156 Chapter 5 . Vectors: Spin 1 Fields

QED/FIELD THEORY OVERVIEW: PART 1

Wholeness Chart 5-4. From Field Equations to Propagators and Observables Heisenberg Picture, Free Fields

Spin 0 Spin ½ Spin 1

2 Classical L 1 AA 1  A   A  L 0  2 None. Macroscopic spinor fields not 0 2 2      Lagrangian K       0   observed.  0 density, free for photons

2nd quantization, Bosons: Quantum field L (or equivalently, H) same as classical, fields are complex, and K =1. Postulate #1 Spinors: Dirac eq from RQM with states → fields. Deduce L from Dirac eq; H from Legendre transf.

QFT Lagrangian 0 † 2 † L      L 1/ 2 i  m      As above for classical. density, free 0    0   

L ↑ into the Euler-Lagrange equation yields ↓

 2   0  2     (i m )  0    A 0 photon   0 Free field equations  2 † (i m  )  0  †  †       0  A A for chargeless (photon)

L0 L 0 Conjugate 00ɺ †0†  0 ɺ 12/ † 12 / 1 ɺ 0  ; 0    i  ;   0    A  momenta ɺ ɺ†

H0 0ɺ 0†† ɺ L 0 0 0   0   0 Hamiltonian 12/1/ 2 12 / 11 ɺ  1 H  ɺ  L H  A  L density ɺ ɺ †   †   2†  0 0 0 0

          Free field     A A  A (photon) solutions †  †   †      

m ipx 1 ikx   (c()p u () p e  x   ( a(k ) e  VE r r  r, p p  Discrete k 2Vk A  † ipx eigenstates † ikx  b(k ) e )  dr()p v r () p e ) 1 (  ()ka () k e ikx (Plane waves, m ipx  r r   (d()p v () p e r, k 2Vk † 1 ikx  VE r r constrained to  x    ( b(k ) e r, p p 2V  † ikx volume V) k k  r()ka r () k e )  c† ()p u () p eipx )  a† (k ) eikx ) r r

3 m d p ipx   3 (cr ()p u r () p e  dk ikx  2  ()x (() ak e r   Ep A   3 2 2   Continuous   k  d† ()p v () p e ipx ) dk  ikx r r (r ()ka r () k e eigenstates † ikx   3  b(k ) e ) 3 r 2 2    m d p ipx k (Plane waves,   3 (dr ()p v r () p e † dk ikx  2   † ikx no volume  ()x (() bk e r   Ep  r()ka r () k e )  3 constraint) 2 2   k † ipx r 0 ,,, 1 2 3 (4 polarization vectors)  cr()p u r () p e )  a† (k ) e ikx ) spinor indices on ur, vr, and  suppressed. r = 1,2.

Section 5.11 Summary Chart 157

r rrr r nd Bosons: xy,t,,t       i  xy  ,  = any field, other commutators = 0. 2 quantization  s    ss  s   Postulate #2 Spinors: Coefficient anti-commutation relations parallel coefficient commutation relations for bosons. Bosons: using conjugate momenta expressions in ↑ yields ↓ Equal time A(x ,t,A )ɺ  ( y ,t )  commutators   x,t,  ɺ†  y ,t    i   xy   Not needed for spinor derivation. (intermediate    step only)  ig  (x  y ) Bosons: Using free field solutions in ↑ with 3D Dirac delta function (e.g., for discrete solutions, 1    x y   eikn xy   e i k n   xy   ), and matching terms, yields the coefficient commutators ↓. 2V n   

Coefficient † † †  †  † ak  ,a  k   b  kk  ,b     crspp  ,c   d rs  pp  ,d    a(k ),a ( k  )  commutators        r s    1,   1 discrete  kk    rs  pp  rrskk  0 123 ,,

continuous  k  k  rs  p  p  r  r s  k  k  Other coeffs All other commutators = 0 All other anti-commutators = 0 All other commutators = 0 The Hamiltonian Operator

3 Substituting the free field solutions into the free Hamiltonian density H0, integrating H0 = ∫H0 d x, and

using the coefficient commutators ↑ in the result, yields ↓. Acting on states with H0 yields number operators.

 N()k1  N() k  1 E Np 1  N  p   1  N k  1 H0  k  a2 b 2   p  r2 r 2   k  r 2  k p,r k ,r

Nk a† k a k N()cp † p c p Number a       r r  r   † N()k  a k a  k  operators † † r rr r Nb k  b  k  b  k  N()drp r p  d r  p 

Creation and Destruction Operators

Evaluating Na (k) a(k) |nk〉 (similar for other particle types) with ↑ and the coefficient commutators yields ↓ † † † † † creation ak , b  k  crp , d r  p  ar k

destruction ak , b  k  crp , d r  p  ar k  Normaliz factors cp   | 0  lowering ak  |nk n|n k k 1 r  r, p as with scalars

† † raising ak  |nk n k 1 |n k  1 crp | 0   r, p as with scalars

N()  Np  N p  tot particle num N()  Nak  N b  k   r  r   N A   N r k  k p,r k ,r tot particle num: lowering           A  raising †  †   †       A 

158 Chapter 5 . Vectors: Spin 1 Fields

Four Currents and Probability

Four currents , † ,  † j  iA  A  AA   (operators) j ,j  i †,   ,  † j , j            †  = 0 for photons A A  j , = 0  

