[co-vector-of-Hermitian-tensor]
ๅฏผๅบ co-vector of Hermitian-tensor
็ฑไบ ๆฏ Hermitian ็ฉ้ต, ๆไปฅ , ๆไปฅ
ๅ ไธบ ็ Hermitian or ๅฏน ๅ ็งฏ็ self-adjoint, ๅฏไปฅ่ฎคไธบ ไฝ็จๅฏน็งฐๅฐไฝ็จๅจไธคไธช slot
vector ็ฉบ้ด็ base ็ปๅบ co-vector ็็ณปๆฐ
ไฝ็จๅฏผๅบๅฐๅฏน co-vector ็ไฝ็จๆฏ
ไนๅฏนๅบ dual base i.e. base of co-vector space ็ๅๆข
parity ๅฏนๅถไธ ๅฏผๅบไฝ็จ
็ฑปไผผๅฐ, ๅฏนไบ anti-Hermitian ไนๅฏไปฅๅฎไน co-vector
ๅฏนไบๅคๅ ฑ่ฝญไบ้ๅผ ้
[spinor-field-motivation]
-
formally ๅฐๅจ้ ๅฏนๅบๅฐ gradient ๅจ้ , ๅฐ ๅฏนๅบๅฐ spacetime-momentum-spinor-representation
-
formally ็จไบ ็ๆ็ co-vector ๅพๅฐ ๅบ
-
action + product rule + ๆฃๅบฆ้ + ่พน็้ถ + ็งฏๅไบๆฌกๅ ==> self-adjoint ็ฎๅญ
[massless-spinor-Lagrangian] alias [Weyl-Lagrangian]
or or
where ๆฏ็จ ็งฏๅ + ็ไบๆฌกๅ
่ตทไฝ็จ็ๅชๆ , ๅ ไธบ ๆฏๆฃๅบฆ้, ็จ Stokes ๅฎ็ + ่พน็้ถ
ๅๅ็ปๅบ linear part
[massless-spinor-equation] alias [Weyl-equation]
or or
็ฑปไผผไบ via , ๅฏน ๅผ ๅๅ็ญไปทไบๅฏน ๅผ ๅๅ
ๅฏไปฅ่งฃ้ไธบ (metric-dual ๅ) ๅบ็ gradient ๅจ้ , ๅคๅๅฐ, ๅจ้ไธๆ้็ไนๆณ
[Weyl-parity]
parity ๅฏนๅถไฝ็จ้ไฝฟ็จ ๆ้
parity ๅฏนๅถ็ eq
or or
[Weyl-eq-plane-wave]
ๅนณ้ขๆณข่งฃ with and
็บฟๆงๆน็จ with ่ฏดๆๆ้้ถ่งฃ, ่งฃ็ฉบ้ดๆฏไธ็ปด็, ่งฃๅฏไปฅๅไธบ with
[massive-spinor-Lagrangian] mass couple ๆ้็ไฝ็จ้, alias [Dirac-Lagrangian]
couple Weyl ๆ้ๅๅ ถ parity to
,
invariant non couple term
non couple term ๅฏน ๅๅ็ปๅบ
ๆ นๆฎ parity ไน้ด็ธไบๆตๆถ
invariant couple term
couple term ๅฏน ๅๅ็ปๅบ
- ๆดไฝๅฏน ๅๅ็ปๅบ
- ๆดไฝๅฏน ๅๅ็ปๅบ
- ๆถ decouple ๅฐไธคไธช parity ๅฏนๅถ็ massless-spinor
่ฟไธคไธช PDE ่ดๅซ
