Abstract

The geometric algebra of space–time is used to write the addition of nonparallel boosts, usually a cumbersome and inelegant process, in a compact form similar to that of the familiar addition law for parallel boosts. This compact expression is related to that used by Vigoureux [ J. Opt. Soc. Am. A 8, 1697 ( 1991); J. Opt. Soc. Am. A 9, 1313 ( 1992); J. Phys. A 26, 385 ( 1993); Am. J. Phys. 61, 707 ( 1993)] for combining reflection coefficients for stratified media, thus generalizing and strengthening the links between the formalisms used in special relativity theory and in optics. The comparison is then shown to hold when the addition schemes are performed to an arbitrary number of iterations.

© 1995 Optical Society of America

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References

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  1. P. Pellat-Finet, “Geometrical approach to polarization optics. I. Geometrical structure of polarized light,” Optik (Stuttgart) 87, 27–33 (1991).
  2. P. Pellat-Finat, M. Bausset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart)90, 101–106 (1992).
  3. T. Opatrný, J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
    [CrossRef]
  4. B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
    [CrossRef] [PubMed]
  5. D. Han, Y. S. Kim, “Special relativity and interferometers,” Phys. Rev. A 37, 4494–4496 (1988).
    [CrossRef] [PubMed]
  6. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  7. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992), Eqs. (2) and (3).
    [CrossRef]
  8. J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoid of amplitudes and a phase ‘Thomas precession’,” J. Phys. A 26, 385–393 (1993), Eqs. (4) and (5).
    [CrossRef]
  9. J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
    [CrossRef]
  10. A. C. Hirschfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986), Eqs. (1) and (4).
    [CrossRef]
  11. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 363–368.
  12. If the Thomas rotation is required, then it is easily accumulated at each iteration, since the rotation is calculated when it is factored out. This is not discussed here but would be easily incorporated into a computer program, for example.
  13. A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991), Eqs. (37a) and (37b).
    [CrossRef]
  14. C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
    [CrossRef]
  15. S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
    [CrossRef]
  16. A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
    [CrossRef]
  17. A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
    [CrossRef]
  18. S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
    [CrossRef]
  19. D. Hestenes, Space–Time Algebra (Gordon & Breach, New York, 1966).The Clifford algebra developed on flat space–time with a Minkowski metric is referred to as the geometric algebra of space–time, a name originally given by Hestenes.
  20. P. Lounesto, “Clifford algebras and Hestenes spinors,” Found. Phys. 23, 1203–1237 (1993).
    [CrossRef]
  21. A. W. M. Dress, T. F. Havel, “Distance geometry and geometric algebra,” Found. Phys. 23, 1357–1374 (1993).
    [CrossRef]
  22. S. L. Altmann, Rotations, Quarternions and Double Groups (Oxford U. Press, Oxford, UK, 1986), Chap. 12.
  23. P. Lounesto, “Scalar products of spinors and an extension of Brauer–Wall groups,” Found. Phys. 11, 721–740 (1981), Example 1.
    [CrossRef]
  24. B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).The pseudonorm is discussed on p. 15 and also pp. 70–71.
  25. Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics, rev. ed. (North-Holland, Amsterdam, 1982), pp. 64–68.
  26. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Eqs. (11.27).

1993 (10)

T. Opatrný, J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoid of amplitudes and a phase ‘Thomas precession’,” J. Phys. A 26, 385–393 (1993), Eqs. (4) and (5).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
[CrossRef]

S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
[CrossRef]

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

P. Lounesto, “Clifford algebras and Hestenes spinors,” Found. Phys. 23, 1203–1237 (1993).
[CrossRef]

A. W. M. Dress, T. F. Havel, “Distance geometry and geometric algebra,” Found. Phys. 23, 1357–1374 (1993).
[CrossRef]

1992 (2)

P. Pellat-Finat, M. Bausset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart)90, 101–106 (1992).

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992), Eqs. (2) and (3).
[CrossRef]

1991 (3)

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991), Eqs. (37a) and (37b).
[CrossRef]

P. Pellat-Finet, “Geometrical approach to polarization optics. I. Geometrical structure of polarized light,” Optik (Stuttgart) 87, 27–33 (1991).

