Naber G.L. The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity

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304 B Spinorial Objects

But the conclusion we draw from the Dirac experiment is that there may

well be more to a system’s “state” than merely its “conﬁguration”. This

additional element has been called (see [MTW]) the version or orientation-

entanglement relation of the system and its surroundings and at times it must

be taken into account, e.g., when describing the quantum mechanical state of

an electron with spin. Where is one to look for a mathematical model for such

a system’s “state” if now there are two, where we thought there was one?

The mathematics itself suggests an answer. Indeed, the universal covering

manifold of SO(3) (i.e., of RP

)isS

and is, in fact, a double cover, i.e.,

the covering map is precisely two-to-one, taking the same value at y and −y

for each y ∈ S

. Will S

do as the “state space”? That this idea is not the

shot-in-the-dark it may at ﬁrst appear will become apparent once it has been

pointed out that we have actually seen all of this before.

Recall that, in Section 1.7, we constructed a homomorphism, called the

spinor map, from SL(2, C)ontoL that was also precisely two-to-one and

carried the unitary subgroup SU

of SL(2, C) onto the rotation subgroup R

(i.e., SO(3)) of L.

Exercise B.3.10 Let I

i 0

0 −i

−10

and k

0 i

i 0

Show that I

,i,jand k are all in SU

and that, moreover, any A ∈ SU

uniquely expressible in the form

A = aI

+ bi + cj + dk,

where a, b, c, d ∈ R and a

+ b

+ c

+ d

= 1. Regard SU

as a subset of R

by identifying

A =

a + bi c + di

−c + di a − bi

with the column matrix col [abcd− cda− b] ∈ R

and deﬁne a map

from SU

into R

that carries this column matrix onto col [abcd] ∈ R

Show that this map is a homeomorphism of SU

onto S

⊆ R

. Finally,

observe that the restriction of the spinor map to SU

is a continuous map

onto R (i.e., SO(3)) which satisﬁes the deﬁning property of a covering map

for SO(3).

Thus we ﬁnd that SU

and the restriction of the spinor map to it constitute

a concrete realization of the universal covering manifold for SO(3) and its

covering map. Old friends, in new attire. And now, how natural it all appears.

Identify a “state” of the system with some ˜y ∈ SU

. This corresponds to some

“conﬁguration” y ∈ SO(3) (the image of ˜y under the spinor map). Rotating

the system through 2π corresponds to a loop in SO(3) which, in turn, lifts

((5.2) of [G]) to a path in SU

from ˜y to −˜y (a diﬀerent “state”). Further

rotation of the system through 2π traverses the loop in SO(3) again, but,

in SU

, corresponds to a path from −˜y to ˜y and so a rotation through 4π

returns the original “state”.

B.3 Homotopy in the Rotation and Lorentz Groups 305

Exercise B.3.11 For each t ∈ R deﬁne a matrix A(t)by

A(t)=

0 e

−

Show that A(t) ∈ SU

and that its image under the spinor map is the rotation

R(t)=

⎡

⎢

⎣

cos t −sin t 00

sin t cos t 00

0010

0001

⎤

⎥

⎦

Hint:Takeθ = φ

=0andφ

= t in Exercise 1.7.7.

Exercise B.3.12 Show that α :[0, 2π] → SU

deﬁned by α(t)=A(t)for

0 ≤ t ≤ 2π is a path in SU

from the identity I

2×2

to −I

2×2

whose image

under the spinor map is a loop Spin ◦α at I

4×4

(which is not nullhomotopic

in R). On the other hand, β :[0, 4π] → SU

deﬁned by β(t)=A(t)for

0 ≤ t ≤ 4π is a loop at I

2×2

in SU

and its image Spin ◦ β isalsoaloopat

4×4

(which is nullhomotopic in R).

