Download Lecture Notes 21: More on Gauge Invariance, Why Photon Mass = 0, "Universal"/Common Aspects of Fundamental Forces

UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
LECTURE NOTES 21
Gauge Invariance of the EM Interaction and Electric Charge/Current Conservation
In P436 Lecture Notes # 19, pages 14-18, we saw that we could write the anti-symmetric
rank-2 EM field tensor F  v in terms of covariant space-time derivatives of the 4-vector EM
potential field A as:
Row #
Ex c E y c Ez c 
 0


v


0
Bz
 Ex c
 By 
A A
v

F v 

A  V c , A
F

where:
and:
  E y c  Bz
0
Bx 
x xv


By
0 
 Bx
  Ez c
Column #



A 
v

 E  V   A A
v




t
Which correctly gave the familiar relations: ( i.F

e. LHS)
   x xv
 B  
A 
( i .e. RHS)


We also saw that F
equation
v
 Av A


 x
  xv


 automatically satisfied the homogeneous Maxwell

F v Fv F
G  v
 0 or equivalently:

 v 0 ,
v
x
x  x 
x

 
 
B
both of which contain the homogeneous Maxwell equations  B  0 and:   E  
.
t
1  v
v
F and   v = totally anti-symmetric rank-4 tensor.
G  v is the dual tensor to F  v : G  
2
Row #
 0

 Bx
v
G 
  By

  Bz
Column #


0
 Ez c E y c 
0
Ez c
 Ex c 

0 
 E y c Ex c
Bx
By
Bz
 Av A
v
F



We also saw that the relativistic 4-potential formulation

xv
x


v
inhomogeneous Maxwell equation  v F 

 satisfied the

F  v
  Av A



x v
x v  x xv
 
inhomogeneous Maxwell equations  E  tot  o


  o J which contains the


  1 E

 o J tot provided we
and   B  2
c t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
v
use the Lorenz gauge condition:  v A 
 v F v 
F  v
  Av A



x v
x v  x xv


 
 x
Prof. Steven Errede
Av
 0 in the inhomogeneous Maxwell equation:
x v
 Av    A 

 v 
  v   o J
 x  xv  x 
Lorenz Gauge Condition:
Av
   1 V

 v Av  v  0    A  2
 0
x
c t


v
Then:  v F 
Lect. Notes 21
D’Alembertian operator:
1 2
 
 
2
2 




xv x v x v xv
c 2 t 2
  
  
F  v
A  v
A   o J 
 2 A  o J  or: 2 A 
v
v
x
xv x
x xv
Av
 0 gives us the freedom to add to the 4-vector
x v
potential field A any arbitrary constant 4-vector, usually (i.e. traditionally) written as the space
time / 4-gradient of an arbitrary scalar space-time point function λ     r , t   :
v
The Lorenz gauge condition  v A 
A  A*  A 

x

0
x
where:
 

i.e. the scalar and vector potentials V  r , t  and A  r , t  respectively are not uniquely


determined / defined by the EM fields E and B up to an arbitrary constant.


We then showed that the EM field tensor F
 A*v A*
F  v  

 x
xv
 
The 4-vector A in “field-space”:
v
is manifestly gauge invariant:
  Av A

  
  x xv

v
  F

Several steps {See P436
Lect. Notes 19, p. 17}

x
A
A
A*

  r , t 

 
n.b. A  r , t   A  r , t  
x
*

x


i.e. The EM field tensor F  v (i.e. the EM fields E and B ) are invariant / unchanged by a gauge
transformation (scale change/rotation in “field-space”) of the EM 4-potential field A :
A*  A 
A  A*  A 
2

x
where:

0
x
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
Now local conservation of electric charge/current – i.e. conservation of charge/current at each and



every point in space-time  r , t  requires that J tot  c tot , J tot obeys the continuity equation:


  J tot


 

J tot
 0 i.e.  J tot   tot

t
x






And since: J tot  J free  J bound and: tot   free  bound , thus: J tot  J free  J bound


Thus, free electric charge/currents J free  c  free , J free

  J free 
J free
x


separately obey:
 

 0 i.e.  J free   free
t


And bound electric charge/currents J bound  c bound , J bound


  J bound

separately obey:

 
J bound




0
 J bound   bound
i.e.

x
t

Electric charge/current is conserved at each/every/all points  r , t  in space-time.
 