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

1988 (1)

D. Han, Y. S. Kim, “Special relativity and interferometers,” Phys. Rev. A 37, 4494–4496 (1988).
[CrossRef] [PubMed]

1986 (2)

A. C. Hirschfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986), Eqs. (1) and (4).
[CrossRef]

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

1981 (1)

P. Lounesto, “Scalar products of spinors and an extension of Brauer–Wall groups,” Found. Phys. 11, 721–740 (1981), Example 1.
[CrossRef]

Altmann, S. L.

S. L. Altmann, Rotations, Quarternions and Double Groups (Oxford U. Press, Oxford, UK, 1986), Chap. 12.

Bausset, M.

P. Pellat-Finat, M. Bausset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart)90, 101–106 (1992).

Choquet-Bruhat, Y.

Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics, rev. ed. (North-Holland, Amsterdam, 1982), pp. 64–68.

DeWitt-Morette, C.

Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics, rev. ed. (North-Holland, Amsterdam, 1982), pp. 64–68.

Dillard-Bleick, M.

Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics, rev. ed. (North-Holland, Amsterdam, 1982), pp. 64–68.

Doran, C.

C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
[CrossRef]

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

Doran, C. J. L.

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
[CrossRef]

S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
[CrossRef]

Dress, A. W. M.

A. W. M. Dress, T. F. Havel, “Distance geometry and geometric algebra,” Found. Phys. 23, 1357–1374 (1993).
[CrossRef]

Grossel, Ph.

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

Gull, S.

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
[CrossRef]

Gull, S. F.

S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
[CrossRef]

Han, D.

D. Han, Y. S. Kim, “Special relativity and interferometers,” Phys. Rev. A 37, 4494–4496 (1988).
[CrossRef] [PubMed]

Havel, T. F.

A. W. M. Dress, T. F. Havel, “Distance geometry and geometric algebra,” Found. Phys. 23, 1357–1374 (1993).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 363–368.

Hestenes, D.

D. Hestenes, Space–Time Algebra (Gordon & Breach, New York, 1966).The Clifford algebra developed on flat space–time with a Minkowski metric is referred to as the geometric algebra of space–time, a name originally given by Hestenes.

Hirschfeld, A. C.

A. C. Hirschfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986), Eqs. (1) and (4).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Eqs. (11.27).

Kim, Y. S.

D. Han, Y. S. Kim, “Special relativity and interferometers,” Phys. Rev. A 37, 4494–4496 (1988).
[CrossRef] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

Lasenby, A.

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
[CrossRef]

Lasenby, A. N.

S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
[CrossRef]

Lounesto, P.

P. Lounesto, “Clifford algebras and Hestenes spinors,” Found. Phys. 23, 1203–1237 (1993).
[CrossRef]

P. Lounesto, “Scalar products of spinors and an extension of Brauer–Wall groups,” Found. Phys. 11, 721–740 (1981), Example 1.
[CrossRef]

McCall, S. L.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

Metzger, F.

A. C. Hirschfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986), Eqs. (1) and (4).
[CrossRef]

Opatrný, T.

T. Opatrný, J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

Pellat-Finat, P.

P. Pellat-Finat, M. Bausset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart)90, 101–106 (1992).

Pellat-Finet, P.

P. Pellat-Finet, “Geometrical approach to polarization optics. I. Geometrical structure of polarized light,” Optik (Stuttgart) 87, 27–33 (1991).

Perina, J.

T. Opatrný, J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

Schutz, B.

B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).The pseudonorm is discussed on p. 15 and also pp. 70–71.

Ungar, A. A.

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991), Eqs. (37a) and (37b).
[CrossRef]

Vigoureux, J. M.

