Download Relations between Massive and Massless one

Relations between Massive and Massless
one-particle states
GE Fengjun (葛峰峻)
LIU Changli (刘长礼)1
Institute of Applied Physics and Computational Mathematics, Beijing China, 100094
Abstract: In Weinberg’s book, The Quantum Theory of Fields Vol. 1, chapter 2,
one-particle states are described detailed. The massive particle with spin s has 2s+1
one-particle states. The massless particle with spin s has only two one-particle states.
There is a large gap between them. The paper proves that massive one-particle
states’ transformation can continuously change into massless one particle states’. In
another words, the SO(3) group describing massive one-particle states can
continuously turn into the ISO(2) group describing mass zero. That massless particle’s
helicity has only two values are proved in theory.
Key words: one-particle states, helicity, SO(3), ISO(2)
1. Introduction
A massive particle with spin j has 2j+1 one-particle states, such as the spin 3/2 9Be’s
nuclei, whose magnetic quantum numbers are 3/2, 1/2, -1/2, -3/2. However, it is
different about a particle with mass zero (for example photon). Photons’ helicity has
only two values: 1,-1. Helicity zero is forbidden. Generally, explanations of massless
particle’s helicity with two values are by gauge invariance. We explain it in another
way.
One-particle states are described detailed in Weinberg’ book [1]. Methods and notes
in this paper are close to his book.
In Weinberg’s book, a massive particle with spin j, the transformation is [1, eqs
2.5.23]
U     p , 
 p 
p
0
0




p , 
Dj  W  , p  
(1)
For massless particle of arbitrary helicity, the Lorentz transformation is as following
[1, eqs 2.5.42]
U     p , 
 p 
p0
0
 p , exp  i  , p  
Both transformations are different except the spin zero.
2. Solutions
1
Corresponding author’s E-Mail: liucl78@pku.org.cn
(2)
We use the Lorentz metric: g=diag{1,1,1,-1}.
Thought of proof is: We define standard momentum as 𝑘 = (0,0, 𝑝, 𝐸) (which is
different from Weinberg’s book [1, pp.66]), where p is the third component of
momentum, and E is the energy. From Einstein’s equation 𝐸 2 = 𝑝2 + 𝑚2 , keeping E
constant, m and p can vary. When mass is zero, the standard k turn the Weinberg
book’s standard one of massless particle ((0,0, 𝐸, 𝐸)); and when p is zero, it becomes
massive particle’s standard one ((0,0,0, 𝐸)). We have to work out the structure of the
little group.
Referring to methods of solving massless particle’s little group in Weinberg’s book [1,
pp.69-74], we solve the little group W which can keep 𝑘 = (0,0, 𝑝, 𝐸) invariant. The
little group can be expressed as 𝑊(𝜃, 𝛼, 𝛽) = 𝑆(𝛼, 𝛽)𝑅(𝜃)[1,pp. 69-70], in which
𝑅(𝜃) is a rotation around the third axis.
 cos 

 sin 
R    
 0

 0
sin 
cos 
0
0
0

0
0

1
0
0
1
0
Matrix S can be written as [1,pp. 69-70]:
 u v 

w s 
S 
 x y 1 

 z r q
q
q
q
1  q 2






Where q=p/E is a dimensionless parameter. Equation q=0 represents massive
particle’s standard momentum, and q=1 represents massless one. The unknown ζ is
satisfied with the equation
 2   2  1  q 2   2  2  0
We let  
1 
1  1  1  q 2  2   2  
2 


1 q
The First two columns of transformation S are solved by g = S 𝑇 gS [1, eqs 2.3.5]
(where g=diag{1,1,1,-1} is the Lorentz metric). In fact all elements of S are solved
through this equation. The solutions are as following
s  1  1  q 2   2
x  / s
y  u
v   1  q 2   / s
z  xq
r  yq
u  1  1  q 2   2 / s 2
w0
For 𝛼 → 0, 𝛽 → 0, 𝜃 → 0, the general little group elements, namely the Lie algebra
of group 𝑊(𝜃, 𝛼, 𝛽) = 𝑆(𝛼, 𝛽)𝑅(𝜃), are
 0


W  ,  ,    I  


 q

0

q


0
0
q 

q 
0 

0 
The little group W can leave stand momentum 𝑘 = (0,0, 𝑝, 𝐸) invariant. Referring
to Weinberg’s book, omitted details can easily be added.
The corresponding Hilbert space operator is
U W  ,  ,    1  i A  i  B  i J 3
where
A  J 2  qK1
B   J1  qK 2
We can see that these generators have the commutations.
 J 3 , A  iB;  B, J 3   iA;  A, B   i 1  q 2  J 3
(3)
From the Lie algebra of W above, we can see that the algebra is o(3) when q=0 (that
means the particle is massive); or the algebra is iso(2) when q=1 (that means the
particle is massless).
The above is written by the first author, and the second author does the following
work.
3. Group Representations and Results
If we let J1 
A
1  q2
, J 2 
B
1  q2
, J 3  J 3 , the Lie algebra (3) becomes
 J3, J1  iJ 2 ;  J 2 , J3   iJ1;  J1, J 2   iJ3
which is o(3) algebra. So the little group W is a SO(3) group.

In physics, the infinity is usually not observed. When q  1 , eigenvalues of J1,2
trends to infinity, and it is not physical states. We need physical states to keep
 finite. That need eigenvalues of A and B are zero. In another
eigenvalues of J1,2
words, only the states which keep eigenvalues of A and B to be zero are physical.
However the representations of the corresponding Lie group W, which have to
include the parameter q=p/E, are different from representations of group SO(3). The
elements [2] are
Dmj m W   Dmj m    jm exp  i J 3  exp  i  B  exp  i J 3  jm


 jm exp  i J 3  exp i  1  q 2 J 2 exp  i J 3  jm
   1
n
n
 j  m ! j  m ! j  m! j  m!
exp  im  
 j  m  n ! j  m  n !n ! n  m  m !
2 j  m  m  2 n
(4)
2 n  m  m


 1  q2 
 1  q2 
 cos

 sin

 exp  im 




2
2




When q trends to 1 and eigenvalues of B is zero, the elements contract
Dmj m     mm exp  im     
(5)
If the representation (4) of massive particle is substituted into transformation (1),
equation (1) is the same as the massive particle in Weinberg’s book when q=0. When
particle mass trends to zero, namely q trending to 1, representation (4) turns into
representation (5), and transformation (1) continuously turn into transformation (2).
Transformation (2) is only the limit state of transformation (1) as mass trending to 0.
From the representation (4), as q → 1, the rotation J 2 around axis y becomes zero.
There are no rotations around axis x and y, and only rotation around z is left. Raising
and lowering operators (𝐽± = 𝐽1 ± 𝑖𝐽2 ) disappear. So the helicity cannot change.
There are only two helicity states of massless particles.
References：
[1] Steven Weinberg, The Quantum Theory of Fields, Vol. 1, Chapter 2, New York:
Cambridge University Press, 1995, pp.49-91
[2] Wigner EP. Group Theory and Its Application to the Quantum Mechanics of
Atomic Spectra. New York: Academic Press Inc., 1959, p167