The spinor-helicity formalism , also Weyl-van-der-Waerden-formalism after Hermann Weyl and Bartel Leendert van der Waerden , is an alternative mathematical formulation of quantum field theories based on the use of spinors and invariants of the special linear group instead of the use based on four-vectors and invariants of the Lorentz group .
Basics
Group theory of the Lorentz group
In 3 + 1 space-time -Dimensionen the real Lorentz group is isomorphic to complex specific linear array in two dimensions . This means that each group element of the Lorentz group can be assigned an element of the complex special linear group and each vector in the real four-dimensional space-time on which the Lorentz group operates, a matrix in the space of complex matrices on which the special linear group operated on. This transition occurs through the four Pauli matrices . Let be a four-vector, then:
![{\ displaystyle SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e4b51615b649a6f78ced10d5313c86da67033b)
![{\ displaystyle \ mathbb {R} ^ {1,3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b)
![2 \ times 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80)
![{\ displaystyle \ sigma ^ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3f88bdd05d08345945334cf491d972c2132493)
![{\ displaystyle k _ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194a4b8223ba71dc660d1198856fc19f5eb44d54)
![{\ displaystyle k ^ {a {\ dot {a}}} = \ sigma _ {\ mu} ^ {a {\ dot {a}}} k ^ {\ mu} = {\ begin {pmatrix} k_ {0 } + k_ {3} & k_ {1} - \ mathrm {i} k_ {2} \\ k_ {1} + \ mathrm {i} k_ {2} & k_ {0} -k_ {3} \ end {pmatrix} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ecd59978b38e0fad378eb09167d59690bcac976)
The Greek indices denote Lorentz indices that run from 0 to 3, while the Latin indices are called spinor indices and run from 1 to 2. The reverse transformation from to works via
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![{\ displaystyle a, {\ dot {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f36b1b6c174bd1e1d1cf0e3c3c7d2aeb20c1735)
![{\ displaystyle M_ {2} (\ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4b88668d8b0abccb0b28518e83ace17c2c4b2a)
![{\ displaystyle \ mathbb {R} ^ {1,3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b)
![{\ displaystyle k _ {\ mu} = {\ frac {1} {2}} {\ bar {\ sigma}} _ {{\ dot {a}} a} ^ {\ mu} k ^ {a {\ dot {a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00bc2fbbeb5352f06debbc8b89c5619f23fb3b48)
The Lorentz variable is translated via
![{\ displaystyle k \ cdot p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac114a14f65c0ece1cd3e5eb4e90d7591b4f68b)
![{\ displaystyle k ^ {\ mu} p _ {\ mu} = {\ frac {1} {2}} \ varepsilon _ {ba} \ varepsilon _ {{\ dot {a}} {\ dot {b}}} k ^ {a {\ dot {a}}} p ^ {b {\ dot {b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ab6892876d68f7945d2dde2013c48f97bd15bb8)
with the totally antisymmetric Levi Civita symbol . In particular,
![\ varepsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173)
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.
The group operation of an element of the Lorentz group with a Lorentz matrix translates as
![{\ displaystyle k ^ {\ nu} \ to k '^ {\ nu} = \ Lambda _ {\ mu} ^ {\ nu} k ^ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cf2e2a9a4965d9ffead5a42dedd154f01e2b67)
![\ Lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733)
![{\ displaystyle k ^ {a {\ dot {a}}} \ to k '^ {a {\ dot {a}}} = \ zeta _ {b} ^ {a} k ^ {b {\ dot {b }}} {\ tilde {\ zeta}} _ {\ dot {b}} ^ {\ dot {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a80db09a63afca33f08e1276fa55a3afbdb83352)
with . The matrices are for rotations around an axis with the angle![{\ displaystyle \ zeta, {\ tilde {\ zeta}} = \ zeta ^ {\ dagger} \ in SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/958f10ede4b8d30f7309b6af15b2589a5d0b0c48)
![\ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![{\ displaystyle \ zeta (R_ {i} (\ theta)) = \ exp (- \ mathrm {i} {\ tfrac {\ alpha} {2}} \ sigma _ {i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6668e334479dc7b073066c4d30987f67cf4fdc1)
and for Lorentz boosts along an axis with rapidity
![{\ displaystyle \ zeta (B_ {i} (\ theta)) = \ exp (- \ mathrm {\ tfrac {\ theta} {2}} \ sigma _ {i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d136555b78ce07d9283044419342170963cf8ca0)
where denotes the matrix exponential .