Explicitly/generically: J  x   c   r , t  , J  r , t 

But:
 v F v 
  v F
v
F  v    Av A
 v

x v
 x  x xv
  F  v 

   v  
x  x  x



   o J = inhomogeneous Maxwell equation
 
   Av A

 v 
 x  x xv

J 

   o   o   J  0
x
 
Experimentally/empirically, we know that electric charge/current is conserved, i.e. we know
J 

that the RHS of this equation is indeed true, that:   J    0 .
x
What about the LHS of this equation? Does:
   Av A
We know that: 
 v 
x

xv
v
 x

  F  v

x   x v


 
 x
   Av A

 v 
 x  x xv

  A 
Av
0




 provided that:
x v
xv  x v 


   0 ???
 
i.e. we adopt/use the
Lorenz gauge condition
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
F  v
  Av A



But:
x v
x v  x xv



Thus:
Fall Semester, 2015

x 
 F  v
 v
 x


 
 x
Lect. Notes 21
   A

 v
 xv  x
Prof. Steven Errede
 
 .
 
However, we are free to interchange the order of the derivatives in this equation, i.e.:
  F  v

x   x v

 A
   A 


but:



 
xv x v  x  

 x

  0 !!!

  F  v 
   A 
This can also be equivalently written as:



0
x v  x  
xv x v  x  
Then we see that the LHS of this equation does  0 and hence the RHS of this equation must
also = 0, i.e.:
 
   Av A 
J 
   v F v   v F v   v 


 o   J   0


o


x x
x x  x xv 
x

provided that the Lorenz gauge condition holds, i.e. that:   A 
A
0 .
x 
J 
0
Stated in this manner, we realize that electric charge / electric current conservation
x 
A
 0 !!! The Lorenz gauge condition
is a direct consequence of the Lorenz gauge condition
x 


is in fact a statement about the gauge invariance of F  v E and B


*

i.e. we have the freedom to define A  A  A 

Clearly, the Lorenz gauge condition   A 
*
But:   A 


!!!
x
A
 0 is satisfied for the specific choice of A .

x
A*
 0 would be the corresponding Lorenz gauge condition for the 4-potential
x 


*

vector field A   A 
x


 

0.
    0 , where:
 provided (if and only if):

x x
x


    r , t  can be any/an arbitrary space-time 4-point function, as long as it satisfies:

2
 
1   r ,t 



2
2
 0 where:
0.
     r ,t     r ,t   2
2

x x
t
x
c
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
*
  A 
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
A*
 
A

2


A

0 .


0



0


because:
and:

x 
x 
x  x
A* A


   0 where: A*  A 
 0 because of the gauge invariance
Thus:
with:

x
x
x
x


* v
v
associated with choice of A leaving F  v (i.e. the EM fields E and B ) unchanged: F  F .
J 


J

 0 i.e. electric charge /
As a consequence of the gauge invariance of A : 
x 

electric current is locally conserved at each/every point  r , t  in space-time.

Put conversely, if: A*  A 


 0 , then because this A -field is not locally
, with:
x
x
A* A

   0 e.g. because
gauge invariant at each/every space-time point  r , t  , i.e.
x 
x




 
  2   0 {eek!!!} Then: F * v  F  v (i.e. E *  E and B*  B ) which would be
  
x x
terrible !!! And worse yet:
J 
 0  Electric charge / electric current would not be conserved !!!
x 
Let us consider what happens to the term A A when we make a gauge transformation:
A  A*  A 

. Then:
x


A* A*   A 

x



  A  
x

A
  

         

 A     A  
   
  A A  



x
x
x








  x 

If A* A* is to = A A , then these terms must = 0. But:


