Representation theory of the Lorentz group

In physics , the representation theory of the Lorentz group is needed to describe elementary particles in relativistic quantum mechanics and to describe fields in quantum field theory .

Lorentz group

The Lorentz group is the group of linear maps of space-time that leave the Minkowski metric invariant ${\displaystyle t^{2}-x^{2}-y^{2}-z^{2}}$ ${\displaystyle \mathbb {R} ^{1+3}}$

${\displaystyle {\text{O}}(3,1):=\left\{A\in \mathbb {R} ^{4\times 4}:A{\begin{pmatrix}1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}}A^{T}={\begin{pmatrix}1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}}\right\}}$.

It has four interrelated components . The connected component of the neutral element is called . This component is superimposed by two . ${\displaystyle {\text{SO}}^{+}(3,1)}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

In particular, its Lie algebra is isomorphic to the Lie algebra sl (2, C) . ${\displaystyle {\mathfrak {so}}(3,1)}$

Finally dimensional representations

Representations of Lie algebra

The representation theory of sl (2, C) shows that every -linear, irreducible and finite-dimensional representation of is a so-called spin representation for a . This representation is -dimensional and there is an irreducible representation that is unique except for isomorphism for every integer or half- integer value . ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle m}$ ${\displaystyle m\in {\tfrac {1}{2}}\mathbb {Z} }$ ${\displaystyle (2m+1)}$ ${\displaystyle m}$ ${\displaystyle \pi _{m}}$

It then follows that every linear, irreducible and finite-dimensional representation of is of the form with integer or half-integer values . Here, the tensor product of two Lie algebra representations is defined by ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )={\mathfrak {so}}(3,1)}$ ${\displaystyle \pi _{m,n}:=\pi _{m}\otimes {\overline {\pi }}_{n}}$ ${\displaystyle m,n\in {\tfrac {1}{2}}\mathbb {Z} }$

${\displaystyle \pi _{m,n}(X):=I_{2m+1}\otimes {\overline {\pi }}_{n}(X)+\pi _{m}(X)\otimes I_{2n+1}}$

and denotes the too complex conjugate representation. (The corresponding Lie group representation is the tensor product of the first Lie group representation with the complex conjugate of the second.) ${\displaystyle {\overline {\pi }}_{n}}$ ${\displaystyle \pi _{n}}$

The representation is -dimensional and irreducible. ${\displaystyle \pi _{m,n}}$ ${\displaystyle (2m+1)(2n+1)}$

Projective representations

Every Lie algebra representation determines (according to Lie's second theorem ) a (real) representation of and thus a projective representation of . ${\displaystyle \pi _{m,n}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle \rho _{m,n}}$ ${\displaystyle {\text{SO}}^{+}(3,1)}$

If so, a projective representation of the entire Lorentz group can be continued. ${\displaystyle m=n}$ ${\displaystyle \rho _{m,n}}$ ${\displaystyle {\text{O}}(3,1)}$

This is not possible for , but at least one can then continue to an irreducible projective representation of . ${\displaystyle m\not =n}$ ${\displaystyle \rho _{m,n}\oplus \rho _{n,m}}$ ${\displaystyle {\text{O}}^{+}(3,1)}$

Representations

${\displaystyle {\text{SL}}(2,\mathbb {C} )}$is a double superposition of , where and are mapped onto the neutral element . A representation of corresponds to a representation (and not just a projective representation) of when it is also mapped onto the identity matrix . ${\displaystyle {\text{SO}}^{+}(3,1)}$ ${\displaystyle I_{2}}$ ${\displaystyle -I_{2}}$ ${\displaystyle I_{4}\in {\text{SO}}^{+}(3,1)}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SO}}^{+}(3,1)}$ ${\displaystyle -I_{2}=\left({\begin{smallmatrix}-1&0\\0&-1\end{smallmatrix}}\right)\in {\text{SL}}(2,\mathbb {C} )}$

It is easy to check that this is the case for the representations if and only if and are whole numbers. ${\displaystyle \rho _{m,n}}$ ${\displaystyle m}$ ${\displaystyle n}$

If so , then you get a representation of the full Lorentz group . ${\displaystyle m=n\in \mathbb {Z} }$ ${\displaystyle O(3,1)}$

Examples

In the following denote the projective representation of . ${\displaystyle (m,n)}$ ${\displaystyle (2m+1)(2n+1)}$ ${\displaystyle \rho _{m,n}}$ ${\displaystyle SO^{+}(3,1)}$

• (0, 0) is the scalar Lorentz representation used in relativistic scalar field theories .
• ${\displaystyle \left({\tfrac {1}{2}},0\right)}$is the projective representation of the left-handed Weyl spinors , that of the right-handed Weyl spinors. These two representations are not linear representations of the group . ${\displaystyle \left(0,{\tfrac {1}{2}}\right)}$ ${\displaystyle {\text{SO}}^{+}(3,1)}$
• ${\displaystyle \left({\tfrac {1}{2}},0\right)\oplus \left(0,{\tfrac {1}{2}}\right)}$is the Bispinor representation.
• ${\displaystyle \left({\tfrac {1}{2}},{\tfrac {1}{2}}\right)}$is the four-vector representation. The quadruple momentum of a particle is transformed according to this representation.
• ${\displaystyle (1,0)}$is the projective representation in the space of the self-dual 2-forms and the projective representation in the space of the anti-self-dual 2-forms. These two representations are not linear representations of the group . ${\displaystyle (0,1)}$ ${\displaystyle {\text{SO}}^{+}(3,1)}$
• (1, 0) ⊕ (0, 1) is the representation of a parity- invariant field of 2-shapes (i.e. curvature shapes ). The electromagnetic tensor field transforms according to this representation.
•   ${\displaystyle \left(1,{\tfrac {1}{2}}\right)\oplus \left({\tfrac {1}{2}},1\right)}$corresponds to the Rarita Schwinger field .
• (1, 1) is the spin-2 representation of a traceless symmetrical tensor field.

literature

• Brian C. Hall: Lie groups, Lie algebras, and representations. An elementary introduction. (= Graduate Texts in Mathematics. 222). Springer-Verlag, New York 2003, ISBN 0-387-40122-9 .
• Sigurður Helgason: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. (= Mathematical Surveys and Monographs. 83). Corrected reprint of the 1984 original. American Mathematical Society, Providence, RI 2000, ISBN 0-8218-2673-5 .
• Anthony W. Knapp: Representation theory of semisimple groups. An overview based on examples. (= Princeton Landmarks in Mathematics ). Reprint of the 1986 original. Princeton University Press, Princeton, NJ 2001, ISBN 0-691-09089-0 .
• ER Paërl: Representations of the Lorentz group and projective geometry. (= Mathematical Center Tracts. No. 25). Mathematical Center, Amsterdam 1969.
• W. Rühl: The Lorentz group and harmonic analysis. WA Benjamin, New York 1970, OCLC 797189612 .
• Steven Weinberg: The quantum theory of fields. Vol. I: Foundations. Cambridge University Press, Cambridge 2005, ISBN 0-521-55001-7 .

Individual evidence

1. ↑ Knapp, op.cit., Chapter II.3