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

note-math

Prop Generally, for , are not equivalent in the sense of changing coordinates: There does not exist ,

Proof

Eigenvalues remain unchanged under coordinate transformation

Generally, have different eigenvalues

Example

Prop are equivalent, are equivalent

use

Its complex conjugate [conjugate-representation]

The above only works for

Second-order tensor, with one undergoing complex conjugation

can be decompose to

[Hermitian-tensor]

[anti-Hermitian-tensor]

The conjugate modification in the other direction is similar

[Hermitian-tensor-induced-linear-map] The induced action of on :=

[matrix-description-of-Hermitian-tensor]

Use tensor base

Corresponding to the matrix representation

Or written in Dirac notation

The version without complex conjugation

notation-overload: The space of matrix representation is also denoted as

Hermitian matrix

anti-Hermitian matrix

For , since , the dimension of anti-Hermitian is higher than Hermitian

[Hermitian-tensor-induced-linear-map-matrix] The matrix representation of

preserves the decomposition into Hermitian and anti-Hermitian

For general , there is also

The "matrix" representation of needs to be handled separately, the composition of cannot be represented as usual matrix multiplication. It can still make well-defined

[spacetime-momentum-spinor-representation]

( represents "momentum" or "velocity" or tangent vector)

Bijection

metric

let and , action

Because multiplication is non-commutative, the general definition of for is problematic. However, the definition of does not require general multiplicative commutativity. At this time, is defined as . This is not a good notation because it may not be generalized to

are also the spinor lifts of . Similarly, is also the spinor lift of

Example [Pauli-matrix] alias [sigma-matrix]

for

  • time-like
  • light-like
  • space-like

(when generalized to , it corresponds to all imaginary elements)

  • is orthonormal base

Question What is the cognitive motivation for these constructions of ?

  • acts on lifts to act on
  • action, denoted as
  • action, denoted as

[square-root-of-Lorentz-group]

act on is some kind of "square" of , i.e. or represents the "real part" or "symmetric part"

Thus are some kind of "square root" of act on

[square-root-of-spacetime-metric-1] (inspired by ref-14, ch.11)

. Note that it is not alternating for

metric with is some kind of "square" of , i.e.

quadratic-form is

cf. Pauli-matrix

The calculation shows that is correct for . For , use sum

orthogonal of sigma matrix can also be obtained through calculation, thus

Thus is some kind of "square root" of metric

Question Still no direct intuitive calculation equation โ€ฆ

[spacetime-momentum-aciton-spinor-representation]

let .

where is Lorentz-group-spinor-representation

is spacetime-momentum-spinor-representation

Then there is a homomorphism

Proof Use the correspondence of 3 rotations, 3 boosts

[spinor-representation-adjoint]

Proof

use 3 boost, 3 rotation

use

,

Prop Use spacetime-momentum-spinor-representation for , + projection gives projective-lightcone

Therefore the following are equivalent

  • act on via
  • act on via

Proof

with (requires associative law of multiplication?)

Given

in ,

let

Also need to calculate

In order to get , compare norm, phase

norm

phase

so let with

Generally . Compare to get

[parity]

parity corresponds to vs representation, or vs , cf. conjugate-representation

let .

parity corresponds to space inversion

corresponds to time inversion

parity corresponds to trace or determinant reversal

[determinant-reversal]

let

determinant reversal with

[trace-reversal] := . or .

==> determinant reversal is the same as trace reversal

[square-root-of-spacetime-metric-2] a "square root" of metric

let .

give

Also have

for Pauli-matrix

  • or

  • , for (because parity is spatial inversion)

A better explanation of this "square root"?

Direct matrix multiplication without parity will give the square root of the metric, with ,

[square-root-of-Lorentz-Lie-algebra] "square root" of spacetime Lie-algebra

where is Lorentz-Lie-algebra

Proof

  • is ฮด rotation in where is any cyclic

  • where

Question A better explanation? Representation?

[property-of-parity]

  • i.e. parity preserve Hermitian

[parity-Euclidean-invariant] parity commutes with spatial action . In , it manifests as and commutes with . let

Generally, they do not commute, for example, certainly does not commute with the time-changing part in

let

or

or

[parity-reverse-boost] The effect of parity on the Lie-algebra is that it does not change ฮด rotation, but multiplies ฮด boost by

[Euclidean-spinor]

replace lightcone with just sphere acted by and

replace with , with

use trace-free Hermitian