![\ exp](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1185b570f67b4221307626254f64f9e619e769)
This can be generalized to the complex Lorentz group that operates on . Then the isomorphism applies and does not necessarily have to be the same and it does not have to apply.
![{\ displaystyle SO (1,3, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc1dab6b748f01078d6ca7829541fcd0db73c5d)
![{\ displaystyle \ mathbb {C} ^ {1,3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b40cb1e94241994b2711c71d50292aa61b76a0)
![{\ displaystyle SO (1,3, \ mathbb {C}) \ cong SL (2, \ mathrm {C}) \ times SL (2, \ mathrm {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d79626ffc2b1a7ceb8a72c1a3cffabdd56b27a20)
![{\ displaystyle {\ tilde {\ zeta}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4152ca4eb43a81b79a495f535b0f5598ca4fcc0d)
![{\ displaystyle \ zeta ^ {\ dagger}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec29f5237a8e30e89d315417b4fb186a202e22c)
![{\ displaystyle \ kappa = {\ tilde {\ kappa}} ^ {\ dagger}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d862aea5a7b09f1e92c30a2286464b48f3ae9a2)
notation
From it follows that a light-like vector translates into a matrix without full rank . Since is the dimension of two, it follows provided that is. Therefore, the dyadic product can be written:
![{\ displaystyle k \ cdot k = \ det (k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e80e5a60d0c4a2185170d43e96cee010189d73)
![{\ displaystyle k ^ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646660a1fce65721c8e043b6dfe9d3c66c2889f9)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle \ operatorname {rg} k = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca50e3a58176206b10b40a250bbc5959237006d)
![k \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4367cc52bf0bebde550f396dde7bf07fa67bdd)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle k ^ {a {\ dot {a}}} = \ kappa ^ {a} {\ tilde {\ kappa}} ^ {\ dot {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3793231ac6a107ca219411af4ab659c316774f3f)
Both and are two-dimensional objects called spinors . The spinor is called holomorphic spinor, the spinor antiholomorphic spinor. An explicit representation of these spinors reads:
![\ kappa](https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d)
![{\ displaystyle {\ tilde {\ kappa}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ac4afc61631f192d7127c03745b279863cd51e)
![\ kappa](https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d)
![{\ displaystyle {\ tilde {\ kappa}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ac4afc61631f192d7127c03745b279863cd51e)
![{\ displaystyle \ kappa ^ {a} = {\ frac {1} {\ sqrt {k_ {0} + k_ {3}}}} {\ begin {pmatrix} k_ {0} + k_ {3} \\ k_ {1} + \ mathrm {i} k_ {2} \ end {pmatrix}} \ qquad {\ tilde {\ kappa}} ^ {\ dot {a}} = {\ frac {1} {\ sqrt {k_ { 0} + k_ {3}}}} {\ begin {pmatrix} k_ {0} + k_ {3} & k_ {1} - \ mathrm {i} k_ {2} \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93d468db3fd194d3c9304b9ebbd858f30ec23d0b)
In particular, the spinors can be respectively rescaled by a factor without this changing the matrix . It can be seen that the two spinors are adjoint as long as the vector is real. In the following it is assumed that all vectors that occur are light-like.
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
![t ^ {{- 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82182729910430d9930cc0cbf96a3f43050b381c)
![{\ displaystyle k ^ {a {\ dot {a}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2753e0b9e1182091df9c8ec35d91d3126b8da5c0)
![{\ displaystyle k ^ {\ mu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646660a1fce65721c8e043b6dfe9d3c66c2889f9)
A scalar product of two four-vectors can therefore be written as
![{\ displaystyle k \ cdot p = \ kappa ^ {a} \ varepsilon _ {ab} \ pi ^ {b} {\ tilde {\ kappa}} ^ {\ dot {a}} \ varepsilon _ {{\ dot { a}} {\ dot {b}}} {\ tilde {\ pi}} ^ {\ dot {b}} \ equiv \ langle \ kappa \ pi \ rangle [\ pi \ kappa]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfae8e92f5fd129c71157a9c455851258d6ef7f)
to be written. In this sense, the Levi-Civita symbols take on the role of the metric in the . It applies
![{\ displaystyle SL (2, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e4b51615b649a6f78ced10d5313c86da67033b)
-
and
Analogous to the Bra-Ket notation , the notation for the spinors is:
![{\ displaystyle \ kappa ^ {a} = \ kappa \ rangle \ quad \ kappa _ {a} = \ langle \ kappa \ quad {\ tilde {\ kappa}} ^ {\ dot {a}} = [\ kappa \ quad {\ tilde {\ kappa}} _ {\ dot {a}} = \ kappa]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be37b6088055856e38e0fa909446fb024e6cf545)
In particular, is due to the antisymmetry of the Levi-Civita symbol
-
.