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoid of amplitudes and a phase ‘Thomas precession’,” J. Phys. A 26, 385–393 (1993), Eqs. (4) and (5).
[CrossRef]

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992), Eqs. (2) and (3).
[CrossRef]

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

Yurke, B.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

Am. J. Phys. (3)

J. M. Vigoureux, Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[CrossRef]

A. C. Hirschfeld, F. Metzger, “A simple formula for combining rotations and Lorentz boosts,” Am. J. Phys. 54, 550–552 (1986), Eqs. (1) and (4).
[CrossRef]

A. A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824–834 (1991), Eqs. (37a) and (37b).
[CrossRef]

Found. Phys. (7)

C. Doran, A. Lasenby, S. Gull, “States and operators in the spacetime algebra,” Found. Phys. 23, 1239–1264 (1993).
[CrossRef]

S. F. Gull, A. N. Lasenby, C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,” Found. Phys. 23, 1329–1356 (1993).
[CrossRef]

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “A multivector derivative approach to Lagrangian field theory,” Found. Phys. 23, 1295–1327 (1993).
[CrossRef]

S. Gull, A. Lasenby, C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,” Found. Phys. 23, 1175–1201 (1993).
[CrossRef]

P. Lounesto, “Scalar products of spinors and an extension of Brauer–Wall groups,” Found. Phys. 11, 721–740 (1981), Example 1.
[CrossRef]

P. Lounesto, “Clifford algebras and Hestenes spinors,” Found. Phys. 23, 1203–1237 (1993).
[CrossRef]

A. W. M. Dress, T. F. Havel, “Distance geometry and geometric algebra,” Found. Phys. 23, 1357–1374 (1993).
[CrossRef]

J. Math. Phys. (1)

A. N. Lasenby, C. J. L. Doran, S. F. Gull, “Grassman calculus, pseudoclassical mechanics and geometric algebra,” J. Math. Phys. 34, 3683–3712 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoid of amplitudes and a phase ‘Thomas precession’,” J. Phys. A 26, 385–393 (1993), Eqs. (4) and (5).
[CrossRef]

Optik (Stuttgart) (2)

P. Pellat-Finet, “Geometrical approach to polarization optics. I. Geometrical structure of polarized light,” Optik (Stuttgart) 87, 27–33 (1991).

P. Pellat-Finat, M. Bausset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart)90, 101–106 (1992).

Phys. Lett. A (1)

T. Opatrný, J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

Phys. Rev. A (2)

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

D. Han, Y. S. Kim, “Special relativity and interferometers,” Phys. Rev. A 37, 4494–4496 (1988).
[CrossRef] [PubMed]

Other (7)

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), pp. 363–368.

If the Thomas rotation is required, then it is easily accumulated at each iteration, since the rotation is calculated when it is factored out. This is not discussed here but would be easily incorporated into a computer program, for example.

D. Hestenes, Space–Time Algebra (Gordon & Breach, New York, 1966).The Clifford algebra developed on flat space–time with a Minkowski metric is referred to as the geometric algebra of space–time, a name originally given by Hestenes.

S. L. Altmann, Rotations, Quarternions and Double Groups (Oxford U. Press, Oxford, UK, 1986), Chap. 12.

B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).The pseudonorm is discussed on p. 15 and also pp. 70–71.

Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, Manifolds and Physics, rev. ed. (North-Holland, Amsterdam, 1982), pp. 64–68.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Eqs. (11.27).

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Figures (2)

Fig. 1
Fig. 1

Ray diagram for three media and hence two interfaces. The planes labeled Zj are included for the reference of phase, and the point Q is arbitrarily chosen as zero phase. E denotes the various field amplitudes, with E0 as the incident amplitude.

