1. notice
  3. English
  5. logic-topic
  6. 1. logic
  7. 2. basic
  8. 3. map
  9. 4. order
  10. 5. combinatorics
  12. calculus
  13. 6. real-numbers
  14. 7. limit-sequence
  15. 8. division-algebra
  16. 9. Euclidean-space
  17. 10. Minkowski-space
  18. 11. polynomial
  19. 12. analytic-Euclidean
  20. 13. analytic-struct-operation
  21. 14. ordinary-differential-equation
  22. 15. volume
  23. 16. integral
  24. 17. divergence
  25. 18. limit-net
  26. 19. topology
  27. 20. compact
  28. 21. connected
  29. 22. topology-struct-operation
  30. 23. exponential
  31. 24. angle
  33. geometry
  34. 25. manifold
  35. 26. metric
  36. 27. metric-connection
  37. 28. geodesic-derivative
  38. 29. curvature-of-metric
  39. 30. Einstein-metric
  40. 31. constant-sectional-curvature
  41. 32. simple-symmetric-space
  42. 33. principal-bundle
  43. 34. group
  44. 35. stereographic-projection
  45. 36. Hopf-bundle
  47. field-theory
  48. 37. point-particle-non-relativity
  49. 38. point-particle-relativity
  50. 39. scalar-field
  51. 40. scalar-field-current
  52. 41. scalar-field-non-relativity
  53. 42. projective-lightcone
  54. 43. spacetime-momentum-spinor-representation
  55. 44. Lorentz-group
  56. 45. spinor-field
  57. 46. spinor-field-current
  58. 47. electromagnetic-field
  59. 48. Laplacian-of-tensor-field
  60. 49. Einstein-metric
  61. 50. interaction
  62. 51. harmonic-oscillator-quantization
  63. 52. spinor-field-misc
  64. 53. reference
  66. ไธญๆ–‡
  67. 54. notice
  69. ้€ป่พ‘
  70. 55. ้€ป่พ‘
  71. 56. ๅŸบ็ก€
  72. 57. ๆ˜ ๅฐ„
  73. 58. ๅบ
  74. 59. ็ป„ๅˆ
  76. ๅพฎ็งฏๅˆ†
  77. 60. ๅฎžๆ•ฐ
  78. 61. ๆ•ฐๅˆ—ๆž้™
  79. 62. ๅฏ้™คไปฃๆ•ฐ
  80. 63. Euclidean ็ฉบ้—ด
  81. 64. Minkowski ็ฉบ้—ด
  82. 65. ๅคš้กนๅผ
  83. 66. ่งฃๆž (Euclidean)
  84. 67. ่งฃๆž struct ็š„ๆ“ไฝœ
  85. 68. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  86. 69. ไฝ“็งฏ
  87. 70. ็งฏๅˆ†
  88. 71. ๆ•ฃๅบฆ
  89. 72. ็ฝ‘ๆž้™
  90. 73. ๆ‹“ๆ‰‘
  91. 74. ็ดง่‡ด
  92. 75. ่ฟž้€š
  93. 76. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  94. 77. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  95. 78. ่ง’ๅบฆ
  97. ๅ‡ ไฝ•
  98. 79. ๆตๅฝข
  99. 80. ๅบฆ่ง„
  100. 81. ๅบฆ่ง„็š„่”็ปœ
  101. 82. Levi-Civita ๅฏผๆ•ฐ
  102. 83. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  103. 84. Einstein ๅบฆ่ง„
  104. 85. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  105. 86. simple-symmetric-space
  106. 87. ไธปไธ›
  107. 88. ็พค
  108. 89. ็ƒๆžๆŠ•ๅฝฑ
  109. 90. Hopf ไธ›
  111. ๅœบ่ฎบ
  112. 91. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  113. 92. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  114. 93. ็บฏ้‡ๅœบ
  115. 94. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  116. 95. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  117. 96. ๅ…‰้”ฅๅฐ„ๅฝฑ
  118. 97. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  119. 98. Lorentz ็พค
  120. 99. ๆ—‹้‡ๅœบ
  121. 100. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  122. 101. ็”ต็ฃๅœบ
  123. 102. ๅผ ้‡ๅœบ็š„ Laplacian
  124. 103. Einstein ๅบฆ่ง„
  125. 104. ็›ธไบ’ไฝœ็”จ
  126. 105. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  127. 106. ๆ—‹้‡ๅœบๆ‚้กน
  128. 107. ๅ‚่€ƒ

note-math

cf. Action of spinor field

  • spinor-current-translation-of-spacetime

massless spinor Lagrangian

massive spinor Lagrangian

Since only the part of the spinor field action plays a role, a type theory can also be used.

Similar to the case of scalar field, try to calculate the energy-momentum-tensor

The solution of the spinor eq or makes

==> [energy-momentum-tensor-massless-spinor]

or component form

For massive, the derivative of the mass term with respect to is zero + the action with solution is still zero , so that the energy-momentum tensor is not affected by the mass term.

[energy-momentum-tensor-massive-spinor]

or component form

All are quantities with zero divergence

  • spinor-energy

Fix coordinates and consider to be an integrable quantity in . Define energy for massless-spinor

Similar to the case of scalar field, time invariant by

Example Plane wave, similar to the case of scalar field

massive-spinor energy

  • spinor-current-rotation-boost-of-spacetime

Using the product rule of differentiation, separate into two parts

  • The domain differentiation part is still similar to the case of scalar field

or

or

  • The codomain differentiation part

The ฮด diffeomorphism given by the Lorentz-Lie-algebra is . (cf. square-root-of-Lorentz-Lie-algebra)

+ product rule + gives

forall ==> integrand is zero ==> divergence-free quantity

or

  • [spinor-angular-momentum]

The angular momentum of domain and codomain combined, spinor-angular-momentum is

or

The case of massive-spinor is similar. It should be possible to prove by calculation that angular momentum is not affected by the mass term.

or

  • current-gauge-spinor

let be a solution to the spinor eq. Phase change and its ฮด change belong to boundary fixed variations near the solution, so

Using product rule + divergence-free quantity + boundary zero

for all value function , so

[current-gauge-spinor] is called the 4-current of massless-spinor

Similarly, for massive-spinor, the 4-current is

[conserved-spatial-integral-charge-spinor] Fixing coordinates, consider as a quantity integrable over

time invariant by

In fact, one can choose a spacetime decomposition coordinate and write the spinor eq in the form of unitary evolution of charge where is self-adjoint for quadratic form + integral

(After dropping ) charge alias probability density or particle number density or electric charge density

The case of massive-spinor is

Similar to the massless case, the spinor eq can be written in the form of charge unitary evolution.

conserved-current on mainfold โ€ฆ