A space- or time-like vector can always be by means of
![{\ displaystyle k = k ^ {\ flat} + {\ frac {k ^ {2}} {2k \ cdot p}} p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db1678f27f0bb659a16cc8b7676c05c1335d449)
can be decomposed into two light-like vectors. In this example it is called auxiliary vector.
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
Physical implication
Fermions
With the help of the spinor-helicity formalism, the solution of the Dirac equation is trivial. The Dirac equation is:
![{\ displaystyle (- \ mathrm {i} \ gamma ^ {\ mu} \ partial _ {\ mu} -m) \ psi = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f57284cc0312e824519c516079d40b53ffaac6)
Where is the momentum of the particle and the Dirac matrices . The approach or leads to
![p_ \ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/607f1ff42af33677dd20a66d6f81701f108a81ee)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![{\ displaystyle \ psi = u \ exp (- \ mathrm {i} px)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97ea45f3cd684d2bd5389d0b4fd4c3a9de54d19)
![{\ displaystyle \ psi = v \ exp (\ mathrm {i} px)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b60bfab5e775820ce652f6fc2231b4a014b9943)
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and
for particles or antiparticles. In the Weyl representation, the following applies
![{\ displaystyle \ gamma ^ {\ mu} = {\ begin {pmatrix} 0 & \ sigma ^ {\ mu} \\ {\ bar {\ sigma}} ^ {\ mu} & 0 \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec1dea359822388b461dacc0c7bed6394aa20a2)
On the mass shell is also space-like and must therefore be decomposed, so that the Dirac equation in the spinor-helicity formalism
![{\ displaystyle p ^ {2} = m ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7e338af54fca21f40cc6a819c94c9e485c75ad)
![{\ displaystyle {\ begin {pmatrix} \ mp m & \ pi \ rangle [\ pi + {\ frac {m ^ {2}} {\ langle \ pi \ mu \ rangle [\ mu \ pi]}} \ mu \ rangle [\ mu \\\ pi] \ langle \ pi + {\ frac {m ^ {2}} {\ langle \ pi \ mu \ rangle [\ mu \ pi]}} \ mu] \ langle \ mu & \ mp m \ end {pmatrix}} {\ begin {pmatrix} u \ rangle \\ u] \ end {pmatrix}} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ffc1b991b1cb0c503fa39e026987ec31c35a9e)
with an auxiliary "vector" is. It follows
![{\ displaystyle \ mu \ rangle [\ mu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501bf4eb23391988ff3de11e912a75c01bf816ff)
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and
as solutions to the eigenvalue problem. The Dirac spinors are normalized so that . In the massive case, the spinors are particularly dependent on the choice of the auxiliary vector; not in the massless case.
![{\ displaystyle {\ bar {\ psi}} \ psi = 2m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47ba9257f8d676effcba31c07e576ea4a8b6f17a)
Vector bosons
The Maxwell equations
![{\ displaystyle p _ {\ mu} \ epsilon ^ {\ mu} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0432c45f0b5a10a9a760b5314fa4b50bb56ac34)
with the polarization vector has the two solutions
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
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and
where applies. The normalization was chosen so that the two solutions are orthonormal.
![{\ displaystyle p ^ {2} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbf6836af07f8854f7b20cf2704b8e8e853dabf)
The Proca equation for massive vector bosons has the additional solution
![{\ displaystyle \ epsilon = {\ frac {1} {m}} \ left (\ pi \ rangle [\ pi - {\ frac {m ^ {2}} {\ langle \ pi \ mu \ rangle [\ mu \ pi]}} \ mu \ rangle [\ mu \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3ce85e22585c11fe60ce89fba708bb99f8bb00)
which corresponds to the longitudinal polarization mode.
Individual evidence
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↑ Eduardo Conde and Andrea Marzolla: Lorentz Constraints on Massive Three-Point Amplitudes . arxiv : 1601.08113 .
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^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 537 (English).
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↑ Timothy Cohen, Henriette Elvang and Michael Kiermaier: On-shell constructibility of tree amplitudes in general field theories . arxiv : 1010.0257 .