Fig. 2
Fig. 2

Illustration of the composition of boosts. (a) The coordinate systems Σ′ and Σ″ are such that at t = 0 observers using either system agree that the coordinate axes are parallel and the origins coincide. Clearly, then, these observers measure the same value for ϕ2. (b) Same as (a), except that a coordinate axis is chosen to be parallel to v ̂ 2 and hence ϕ1 = 0. The origins of all three coordinate systems are coincident at t = 0. (c) The coordinate system Σ″ appears to an observer using the coordinate system Σ to be rotated by . This is the Thomas rotation. An observer using Σ′would measure the angle made between the two boosts as ϕ2, whereas an observer using Σ would measure ϕ2′. This is the aberration effect.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

V 2 υ 1 υ 2 = υ 1 + υ 2 1 + υ 1 υ 2
R 2 = R 1 R 2 = R 1 + R 2 1 + R 2 R ¯ 1 ,
R j r j , j + 1 exp [ i ( β 1 + β 2 + + β j ) ] , R ¯ j r j , j + 1 exp [ + i ( β 1 + β 2 + + β j ) ] ,
R 2 = R 1 R 2 = R 1 + R 2 1 + R 2 R ¯ 1 ,
V 2 = V 1 + V 2 1 + V 2 V ¯ 1 ,
Λ 1 2 = Λ 2 Λ 1 = [ c 2 c 1 v ̂ 2 s 2 v ̂ 1 s 1 c 2 v ̂ 1 s 1 + v ̂ 2 s 2 c 1 c 2 v ̂ 1 s 1 v ̂ 2 s 2 c 1 c 2 c 1 v ̂ 2 s 2 v ̂ 1 s 1 ] ,
c i = cosh ( λ i / 2 ) , s i = sinh ( λ i / 2 ) .
Λ 1 2 = [ cosh ( λ 1 2 / 2 ) p ̂ 2 sinh ( λ 1 2 / 2 ) p ̂ 2 sinh ( λ 1 2 / 2 ) cosh ( λ 1 2 / 2 ) ] × [ n 0 0 n ] .
p ̂ 2 tanh ( λ 1 2 / 2 ) = c 2 v ̂ 1 s 1 + v ̂ 2 s 2 c 1 c 2 c 1 v ̂ 2 s 2 v ̂ 1 s 1 = v ̂ 1 tanh ( λ 1 / 2 ) + v ̂ 2 tanh ( λ 2 / 2 ) 1 + v ̂ 2 tanh ( λ 2 / 2 ) v ¯ ̂ 2 tanh ( λ 1 / 2 ) .
V 1 v ̂ 1 tanh ( λ 1 / 2 ) , V 2 v ̂ 2 tanh ( λ 2 / 2 ) , V n = [ ( V 1 V 2 ) ] V n p ̂ n tanh ( λ 1 2 n / 2 ) ;
V 2 = V 1 V 2 = V 1 + V 2 1 + V 2 V ¯ 1 ,
S 0 n = 1 , S 1 n = i = 1 n V i = V 1 + V 2 + + V n , S 2 n = 1 i < j n V j V ¯ i = V 2 V ¯ 1 + + V n V ¯ 1 + V 3 V ¯ 2 + = 1 i < j n V j V i , S 2 m n = 1 i < j < · · · < υ < w n V w V ¯ υ V j V ¯ i = 1 i < · · · < w n ( 1 ) m V w V i , S 2 m + 1 n = 1 i < j < < w < x n V ¯ x V w V j V ¯ i = 1 i < < w n ( 1 ) m V x V i , S m n = 0 whenever m > n .
V n = m 0 S 2 m + 1 n m 0 S 2 m n .
( e μ | e ν ) = 0 , μ ν , ( e 0 | e 0 ) = 1 , ( e μ | e μ ) = 1 , μ = 1 , 2 , 3 ,