and as "square root of " [square-root-of-spacetime-Laplacian]
ๆดไฝ , square root of KG. ๅฆๆไธไธชๅบๆปก่ถณ Dirac eq, ๅๅฎๆปก่ถณ KG eq. ไป่ Dirac eq ๅฏไปฅ่ฟไผผๅฐ KG eq, ๆ่ ็ปง็ปญ่ฟไผผๅฐ Schrodinger eq. ไฝๆฏๆณจๆ, ๆญคๆถ็ธๆฏไบ ๅผ็ๆ ๅต, ็ไธๅไนๅคๅจไบ, or ่กจ็คบ็่งๅจ้็ฎๅญไผๆๅฝฑๅๅผๅ็่ชๆ้จๅ, ๆ ๅต็ฑปไผผไบ spinor-angular-momentum
ไฝ็จ้ ็ๅ จ้จๅๅฏผๆฐๆฏ้ถ , ็ปๅบ [massive-spinor-equation] , alias [Dirac-equation]
็ฑปไผผไบ via , ๅฏน ๅผ ๅๅ็ญไปทไบๅฏน ๅผ ๅๅ
couple term ๆขๆ , ไฝ็จ้ไป็ถ invariant. ไฝๆฏ eq ไธๅ่ฝๅ่งฃไธบ้ฃ็งๆด็ฎๅ็ๅฝขๅผ
Question forall invariant ็็ฉ้ต ๅคงๆฆๅชๆ
[Dirac-eq-plane-wave]
ๅนณ้ขๆณข่งฃ with and
ๅ่ ๆฏ็บฟๆงๆน็จๆไปฅ่งฃๆฏไธ้พ็, ่งฃ็ฉบ้ดๆฏไบ็ปด, ่งฃๅฏไปฅๅไธบ (ref-17, p.100)
ๅ ฑ่ฝญ็ธไฝๅนณ้ขๆณข ๆปก่ถณ็ๆกไปถๆฏ , ่งฃๅฏไปฅๅไธบ
็ฑปไผผ ็บฏ้ๅบ็ๆ ๅต, ๅฏไปฅๅฏนๅจ้ๆๅจ็ๅๆฒ้ข ่ฟ่ก ๅ ๅ [linear-superposition-of-Dirac-eq]
[squrae-root-of-spacetime-momentum-spinor-representation]
่ฝ็ถๅฏ่ฝๅฏไปฅ็จ Hermite ็ฉ้ต ็็นๅพๅผ ( ), ไฝ่ฟ้่ฟๆฏ็ดๆฅ่ฎก็ฎ. let or
==>
==> ไฝฟ็จ
==> ็ไบๆฌกๆน็จ , ่งฃ
==>
or
ไป็ถ Hermite. ่ฎก็ฎๅฏๅพ
Example
ๅ
ๅฆๆๅ ๅ
1,3 metric ๅนณๆนๆ น or ๅนณๆนๆ น . ไฝไฝ ไนๅฏไปฅ็จ ๅพๅฐ็ๆญฃ็ๅนณๆน
็ฑไบ ๆไปฅๅๆข ไธๆฅ่ช ๆขๅๆ
[motivation-of-gauge-field]
ๅฟฝ็ฅไบไธไบ้ฎ้ข
ๅๅฐๅฝฑๅ ้ฅไธๆฏ่ฏๅฎไน็
ไฝ็ฉถ็ซ้่ฆ ๅบ่ฟๆฏ ๅบ? ๅบๆๅๅฐ ๅบ็้ๆฉๅคชๅคไบ (or ๆๅๅฐ ๅบ), ๅ จ้จๆๅ็้ๆฉๅฝขๆ ๅบ
่ ๆๅๅฐ ๅชๆไธค็ง
ๅจๅผฏๆฒๆตๅฝขไธ, ็่ณๅฏ่ฝ้ฝๆฒกๆๅ จๅฑ็ๅๅผๅๆๅ
ๅบๆๅๅฐ ๅบ็ๆๅๆนๅผ็ๆนๅๅฏนๅบ "ๆนๅ่ง่", ็ปๆ้ ไนไธ ๆฅๆนๅ่ง่
ๅฆๆ้่ฆไฝ็จ้็ๅฎๆๆตๆด็ฎๅ, ๅ็จ ่ง่ๅๆข่ไธๆฏ . ไธๆนๅ Lagrangian ไฝ็จ้, ่ฟไฝฟๅพๅจ่ฎก็ฎๅฎๆๆตๆถๅๅพ็ฎๅ (cf. ็บฏ้ๅบ่ฎก็ฎ ๅฏน็งฐๆง็ 4 ็ตๆต็ๆ ๅต)
