Mixing
tensor product of massless Poincare representations
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Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the connected Poincare group, the semidirect product $ISO_0(1,3)$ of the connected Lorentz group $SO_0(1,3)$ and the 4-dimensional translation group.
rt.representation-theory lie-groups tensor-products unitary-representations
asked 3 hours ago
Arnold Neumaier
1,410620
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up vote
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Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the connected Poincare group, the semidirect product $ISO_0(1,3)$ of the connected Lorentz group $SO_0(1,3)$ and the 4-dimensional translation group.
rt.representation-theory lie-groups tensor-products unitary-representations
asked 3 hours ago
Arnold Neumaier
1,410620
2
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the connected Poincare group, the semidirect product $ISO_0(1,3)$ of the connected Lorentz group $SO_0(1,3)$ and the 4-dimensional translation group.
rt.representation-theory lie-groups tensor-products unitary-representations
asked 3 hours ago
Arnold Neumaier
1,410620
Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the connected Poincare group, the semidirect product $ISO_0(1,3)$ of the connected Lorentz group $SO_0(1,3)$ and the 4-dimensional translation group.
rt.representation-theory lie-groups tensor-products unitary-representations
rt.representation-theory lie-groups tensor-products unitary-representations
asked 3 hours ago
Arnold Neumaier
1,410620
asked 3 hours ago
Arnold Neumaier
1,410620
edited 45 mins ago
asked 3 hours ago
Arnold Neumaier
1,410620
asked 3 hours ago
Arnold Neumaier
1,410620
asked 3 hours ago
Arnold Neumaier
1,410620
1,410620
2
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago
add a comment |Â
2
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago
2
2
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
answered 1 hour ago
Zurab Silagadze
10.7k2368
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
answered 1 hour ago
Zurab Silagadze
10.7k2368
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
add a comment |Â
up vote
2
down vote
accepted
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
answered 1 hour ago
Zurab Silagadze
10.7k2368
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
answered 1 hour ago
Zurab Silagadze
10.7k2368
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
answered 1 hour ago
Zurab Silagadze
10.7k2368
answered 1 hour ago
Zurab Silagadze
10.7k2368
answered 1 hour ago
Zurab Silagadze
10.7k2368
answered 1 hour ago
Zurab Silagadze
10.7k2368
10.7k2368
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
add a comment |Â
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
Thanks! There is a second paper by Lomont and Moses that gives a more explicit reduction: Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group,aip.scitation.org/doi/pdf/10.1063/1.1705287
– Arnold Neumaier
17 mins ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
+1.Also Moussa–Stora (1965), Schaaf (1970), Barut–Rączka (1977, p. 553): $$U^0,,J_1otimes U^0,,J_2congint_0^infty dMsum_J_1-J_2^infty oplus U^M,,J.$$
– Francois Ziegler
34 secs ago
add a comment |Â
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2
What is "massless"? what is "helicity"? representation means irreducible complex finite-dimensional representation? does "$ISO_0(3,1)$" mean the same as "$SO_0(3,1)$"? (SO stands for standard orthogonal, no idea about ISO).
– YCor
2 hours ago
@YCor I means inhomogeneous, the semidirect product with the translation group. massless representations with helicity s are defined in Wigner's classification of irreducible unitary representations of the Poincare group.
– Arnold Neumaier
1 hour ago
@YCor Infinite-dimensional unitary representations on a Hilbert space.
– Robert Furber
23 mins ago