 0 and:   0
x
x

x
A
A*
A*  A 

x
Note that we don’t necessarily expect A A = A* A* on physical grounds
anyway, from the above “field-space” vector diagram.
Thus, any physical quantity involving A A and/or A* A* is manifestly not gauge invariant.
Note, however that both A A and A* A* are properly Lorentz invariant:
In IRF(S') → A    v Av ← In IRF(S)
In IRF(S') → A*    v A*v ← In IRF(S)
A    v Av
A*    v Av*
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
e.g.:
A A     v Av    v Av      v   v  Av Av 
But:
 v v  1

Lect. Notes 21
Prof. Steven Errede
{i.e. the Λ-boost matrices are unitary transformations}
A A  A A Lorentz invariance is obeyed by the A -field.
 
In IRF( S  )
In IRF( S )


Physically, what is A A ? The 4-vector potential: A  V c , A


SI Units: Newtons/Ampere = “p/q”
{momentum per Coulomb!}
We know what Aq is for a point electric charge q, e.g. in its own rest/proper frame,
from the {retarded} Liénard-Wiechert potentials {see P436 Lect. Notes 12, p. 7-8 and/or
Griffiths Example 10.3, p. 433-434}:
 
v r ,t 
 
1 q


Vq  r , t  
Vq  r , t   0
and: Aq  r , t  
2
4 o r
c
n.b.  no time dependence in the rest/proper
frame of the point charged particle.
2


1 2
1 2
1  1  q2
Then: Aq  r  Aq  r    2 Vq  r   Aq  r  Aq  r    2 Vq  r    2 

c
c
c  4 o  r 2

However, recall that the {electrostatic} EM energy density uEM  r  associated with a point
charged particle q probed by a point test charge qT at a separation distance r from q {both at rest} is:
uEM  r   qTVq  r  
Thus, we see that: u
and thus: U
2
EM
 u
v
2
EM
2
EM
1 qT q
referencing: u EM  r     0 , i.e. Vq  r     0
4 o r
 r    qTVq  r  
 r  d  v u
2
EM
2
2
 1 q
2 2 
q 
  qT c Aq  r  Aq  r 
 4 o r 
2
T
 1 
 r  d  q q 

 4 o 
2
T
2
2
1
r
v
2
d   = (rest energy of q)2
This integral has a singularity at r = 0, as we have discussed long ago in P435, thus it should
come as no surprise here {again} that using classical and/or relativistic EM, the calculated rest
energy (i.e. = rest mass mq c 2 ) of the test charge q is formally infinite – this problem remains
even in Quantum Electrodynamics {QED} where the technique of mass (& charge)
renormalization is used to address this problem.
Physically then, we see that  q 2 c 2 A A   mq c 2  , which is a Lorentz invariant quantity,
2
i.e. it has the same numerical value in any/all IRF’s.


2
Thus: A A  A A    mc qc  is known as a mass term because of this.
2
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
Real photons have rest mass m c 2  0 , but since real photons always travel at the speed of
light c {in vacuum/free space} and thus have no rest mass frame, then in the center-of-momentum
frame of a real photon: A A  A A  0 . Since this is a Lorentz invariant quantity, it must be
the same numerical value in all reference frames, e.g. including the lab frame IRF(S).
Thus, we can now see from the point of view of gauge invariance: A*  A   x
that if A* A*    m*c 2 qc   A A    mc 2 qc  this would be a really bad thing!!!
2
2
Manifest gauge invariance {“eich-invarianz”, auf deutsch} of the EM interaction as
represented by the vector field A is intimately connected to the {microscopic} nature of the
force carrier (a.k.a. mediator) of the EM interaction – the photon – an intrinsic spin-1  vector


particle {it carries the E and B fields – i.e. it carries F  v }.
For real photons (i.e. lying on the I  x  x  0 light cone somewhere in 4-D Minkowski
ct vs. x space), the real photon mass m c 2  0 , hence: Etot  p c
  hf  .