Mathematical quantities used to describe various aspects of the system’s

condition are still determined by the state of the system, but now we take this

to mean that they should be expressible as carriers of some representation

of SU

. (Incidentally, any discomfort one might feel about the apparently

miraculous appearance at this point of a group structure for the covering

space should be assuaged by a theorem to the eﬀect that this too is “essen-

tially unique”; see (6.11) of [G].) Although we shall not go into the details

here, it should come as no surprise to learn that the universal cover of the

entire Lorentz group L consists of SL(2, C) and the spinor map so that to

obtain a relativistically invariant description of, say, the state of an electron,

one looks to the representations of SL(2, C), that is, to the 2-valued repre-

sentations of L (see Section 1.7). Quantities such as the wave function of an

electron (which depend not only on the object’s conﬁguration, but also on

“how it got there”) we call spinorial objects and are described mathematically

by carriers of the representations of SL(2, C), i.e., by spinors.

References

[AS] Aharonov, Y. and L. Susskind, “Observability of the sign change of

spinors under 2π rotations”, Phys. Rev., 158(1967), 1237–1238.

[A] Alphors, L., Complex Analysis, McGraw-Hill, New York, 1979.

[BJ] Bade, W. L. and H. Jehle, “An introduction to spinors”, Rev. Mod.

Phys., 25(1953), 714–728.

[B] Bolker, E. D., “The spinor spanner”, Amer. Math. Monthly,

80(1973), 977–984.

[C]Cartan, E.,The Theory of Spinors, M.I.T. Press, Cambridge,

MA, 1966.

[CGK] Cacciatori, S., V. Girini and A. Kamenshchik, “Special relativity in

the 21

century”, Annalen der Physik, 17(2008), 728–768.

[DS] Daigneault, A. and A. Sangalli, “Einstein’s static universe: an idea

whose time has come back”, Notices of AMS, 48(2001), 9–16.

[E] Einstein, A., et al., The Principle of Relativity, Dover, New York,

1958.

[Fa] Fadell, E., “Homotopy groups of conﬁguration spaces and the string

problem of Dirac”, Duke Math. J., 29(1962), 231–242.

[Fe] Feynman, R. P., R. B. Leighton and M. Sands, The Feynman Lec-

tures on Physics, Vol. III, Quantum Mechanics, Addison-Wesley,

Reading, MA, 1966.

[FL] Freedman, M. H. and Feng Luo, Selected Applications of Geome-

try to Low-Dimensional Topology, A.M.S. University Lecture Series,

American Mathematical Society, Providence, RI, 1989.

[GMS] Gelfand, I. M., R. A. Minlos and Z. Ya. Shapiro, Representations of

the Rotation and Lorentz Groups and their Applications, Pergamon,

New York, 1963.

[G] Greenberg, M., Lectures on Algebraic Topology,W.A.Benjamin,

New York, 1967.

[HE] Hawking, S.W. and G.F.R. Ellis, The Large Scale Structure of

Space-Time, Cambridge University Press, Cambridge, England,

1973.

307

308 References

[HKM] Hawking, S. W., A. R. King and P. J. McCarthy, “A new topology

for curved spacetime which incorporates the causal, diﬀerential and

conformal structures”, J. Math. Phys., 17(1976), 174–181.

[H] Herstein, I. N., Topics in Algebra, Blaisdell, Waltham, MA, 1964.

[IS] Ives, H. E. and G. R. Stilwell, “Experimental study of the rate

of a moving atomic clock”, J. Opt. Soc. Am., 28(1938), 215;

31(1941), 369.

[KO] Klein, A. G. and G. I. Opat, “Observability of 2π rotations: A

proposed experiment”, Phys. Rev. D, 11(1975), 523–528.

[K] Kuiper, N. H., Linear Algebra and Geometry, North Holland,

Amsterdam, 1965.

[La] Lang, S., Linear Algebra, Springer-Verlag, New York, 1987.

[LU] Laporte, O. and G. E. Uhlenbeck, “Application of spinor analysis

to the Maxwell and Dirac equations”, Phys. Rev., 37(1931), 1380–

1397.

[LY] Lee, T. D. and C. N. Yang, “Parity nonconservation and a two-

component theory of the neutrino”, Phys. Rev., 105(1957), 1671–

1675.

[Le] Lenard, A., “A characterization of Lorentz transformations”, Amer.