C [ V 4 ( 1 ) ] = { 11 , e 0 , e 1 , e 3 , e 10 , e 20 , e 30 , e 12 , e 23 , e 31 , e 120 , e 230 , e 310 , e 123 , e 1230 } .
1 2 = 1 , i 2 = j 2 = k 2 = 1 , ij = ji = k , jk = kj = i , ki = ik = j .
β / α β α ¯ / | α | 2 , | α | 2 = α α ¯ = a 0 2 + a 1 2 + a 3 2 .
e 0 = [ 1 0 0 1 ] , e 1 = [ 0 i i 0 ] , e 2 = [ 1 j j 0 ] , e 3 = [ 0 k k 0 ] .
11 = [ 1 0 0 1 ] , e 10 = [ 0 i i 0 ] , e 20 = [ 0 j j 0 ] , e 30 = [ 0 k k 0 ] , e 12 = [ k 0 0 k ] , e 23 = [ i 0 0 i ] , e 31 = [ j 0 0 j ] , e 120 = [ k 0 0 k ] , e 230 = [ i 0 0 i ] , e 310 = [ j 0 0 j ] , e 123 = [ 0 1 1 0 ] , e 1230 = [ 0 1 1 0 ] .
( t x ) [ t x x t ] ,
11 e 0 , e 1 , e 2 , e 3 e 10 , e 20 , e 30 , e 12 , e 23 , e 31 e 120 , e 230 , e 310 , e 123 e 1230 scalars vectors bivectors pseudovectors pseudoscalars
υ V 4 ( 1 ) : Λ υ Λ V 4 ( 1 ) , Λ Λ = Λ Λ = ± 11 .
Λ = [ α β β α ] ,
Λ = [ cosh ( λ / 2 ) p ̂ sinh ( λ / 2 ) p ̂ sinh ( λ / 2 ) cosh ( λ / 2 ) ] × [ cos ( θ / 2 ) + r ̂ sin ( θ / 2 ) 0 0 cos ( θ / 2 ) + r ̂ sin ( θ / 2 ) ] ,
Λ R 1 = [ cos ( π / 4 ) + k sin ( π / 4 ) 0 0 cos ( π / 4 ) + k sin ( π / 4 ) ] = [ ( 1 + k ) / 2 0 0 ( 1 + k ) / 2 ] ,
Λ R 2 = [ ( 1 + i ) 2 0 0 ( 1 + i ) 2 ] .
Λ R = [ n ̂ 0 0 n ̂ ] .
n ̂ = cos ( θ / 2 ) + r ̂ sin ( θ / 2 ) = ( 1 + i 2 ) ( 1 + k 2 ) = 1 2 + 3 2 ( i j + k 3 ) .
p ̂ 2 tanh ( λ 1 2 / 2 ) = V 1 + V 2 1 + V 2 V ¯ 1 = ( j μ + k μ ) ( 1 kj μ 2 ) ¯ 1 + μ 4 = [ ( 1 μ 2 ) j + ( 1 + μ 2 ) k 2 ( 1 + μ 4 ) 1 / 2 ] × [ μ 2 ( 1 + μ 4 ) 1 / 2 ] .
tan θ = 1 μ 2 1 + μ 2 θ = 40 ° 54 .
tanh ( λ 1 2 / 2 ) = μ 2 ( 1 + μ 4 ) 1 / 2 υ = 0.6614 c .
( α γ ) ( β γ ) = ( α γ ) ( β γ ¯ ) / ( β γ ) ( β γ ¯ ) = α ( γ γ ¯ ) β ¯ / β ( γ γ ¯ ) β ¯ = α β ¯ / β β ¯ = α / β .
V 1 = S 1 1 S 0 1 = V 1 ,
V 2 = S 1 2 S 0 1 + S 2 2 = V 1 + V 2 1 + V 2 V ¯ 1 .
V n 1 = m 0 S 2 m + 1 n 1 m 0 S 2 m n 1 .
V n 1 V n = V n 1 V n 1 V n V n 1 = A B ¯ / | B | 2 + V n 1 V n A B ¯ / | B | 2 = A B ¯ + V n B B ¯ B B ¯ V n A B ¯ = ( A + V n B ) B ¯ ( B V n A ) B ¯ ( using right cancellation ) = m 0 S 2 m + 1 n 1 + V n ( m 0 S 2 m n 1 ) m 0 S 2 m n 1 V n ( m 0 S 2 m + 1 n 1 ) = m 0 S 2 m + 1 n m 0 S 2 m n = V n .

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