Only because of the fact that m c 2  0 do we have a  1 r 2 Coulomb force law for virtual
photons { m c 2  0 } exchanged between two electrically-charged particles:
4-D space-time:
*
ct
q1
x1
* *
*
*


FCoul  r  
q2
i.e.
x2
x
1  q1q2 

 rˆ
4 o  r 2 
 
 
1 q
E r  
 2  rˆ  V  r 
4 o  r 
1 q

1
V r  
   central potential
4 o  r 
r
The Coulomb / EM force is a conservative   1 r 2  force due to the gauge invariant nature
of the EM interaction – i.e. EM “news” / information propagates at c because m c 2  0 .
The range of the EM force = ∞ for m c 2  0 .
If m c 2  0 , then EM “news” / information would not propagate at the speed of light c.
→ The EM interaction would no longer be gauge invariant !!!
i.e. A*  A   x (i.e. F * v  F  v !!!)
J 
→ Electric charge q / electric currents I would not be conserved – i.e.   J    0
x
2
2
If m c  0 , then Coulomb’s force law would no longer have a purely 1 r nature

i.e. Coulomb’s force law would no longer have a purely 1 r central potential V  r  .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
For m c 2  0 , we would instead have:


FCoul  r  
1  q1q2    r

 e 1   r  rˆ  exponentially damped force law!
4 o  r 2 
 
 
1  q   r
ˆ
1




E r  
e
r
r
V r 


 
4 o  r 2 

V r  
1  q   r
 General e   r form is known as the Yukawa potential.
 e
4 o  r 
Where:   m c   m c 2 c  1  
Etot 
vprop  c
p2 c 2  m2 c 4
and where: h = Planck’s constant, and:   h 2 .
hc = 1240 eV-nm
c = 197.3 MeV-fm
Thus for m  0 , the Coulomb force would be of finite range – falling to 1 e of what it would
have been for m  0 within a characteristic distance scale of    1    m c  c m c 2 :
We can also see this from another perspective:
A
2 


For m c 2  0 , in the Lorenz gauge   A    0 we do have:  A   o J
x

  

A
 0 ← no time dependence, and B    A  0 ← no B -field)
In the static limit (i.e.
t
  
 
1




2
2
2
this equation becomes:  V  r    o c   r      r  But:  E   V  r    V  r 
o


 
1

  E     r  {i.e. Gauss’ law}. Integrate both sides over volume  
o
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede

3 
We know that for a point charge q:  q  r   q  r 
 

1 q
1 q

rˆ and: Vq  r  
Then: Eq  r   Vq  r  
 
2
4 o r
4 o  r 
The range    1    m c  c m c 2 of the EM force = ∞ for m c 2  0 .

But what if m c  0 ?? In the Lorenz gauge:   A 
2
A
0
x 
Then: 2 A   o J  is modified by adding a mass term to this relation:


 A   A   o J
2
2

where:  
m c
c

m c 2
c 2
= scalar / constant
n.b.
Since: 2 
 
     = Lorentz invariant scalar quantity.
x x 
Then: 2   2
also
= Lorentz invariant scalar quantity.

  

Then in the static limit (i.e. A t  0 ← no time dependence, and B    A  0 ← no B -field)
1




we obtain:  2V  r    2V  r    o c 2   r      r 
o


n.b. the 4-potential A (and thus V  r  and A ) now do acquire real physical (i.e. observable)
significance through the mass term  2 A – i.e. changing the gauge A  A*  A 

x
now also causes changes in this mass term !!! {eeek!}
1 q  r


e
The solution for the scalar potential Vq  r  for a point charge q is now: Vq  r  
4 o r
 The range    1    m c  c m c 2 of the EM force ≠ ∞ if m c 2  0 !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
Suppose J   0 , i.e. no electric charges / currents present  “free” field equation for A :
 2 1 2  
For m c  0 :  A  0     2 2  A  0 = homogeneous wave equation.
c t 

 
 i kz t 
 Get harmonic solutions of the form: A  r , t   Ao e 
e.g. for an EM wave propagating in
2
2

the ẑ -direction. The dispersion relation for this situation is:  2   ck  i.e.   ck .
2

1 2 
For m c 2  0 : 2 A   2 A  0    2  2 2  A   2 A  0 =
c t 

 
 i
 Has solutions of the form: A  r , t   Ao e
k 2   2 z  t


inhomogeneous
wave equation.
n.b. EM wave propagation sort of
like that in wave guides !!!
The dispersion relation for this situation becomes:  2   ck     c 
2
2
Define: o  ck →  2  o2   2 c 2 where:   m c   m c 2 c  1   .
There exist experimental upper limits on the photon mass from laboratory experiments and
 
also from geomagnetic data – i.e. the earth’s magnetic field B  r  !
 