J. Phys., 19(1978), 157.

[Lest] Lester, J.A., “Separation-preserving transformations of de Sitter

spacetime”, Abh. Math. Sem. Univ. Hamburg Volume 53, Number

1, 217–224.

[M] Magnon, A. M. R., “Existence and observability of spinor struc-

ture”, J. Math. Phys., 28(1987), 1364–1369.

[MTW] Misner, C. W., K. S. Thorne and J. A. Wheeler, Gravitation,W.

H. Freeman, San Francisco, 1973.

[Nan] Nanda, S., “A geometrical proof that causality implies the Lorentz

group”, Math. Proc. Camb. Phil. Soc., 79(1976), 533–536.

]Naber,G.L.,Topological Methods in Euclidean Spaces, Dover Pub-

lications, Mineola, New York, 2000

]Naber,G.L.,Spacetime and Singularities, Cambridge University

Press, Cambridge, England, 1988.

] Naber, G.L., Topology, Geometry and Gauge ﬁelds: Foundations,

Edition, Springer, New York, 2010.

] Naber, G.L., Topology, Geometry and Gauge ﬁelds: Interactions,

Edition, Springer, New York, 2011.

[Ne] Newman, M. H. A., “On the string problem of Dirac”, J. London

Math. Soc., 17(1942), 173–177.

[O’N] O’Neill, B., Semi-Riemannian Geometry, With Applications to Rel-

ativity, Academic Press, San Diego, New York, 1983.

[Par] Parrott, S., Relativistic Electrodynamics and Diﬀerential Geometry,

Springer-Verlag, New York, 1987.

[Pay] Payne, W. T., “Elementary spinor theory”, Amer. J. Phys.,

20(1952), 253–262.

References 309

[Pen

] Penrose, R., “The apparent shape of a relativistically moving

sphere”, Proc. Camb. Phil. Soc., 55(1959), 137–139.

[Pen

] Penrose, R., “Zero rest-mass ﬁelds including gravitation: asymptotic

behavior”, Proc. Roy. Soc. Lond., Series A, 284(1965), 159–203.

[PR] Penrose, R. and W. Rindler, Spinors and Spacetime,Vols.I–II,

Cambridge University Press, Cambridge, England, 1984, 1986.

[R] Robb, A. A., Geometry of Space and Time, Cambridge University

Press, Cambridge, England, 1936.

[Sa]Salmon,G.,A Treatise on the Analytic Geometry of Three Dimen-

sions,Vol.1,Chelsea,NewYork.

[Sp

] Spivak, M., Calculus on Manifolds, W.A. Benjamin, Inc., Menlo

Park, CA, 1965.

[Sp

] Spivak, M., A Comprehensive Introduction to Diﬀerential Geome-

try, Vols. I–V, Publish or Perish, Houston, TX, 1975.

[Sy]Synge,J.L.,Relativity: The Special Theory, North Holland,

Amsterdam, 1972.

[TW] Taylor, E. F. and J. A. Wheeler, Spacetime Physics,W.H.Freeman,

San Francisco, 1963.

[V] Veblen, O., “Spinors”, Science, 80(1934), 415–419.

[Wald]Wald, R.M., General Relativity, University of Chicago Press,

Chicago, 1984.

[We]Weyl,H.,Space-Time-Matter, Dover, New York, 1952.

[Wi] Willard, S., General Topology, Addison-Wesley, Reading, MA, 1970.

] Zeeman, E. C., “Causality implies the Lorentz group”, J. Math.

Phys., 5(1964), 490–493.

] Zeeman, E. C., “The topology of Minkowski space”, Topology,

6(1967), 161–170.

