

1

1

W

w(x)

w(x) =

1

(E2

+ H2).

2

Pi

Pi = Z

d3xT i0(x, t) = − Z

d3xF ij F 0j = − Z

d3xFij F0j = Z

d3xεijkEjHk = Z

d3x[EH]i,

P = Z

d3xS,

X

f(x, t) =

eikxf(k, t).

k

eikx(x+Lx) = eikxx

= kx =

2π

nx,

Lx

nx

ky

kz

k

n

= (2π)3 k,

n = nx ny

nz =

2π kx 2π ky 2π kz

Lx

Ly

Lz

V

k

→ Z

V d3k

(2π)3 .

X

k

k

→ Z

V d3k

(2π~)3 .

X

3

3

eµ(k, σ)eν (k, σ) = eµ(k, 0)eν(k, 0) − eµ(k, σ)eν (k, σ) = gµν .

σ=0

σ=1

X

X

eµ(k, σ) = gµσ.

3

1

aµ(k) =

√

eµ(k, σ)C(k, σ),

σ=0

√

X

1/ 2ω

3

kµ

eµ(k, σ)C(k, σ) = 0.

σ=0

X

z

k

A(x, t) = X X √1 e(k, σ) C(k, σ)e−ikx + C (k, σ)eikx

.

k

σ=1,2

2ω

˙

1

Ei = −F0i

= −Ai,

Hi =

−

2

εijkFjk = εijk∂j Ak = [ A]i.

H = W =

1

Z d3x(E2 + H2) =

k

1

2

σ=1,2 2ω[C (k, σ)C(k, σ) + C(k, σ)C (k, σ)],

X X

CC C C

CC C C

ωω′(e(k, σ)e(k′, σ′)) + [ke(k, σ)][k′e(k′, σ′)]

k′ = −k

2ω2δσσ′

k′ = k

P = Z d3x[EH] = k

1

σ=1,2 2k[C (k, σ)C(k, σ) + C(k, σ)C (k, σ)],

X X

k

H(k) = [nE(k)] = [E(k)H(k)] = nE2(k) − E(k)(nE(k)) = nE2(k), n =

k

,

|k|

µ = −1, 0, +1

χµ(α)

3

X

S3χµ(α) = µχµ(α), S2χµ(α) = 2χµ(α),

χ

(α)χµ′ (α) = δµµ′ .

µ

α=1

χ−1

= √

−i

,

χ0

=

0 ,

χ+1 =

√

i

.

1

1

0

1

1

−

0

1

0

2

2

µ = 0

χµ

f0 = fz, f±1

1

=

√

(fx

ify),

2

fµ = (−1)µf−µ,

1

f0 = fz, f±1 =

√

(fx

± ify).

2

X

{χ}

f =

fµχµ,

µ=−1,0,+1

X

fg =

fµgµ.

µ=−1,0,+1

k

l

Ylm

µ

YjlM(µ) (n) = ClmjM1αYlm(n)χµ(α),

m + α = M,

S = 0

S = 2

S = 0

l = 1

j = 1

l = S = 2

Φα1α2 (k)

Φα1α2 (k)nα2

Φα1α2 (k)nα2

= Y

(−1)

(n)

= nα1 Y1M (n).

1M

α1