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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 , Dj 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: [email protected] (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 w0 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 mm 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