ๆนๅ่ง่ไธๅ ผๅฎนไบๅจไธๅๆ ไธญๆฑๅ็ฉบ้ด็ๅฏผๆฐ, ๆไปฅ่ฆๅผๅ ฅ้ขๅค็็ปๆ โ connection
ๆๅพๅคๅฏ่ฝ็่็ป. ๆง่ดจๅฅฝ็่็ปๆฏๆฒ็ๆๅฐ็ cf. electromagnetic-field
ๅผฏๆฒๆถ็ฉบ็ ไธๅฏไปฅ็ดๆฅๅจ principal bundle (ๆญฃไบคๆ ๆถไธ) ็ไธๅๆ ไธญๅฎไน. ไฝฟ็จ ๅฏนๅบ, ๆข ไธๅๆ ๆถ, ่ชๅจๅฏนๅบๅฐๆข ไธๅๆ
ๅจๅผฏๆฒๆถ็ฉบ, ้่ฆๅค็ๆ้ๅบ็ๅฏน metric ็ๅๅๅฏผๆฐ, ๅฎๅฏผๅบ่ชๅฏนๅๅ้ๅบ็ metric-connection
ๅฏนไบๆ้, ๅฏ่ฝ่ฆ็จ orthonormal frame ่ไธๆฏ coordinate frame i.e. ็จ principal bundle. ่ฟๅฏน่ฎก็ฎๅๅๅฏผๆฐๅผๅ ฅไบๆฐ็้บป็ฆ?
็่ณๅฆๆๆถ็ฉบๅบๆตๅฝข็ๆๆ้ๅนณๅก, ๅฏนไบ่ง่ๅบ่ฟๅฏ่ฝๅญๅจไธๅ็ bundle type
ไธไธช้ฎ้ขๆฏ, ไธๅไบๆ้ๅบ, ่ง่ไธไผผไนไธ่ฝ็ดๆฅ่็ณปๅฐๅไธ
ไผผไน่ฆๅๆถ่่ๅบไบๅบๆตๅฝข็ๆๆ็ฑปๅ็ bundle type
ๅจ homotopy ๆไนไธ, ๅชๆไธ็ง็ฑปๅ็ bundle type
[spinor-field-gauge-imaginary-automorphism]
่ฝ็ถไปฃไปทๆฏไฝฟ็จ ็ปดๆถ็ฉบ, ๅ็ฉบ้ด , ไฝๅฆๆ่่ๅ ซๅ ๆฐ , ๅฆๆ้ๅไธไธชๅไฝ่ๆฐๅ ๆฅๆ้ spinor Lagrangian, ้ฃไนๅ ซๅ ๆฐ็ ่ๆฐๅ ๅฏน็งฐ็พค ๅฏน ็ isotropy ๅฏผๅบๅ่งฃ , isotropy ็พคๅๆๅฐ , ไธๅฏผๅบๅจ ๅจ ไธ็ไฝ็จ. cf. (ref-19, th.4)
ๆณจๆๅ ซๅ ๆฐ่ๆฐๅ ๅฏน็งฐ็พค ไธๆฏ . ๆไปฅ่ฟไธๆฏไผ ็ปๆไนไธ็ group of gauge connection
ๆฎ่ฏด ๆฏๅผบ็ธไบไฝ็จ็่ง่็พค. ่ฟ้็ๅ ซๅ ๆฐๆนๆณ็ไผๅฟๆฏ, ไธๅ้่ฆ้ขๅคๅ่ฎพ ๅ ็ๅ็ฌๅญๅจไธๅญ็ฉบ่ๆฅๅฐๅฏน ๅ Dirac spinor ่ฟ่กๅผ ้ๆไฝ. ๅ ็ๅผ ้ๆฅ่ช ็ connection ไฝ็จไบๆ้ or .
ไธญๅๆๅฐ ็ isotropy ไผๆนๅ ็ธไฝ or ่ง่ๅๆข็ ่ง่ๅบ, ๅฏผๅบ ไฝ็จๅจ ้จๅ. ๆฏๅฆๅบ่ฏฅ่ฏด, ๆไธ็ง ่ง่ๅๆข็ or ่ง่ๅบ็ ่ง่ๅๆข็ ่ง่ๅบ, ็ถๅๅๆฌกๅ็ต็ฃๅบ้ฃๆ ทๅผๅ ฅๆๅฐๆฒ็็ YangโMills eq?
also cf. (ref-20, th.6)
Question ๅๅ ๆฐ ็ๆ ๅตๅฆไฝ? ็ๆ ๅต็่ๆฐๅ ๅฏน็งฐ็พคๆฏ . ๅบๅฎไธไธช่ๆฐๅ ไนๅ, ๆฏไปไน?