Emphasis in field theory is usually on the number of particles ( N(k) operator), and particle probability densities are rarely used. For completeness, however, and to make the connection with quantum mechanics, they are included below. (Antiparticles would have negative values of those below!)  x,t   Single particle x,t  j0  x ,t    x  ,t  probability 0 As at left, but with Dirac j above. = 0 for chargeless particles. density (not Note integration over x , not x operator) 1 For type a plane wave,   V Charge, not Scalar type b particle → negative . Photons →  = 0. probability Led to conclusion that j0 is really proportional to charge probability density. Observables Observable operators like total energy, three momentum, and charge are found by integrating corresponding

density operators over all 3-space. (For spin ½, electrons assumed below with q = - e)

P ENp  N p P  N k H P0 k  Nak  N b  k  0  p  r  r   0  k r   k p,r k ,r

P = 3- P kN k i P kNa k  N b  k  P pNr p  N r  p   r   momentum k p,r k ,r

s    qj q , j q jconstant   s 0 0 for photons  s0 d 3 x   s0 d 3 x  Q 0 for photons q Nak  N b  k   e Nrp   N r  p   k p,r

Spin operator 1 i 0    i   i  1 , 2 , 3 for RQM states N/A 2 0 i  magnitude = 1 for photons, and QFT fields i  2D Pauli matrices

Helicity operator  i p for RQM states N/A helicity eigenstates p and QFT fields

Spin operator † 3 N/A   d x magnitude = 1 for photons, for QFT states 

 ip  Helicity operator †  d 3 x N/A    helicity eigenstates for QFT states p 

Section 5.11 Summary Chart 159

Bosons, Fermions, and Commutators Operations on states with creation, destruction, and number operators above yield the properties below.

Properties of na (k) = 0,1,2,…, ∞ nr (p) = 0,1 only nr (k) = 0,1,2,…, ∞ states: So, spin 0 states bosonic. So, spin ½ states fermionic. So, spin 1 states bosonic. If anti-commutators used Commutators lead to 2 or more instead of commutators with identical particle states co-existing in Klein-Gordon equation same multiparticle state. Anti- solutions, then observable (not commutators lead to only one given Bosons can only counting vacuum energy) single particle state per multi-particle employ Hamiltonian operator would state. commutators have form Therefore, commutators cannot be Fermions can Same as spin 0. H 0 = 0 and H 0 | 〉 = 0 ,i.e., used with spin ½ fermions. only employ 0 0 k anti- all scalar particles would have This is further proof that we need commutators zero energy. commutators with bosons. Hence, we cannot use anticommutators with spin 0 bosons. The Feynman Propagator Creation and destruction of free particles (& antiparticles) and their propagation visualized below.

Feynman diagrams

† † † If ty < tx , Tx   y    x   y  , i.e., the   y operates first, and should be placed on the right. Step 1   † † Time ordered If tx < ty , Tx   y    yx     , i.e, the  (x) operates first, and should be placed on the right. operator T Note that  (x) commutes with †  y for x ≠ y . [Fermion fields anti-commute.]

† 0|T x  y| 0  iS    Transition 0|T x    y|  0  i  F      F  0|TA  xA   y  | 0  iD F amplitude The above vacuum expectation values (transition amplitudes) represent both (double density 1) creation of a particle at y, destruction at x, and in x and y) transition amplitude = Feynman propagator 2) creation of an antiparticle at x, destruction at y } Step 2 Propagator in By adding a term equal to zero to the Feynman propagator above, it can be expressed as terms of two vacuum expectation values (VEVs ) of two commutators (anti-commutators for fermions) commutators

iS x y  iF  x  y   F    iDF  x y   †         0x,    y  0 tt 0x,     y  0 tty x 0A x,A   y   0 t t   y x      y x †          0y,    x  0 ttx y 0y,     x  0 ttx  y 0A y,A   x  0 tx t y       

160 Chapter 5 . Vectors: Spin 1 Fields

Step 3 With the coefficient commutation relations, the above two commutators/anti-commutators As 3-momentum (for each spin type) can be expressed as two integrals over 3-momentum space integrals

†            Definition of x,    y    i   xy   x,    y  iS   xy   A x,A   y    iD  xy          symbols for †              commutators y,    x  i   xy   y,     x   iS   xy   A y,A   x  iD  xy          ∓ip x y  ∓ik x y    1 p m e 3  1 e 3 iS  d p i  d k 3  iD  g  i   3  2 2   Ep 2 2   k

+ +  + – –  – ± ±  , S , D represent particles;  , S , D represent anti-particles. Symbols S = S 

Although fields such as  are operators, because of their coefficient commutation relations, each integral above is a number, not an operator. The expectation value of a number X is simply the same number X. 〈0|X|0 〉=X 〈0|0 〉=X. So, the Feynman propagator will also be simply a number (no brackets needed.)

Step 4 Contour integral theory (integration in the complex plane) permits the above two integrals As contour (for each spin type) over real 3-momentum space to be expressed as contour integrals. integrals

iS  iD    i  ip x  y   ik x  y  ik x  y  ∓i p m  e 4 ∓ig e 4 ∓i e 4 d p d k d k 4  2 2 4C 2 2 4  2 2 2 C p m 2  k   2  C k     photon = 0 Step 5 Taking certain limits with contour integrals in the complex plane yields a single form for the As one integral Feynman propagator that works for any time ordering and will prove more convenient.

 x y  S x y   F   F   DF  x y   in physical ip x  y ik x  y     ik x  y  1 e 4 1 p m  e g e space d k d4 p d4 k 4 2 2 4 2 2 4 2 2  k  i  2  p m  i  2  k i 

1 p m  in momentum  k  S p   g F   2 2 F   2 2 DF  k   space k  i  p m  i  k2  i 