Symbols

M Minkowski spacetime, 9

O,... observers, 2

Σ,

Σ,... spatial coordinate systems, 2

c speed of light, 3

S,... frames of reference, 3

, ˆx

,... spacetime coordinates, 3

transpose of Λ

η 10

g(v, w)=v · w value of the inner product g on (v, w), 7

⊥

orthogonal complement of W,7

Q quadratic form determined by g,7

= Q(v)=v · v 7

}, {ˆe

},... orthonormal bases, 8

= δ

4 × 4 Kronecker delta

= η

entries of η,10

)nullconeatx

,11

null worldline through x

and x,11

Λ=[Λ

] matrix of an orthogonal transformation, 13

[Λ

]inverseof[Λ

]

general homogeneous Lorentz group, 14

)timeconeatx

,16

) future and past time cones at x

,16

) future and past null cones at x

,17

L Lorentz group, 19

R rotation subgroup of L,20



ˆu,... velocity 3-vectors, 21

β relative speed of S and

S, 21, 26

γ (1 − β

)

−

,21

311

312 Symbols



d,... direction 3-vectors, 22

Λ(β)boost,26

θ velocity parameter, 27

L(θ) hyperbolic form of Λ(β), 27

τ(v) duration of v,43

Δτ = τ(x − x

)43



(t) velocity vector of the curve α,47

L(α)propertimelengthofα,47

τ = τ(t) proper time parameter, 50

U = α



(τ) world velocity of α,50

A = α



(τ) world acceleration of α,51

γ(



, 1) = U 52

S(x − x

) proper spatial separation, 56

 chronological precedence, 58

< causal precedence, 58

conjugate transpose of A

2×2

set of complex 2 × 2 matrices

Hermitian elements of C

2×2

,69

Pauli spin matrices, 69

SL(2, C ) special linear group, 69

image of A under spinor map, 71

A(θ)mapsontoL(θ) under spinor map, 72

special unitary group, 72

−

past null direction through x,74

−

celestial sphere, 74

(α, m) material particle, 81

m proper mass of (α, m), 81

P = mU world momentum of (α, m), 81



relative 3-momentum, 81

(



,mγ)=P 81

E total relativistic energy, 82

(α, N ) photon, 84



ˆe,... direction 3-vectors of (α, N), 84

N world momentum of (α, N), 84

, ˆ,... energies of (α, N), 84

ν, ˆν,... frequencies of (α, N), 84

λ,

λ,... wavelengths of (α, N), 84

h Planck’s constant, 84

(A,x,

A) contact interaction, 87

mass of the electron

≈ is approximately equal to

(α, m, e) charged particle, 93

e charge of (α, m, e), 93



electric ﬁeld 3-vector, 95



magnetic ﬁeld 3-vector, 95

Symbols 313

rng T range of T ,97

ker T kernel of T ,97

tr T trace of T

) open Euclidean ε-ball about x

, 117

∂f

∂x

118

→

F (p) assignment of a linear transformation to p, 118

div F divergence of p

→

F (p), 118

F bilinear form associated with F , 119

F exterior derivative of

F , 120



abcd

Levi-Civita symbol, 121

∗

F dual of

F , 121

= L(e

) components of the bilinear form L, 136

GL(n, R) real general linear group of order n, 138

GL(n, C) complex general linear group of order n, 138

D a group representation, 138

= D(Λ) image of Λ under D, 138

∗

dual of the vector space M, 139

} basis for M

∗

dual to {e

}, 139

∗

element u → v · u of M

∗

for v ∈M, 139

= η

aα

components of v

∗

in {e

}, 139

⊗ tensor (or outer) product, 140

vector space of world tensors on M, 140

···a

···b

components of L ∈T

, 141

∗∗

=(M

∗

)

∗

second dual of M, 141

∗∗

element f → f(x)ofM

∗∗

for x ∈M, 141

Spin the spinor map, 142

space of polynomials in z and ¯z, 144

(

)

spinor representation of type (m, n), 144

A, B, C,... spinor indices taking the values 1, 0

Z,... conjugated spinor indices taking the values

G =





element of SL(2, C), 148

G =

conjugate of G, 148

ß spin space, 153

<, > skew-symmetric “inner product” on ß, 153

}, {ˆs

},... spin frames, 153

,... components of φ ∈ ß, 153

∗

dual of ß, 155

}, {ˆs

},... dual spin frames, 155

2 × 2 Kronecker delta

∗

element ψ →<φ,ψ>of ß

∗

for φ ∈ ß, 155

,... components of φ

∗

∈ ß

∗

, 156





transposed inverse of





, 156

∗∗

element f → f(φ)ofß

∗∗

for φ ∈ ß, 157