็ๆ ๅต็่ๆฐๅ ๅฏน็งฐ็พคๆฏ . ๆพ็ถๅบๅฎไธไธช่ๆฐๅ ๅฐฑๅๆ
[spin-connection]
ๅจๅไธ metric ๅฏผๅบ็ frame bundle ๅ metric-connection ๅฏผๅบ็ frame bundle ็ connection ่กจ็ฐไธบ is locally type , ไฝ็จไบ ๅๅ้ๅบ by
ๅฏผๅบๅฐ spin-connection ็ๆนๅผๆฏ, ๅจ orthonormal-frame ๅฐ induced metric-connection ็ ้จๅๅฏนๅบๅฐ , ๅพๅฐ ไธ็ connection, locally type , ไฝ็จไบ spinor field by with
่ฝ็ถ่ชๆ่กจ็คบ็ Pauli matrix ็ๅฎไน้่ฆ , ไฝๆฏ่ฟไนๅ ๅ Lie algebra ้ฝๅฏไปฅ ็ "ๅนณๆน" ๆฅ่กจ็คบ
spin-connection also denoted by
[motivation-of-scalar-field] can scalar field be related to tautological bundle of projective-lightcone ?
ๆ นๆฎ ๆถ็ฉบๆตๅฝข็ๆ้ๅบ็ๆฆๅฟต, "ๆ่ฝฌ 720 ๅบฆ", "parity" ๅบ่ฏฅๅ็ๅจๅ็ฉบ้ดๆ้ ่ไธๆฏๅ็ๅจๆถ็ฉบๆตๅฝข
ๅ ไธบๆถ็ฉบๆตๅฝข็ๅ็ฉบ้ด้ฝๆฏ , ๆไปฅๆ้ๅบๅฏไปฅๆจๅนฟๅฐไธ่ฌๆถ็ฉบๆตๅฝข?
[spinor-on-Lorentz-manifold] Question
massless-spinor-action
massless-spinor-equation
ๆ่ฟๆฒก้ช่ฏ่ฟ็งๅฎไนๅจๆฆๅฟตไธๆฏๅฆๅ็. ๅฏนๆฏๅนณ็ดๆถ็ฉบ, ๅฐ่ฏ่ฏๆๆ่ฏๅฆ
- ๆฏ self-adjoint
- ๅชๆ ๅฏนไฝ็จ้ๅๅ่ตทไฝ็จ
- i.e. square-root-of spacetime Laplacian (ๆดๆฅ่ฟๅๅ้ๅบ็ Laplacian ่ไธๆฏ็บฏ้ๅบ็)
massive-spinor-Lagrangian
massive-spinor-equation
Question ๅช่ฆๅฑ้จไป quotient ๅๅฐ , ๅฐฑ่ฝๅ้ฟ่ฟ็ปญๅ จๅฑๅๅผ lift to ็้ฎ้ข
ๆไปฌ็ฅ้ KG eq ๆ non-relativity ่ฟไผผๆ้ . massive-spinor ๆ้ ๅญๅจ non-relativity ่ฟไผผๆ้ ๅ?
static ไธ้่ฆ non-relativity ่ฟไผผๆ้ , ๅฐฝ็ฎกๆ ็ๅญๅจ, ๅฐฑๅ static ็ต็ฃๅบๆน็จ้ฃๆ ทไนไธ้่ฆ non-relativity ่ฟไผผๆ้ . ่ฟๅฏน KG ๆน็จไนๆฏๅฆๆญค
let static
static massless spinor eq
static massive spinor eq
ๅฎไปฌๅฏไปฅ couple ๅฐ static electromagnetic gauge potential or just static electric or just static magnetic
ๅจๅญๅจ electromagnetic potential ๆถ, massless ็ parity dual ๅฏ่ฝไธๅ, ไพๅฆ just static electric
electromagnetic potential = 0 ๆถ, parity dual eq ๆฏ็ธๅ็