 
 2 r 2  e  r 2 2  e  r 

 
  m
If m c 2  0 : B  r   o  3rˆ  rˆm   m  1   r 


4 
3  r3 3
r 

If m c 2  0 , this formula tells us that the magnetic field at the surface of the earth
r  R  6370 km would have a “normal”, pure magnetic dipole field component plus an added
constant magnetic field – the   m c   m c 2 c  1   term(s) in the above formula would

give rise to an apparent external B -field contribution which would be {overall} anti-parallel to

the magnetic dipole moment m of the earth.

Satellite measurements and surface observations of B constrain this “external” constant
magnetic field to be less than  4 103   that of a pure magnetic dipole field at the earth’s
magnetic equator {at 90% confidence level (CL)}, which corresponds to a 90% CL lower limit
on the m c 2  0 photon attenuation length scale of     1  108 m , or since   m c 2 c ,
this corresponds to a 90% CL upper limit on the photon rest mass of m c 2  2  1015 eV , or
equivalently m  4  1051 kg .
Recently (1998), the absence of an EM torque on a toroid balance experiment set 90% CL
upper limits on the photon rest mass of     1  109 m , m c 2  2  1016 eV , or equivalently
m  4  1052 kg . {See Lakes, et. al., PRL 80, 1826 (1998).}
10
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
There also exist upper limits on the photon rest mass m c 2 from the absence of frequency
shifts of the Schumann earth-earth’s ionosphere resonances!
If m c 2  0 , e.g. the Schumann n = 0 resonance formula 0       1 c  r  12 h 
is modified, becoming: 2  02   2 c 2 . The absence of shifts in the observed Schumann n = 0
resonance frequencies, attributable to m c 2  0 effects give less stringent, although still
respectable 90% CL limits of     1  105 m , m c 2  2 1012 eV , or m  4 1048 kg .
Note that if m c 2  0 then {massive} real photons would contribute to dark matter / dark
energy, affecting e.g. galaxy formation, large-scale structure of the universe, lead to significant
changes/alterations the ~ 3K black-body/microwave radiation spectrum left over from the
“Big Bang”, suffer red-shifts in light emitted from stars and other gravitational bodies, etc…
The Nuclear / Strong Interactions:
In ~ 1935, Hideki Yukawa proposed a theoretical model for the nuclear (i.e. the strong) force
between nucleons (i.e. protons and neutrons) mediated {at low energies} by massive scalar
(i.e. spinless) particles – the pi mesons   ,  0 (intrinsic spin- 0 ), with m  135  140 MeV c 2 .
In Yukawa’s model, the strong field associated with the “free”-field propagation of massive,
scalar/spin-0 pi mesons {pions} in space-time was described mathematically via the equation:
m c m c2


2   r , t    2  r , t   0 where   r, t  = scalar field of the spinless pion and:     

c
In nearly complete analogy to that associated with the EM “free-field” propagation of “massive”
vector photons (i.e. intrinsic spin-1 ) in space-time, as mathematically described via the equation:
m c m c2


2 A  r , t    2 A  r , t   0 where A  r, t  = vector field of the spin-1 photon and:     

c


In the “static” limit, the solution to 2   r , t    2  r , t   0 is the {scalar} Yukawa potential:
e  r

 r   K
r
where: K 
g
= constant and g = strong / nuclear charge
4
The range of the strong / nuclear force is extremely short, due to the mass of the pion:
1
 strong   strong

c
197.3 MeV -fm

 typical size of a nucleus  1.5 fm  1.5 1015 m !!!
2
m c
~ 135 MeV
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
We can also repeat this for the strong interactions @ high energies, in the context of QCD
(Quantum Chromo-Dynamics) – the mediator of the strong force at high energies is a massless
vector / spin-1 particle known as the gluon, which carries {net} strong charge (n.b. unlike the
photon, which carries no {net} electric charge) !!!
For QCD/the strong interactions,  an octet of gluons – because there exist 3 distinct strong
“color charges” (r,g,b) and 3 distinct strong “color anti-charges”  r , g , b  . The octet of massless
gluons thus carry {orthogonal} color-anticolor strong charge combinations e.g. rg , gb , rb 
In analogy to the vector field A for the massless spin-1 photon {the mediator of the EM
interaction} the massless spin-1 color-anticolor charged gluon {the mediator of the strong
interaction at high energies} is represented by the vector field Ga , where  = 0:3 is the usual
space-time index and the index a = 0:7 denotes which color-anticolor combination of the color
octet this particular gluon has.
2 

In analogy to  A   o J for the EM interaction, for QCD / the strong interaction at high
2

s 
energy we have:  Ga   o J a where J a is the relativistic/space-time 4-D strong color charge
/ strong color current density associated with the color-anticolor index a and os is the strong
color magnetic permeability associated with the QCD vacuum, noting that the “speed of light” c
is the maximum speed for any/all of the four fundamental forces / interactions of nature, thus:
 os os .
c 1
Virtual Gluon Exchange
Between Quarks
in 4-D space-time:
Virtual Photon Exchange
Between Electric Charges
in 4-D space-time:
*
g*
ct
q1
g* g*
g*
q2
ct
q1
g*
x1
12
x2
x
x1
* *
*
*
q2
x2
x
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
The Electroweak Interactions:
In the 1970’s, high-energy physicists discovered that the electromagnetic and weak
interactions were in fact not unrelated to each other – they have common electroweak fields !!!
At very high energies (e.g. the situation in the very early universe, just moments after the
“Big Bang”), the mediators (i.e. force carriers) of the electroweak interaction are all massless
spin- 1 vector particles:
 W1 


 a weak isospin triplet field:  W0  where: W1  antimatter field of the W1 and:
 
 W1 
 a weak isospin singlet field: B0
At very high energies, there are four inhomogeneous Maxwell-type equations – one for each of
the four electroweak spin-1/vector fields:

2 W1   ow J w
0
2 W0   ow J w0

2 W1   ow J w
0
2 B0   oB J B0
(massless W+)
(massless W0)
(massless W)
(massless B0)
The “standard model” of electroweak interactions also predicts that there exists a (complex)
scalar doublet {a.k.a. complex Higgs field}, thought to pervade all space-time in our universe –
{the new æther!} which, e.g. as the very early universe cooled, this scalar field underwent a
phase transition ~ analogous to that of the phase transition associated e.g. with a ferromagnetic
material cooling through its Curie temperature. During this EWK phase transition, the scalar
Higgs field  is “absorbed” by the longitudinal {i.e. spin-0} components of the W+, W spin1 vector bosons, thus each acquired mass, i.e. becoming the massive W  bosons, whose
relativistic fields (at low energies/below the phase transition) are: W  12 W1  W1  .
Since there are two neutral vector bosons, below the energy associated with the EWK phase
transition, the two neutral bosons will in general be orthogonal linear combinations of the two
original W0 and B0 fields (i.e. they will be {orthogonal} mixtures of these two neutral fields).
Because of the differences between weak isospin triplet vs. isospin singlet, only one of the
orthogonal linear combinations of the W0 and B0 fields “absorbs” the scalar Higgs field as its
longitudinal / spin-0 component, thus becoming the massive Z0 boson:
Z    B0 sinW  W0 cos W . The other neutral vector boson, also as a linear (but orthogonal)
combination of the original W0 and B0 fields remains massless – it does not couple {directly}
to the scalar Higgs field – and becomes the photon: A  B0 cos W  W0 sinW !!!
Note also that, due to the intrinsic nature of the weak isospin triplet, the {now} massive
electrically-charged W  and electrically-neutral Z 0 spin-1 vector bosons all carry weak charge.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
The electrically-neutral weak isospin singlet particle {the photon} does not carry any weak charge,
nor does it carry electric charge.
Thus, at low energies: ECM  0  0  246 GeV (below the energy of the EWK phase transition)
where: 0  0  246 GeV = vacuum expectation value of the Higgs field, the inhomogeneous at
At low energies, Maxwell equations for the four spin- 1 EWK vector boson fields become:

2 W  W2  W   oW JW 
0
M W   80.4 GeV c 2 ,
W  M W c 2 c


2 Z    Z2 0 Z    oZ J Z0
M Z 0  90.2 GeV c 2 ,  Z 0  M Z 0 c 2 c
2 A   o J 
M   0.0 GeV c 2 ,   M  c 2 c  0
The Gravitational Interaction:
We an also {attempt} to do this same thing for gravity – the massless graviton is a spin- 2
particle {it must be spin- 2 because the gravitational force is attractive, only}. Then, like the
weak W, Z0 vector bosons and the spin- 1 gluon, which carry their own respective charges
{thus self-coupling to themselves (!!!)}, the graviton similarly carries gravitational charge, and
thus also self-couples to itself. However, because gravity is a spin- 2 tensor field, we thus need
two indices  , v in order to fully describe the relativistic space-time structure of the gravitational
field g  v and its associated rank-2 tensor “matter” charges / “matter” currents J  v .
Thus, the inhomogenous Maxwell equation for gravity is:
2 g  v   oG J  v
M g  0.0 GeV c 2 ,  g  M g c 2 c  0
Today, a more modern approach describes the nature of fundamental interactions using
relativistic versions of the classical dynamics Lagrangian: L  T  V and the Euler-Lagrange
equation:
Classical Dynamics Euler-Lagrange Equation:
d  L  L
0


dt  qi  qi
qi = generalized coordinates
 

, x  = Lagrangian density
Classical Dynamics: L  qi , qi , t   Relativistic Dynamics: L   ,

 x

Relativistic Dynamics Euler-Lagrange Equation:
 L
 
L


0
x     x   


SI units of Lagrangian density L (Joules/m3)  SI units of Lagrangian L  v L d (Joules).
14
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 21
Prof. Steven Errede
Thus, writing the interactions associated with the four fundamental forces in terms of their
relativistic Lagrangian densities:
n.b. If plug LQED into Euler-Lagrange equation
1.) EM Interaction:
LQED
1
  F v F  v  J  A 
4
can show:   F  v    Av   v A  o J v
and   J   0 !!!
EM fields Current/charge interacting with the EM field!
1 a v
 a
2.) Strong Interaction: LQCD   G v Ga  J a G {and other terms}
4
1 a v 1
v

a
3.) Weak Interaction: LWeak   W vWa  B v B  J waW  J z B {and other terms}
4
4
4.) Gravitational Interaction: Lgrav 
c4
16 GN
g g  v R v  g  where:
g   Det  g  v 








and: R v  g      v   v       v   v   and:   v  12 g    g v   v g   g  v  .
OPEN QUESTIONS:
 Why are there four fundamental forces of nature? Why not just one???  more???
 Why are all fundamental forces mediated by integer spin-particles? {bosons}
Why are all fundamental matter particles spin-1/2  ? {quarks & leptons: fermions}
Why are no forces mediated by spin-1/2  , spin-3/2  ? (   SUSY {Supersymmetry}???)
 What precisely is charge (electric, strong, weak)?
 What precisely is intrinsic spin angular momentum (spin-1/2 fermions, spin-1 bosons)?
 Why are the W  , Z 0 bosons massive, while the  is massless?  “EWK symmetry-breaking”
Carry weak charge
Carries no net electrical charge
The gluon is massless, but carries strong charge (which is confined/screened beyond ~ 1 fm)
 Relativity and the microscopic world of particle physics / fundamental forces,
and quantum mechanics are indeed at work in the “everyday” world all around us !!!
We simply weren’t aware of this before !!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15