The separating topology for the Lorentz group L

The separating topology for the Lorentz group L

Citation for published version (APA):

Vroegindeweij, P. G. (1975). The separating topology for the Lorentz group L. Journal of Mathematical Physics,

16(6), 1210-1213. https://doi.org/10.1063/1.522669

DOI:

10.1063/1.522669

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Published: 01/01/1975

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The separating topology for the Lorentz group

L

P. G. Vroegindeweij

Department of Mathematics. Technological University. Eindhoven. The Netherlands (Received 7 June 1974)

Some properties of the Lorentz group L are presented if it is endowed with a topology induced by one of the topologies for the Minkowski space M, proposed by E. C. Zeeman.

1.

PRELIMINARIES

Let M denote Minkowski space, the four-dimensional real vector space

R\

provided with the indefinite qua-dratic form

Q(x)=x~ -

xi -

x~ - xi,

where x = _(xo, Xi, x2 , x3 ) E M. The vectors x of Mare

called timelike if Q(x) >0, lightlike (or isotropic) if

Q(x) = 0, and spacelike if Q(x) <0.

L is the full Lorentz group (all linear maps leaving Q invariant). L' is the orthochronous Lorentz group that is the subgroup of L whose elements preserve the sign of the first coordinate. L: is the subgroup of L' whose

elements I have the property detl = + 1.

Using the canonical basis of R4 we introduce the parity

p by

p=(p), O";i, j,,;3, Poo=1, Pjj:=-l, 1,,; i,,; 3,

andp/j=O for all i*j. We shall also use the time

reversal

t """

-po

Notice that L/L:~ V4 where V4 denotes Klein's

four-group. By

0;

we mean the centralizer of

p

in

L:

that is to say the subgroup of

L:

whose elements

r

have the property

PrP

-1 == r. The elements of

0;

are called pure

rotalions.

Z is the subgroup of L; whose elements

z

have the form z=

o

o

sinha 0 0 cosha 0 0

o

o

1 0

°

1

We introduce furthermore H5, being that subset of L; whose elements h have the property php'l=h'l. Notice that h=th (th is the transposed of h); h is called

hyper-bolic screw. L; has no proper invariant subgroups, cf.

Ref. 1.

Let 5L(2, C) be the group of unimodular 2 x2 matrices over the complex numbers. As is known, there is a surjective homomorphism cp which induces an isomorphism

5L(2, C)/Z2~ L:

where Z2 is the set

1210 Journal of Mathematical Physics, Vol. 16. No.6, June 1975

The homomorphism cp can be described in the following way (cf. Ref. 2): Let x = _(xo, Xu x2 , x3 ) E R4 and let

x

de-note the Hermitian matrix

Consider the bijectionj:R4

- M(2,C), given by f(xl=x.

Let IE L: and s === cp.l(Z)E S L(2, C). We have the relation

Ix

= sxs*, where

x-Ix

is a Hermitian map. Using matrix language, we may write

(

YO+~1 Y2-iy~=(a

f3\(xo+.x1

X2-iX~(~~\

Y2+ ZY3 Yo-Yt)

~

O)V!z+lX3

XO-Xl)~

6)

where :I'

=

Ix == (Yo, Y l> Y2' Y3) and s

=~

~)-We shall also use the group G, that is the group generated by L, the group T of translations of M and the group of multiplications by a positive scalar of the vectors of M. G' is the subgroup of G that we obtain by considering L t instead of L.

There is a partial order « on M given by x« y if and only if Q(y - x) > 0 and Xo < Yo' Another partial order

< on M is given by x < Y if and only if Q(y - x) '" 0 and

xo"; Yo' We still need the relation <', given on M by x

<. y if and only if Q(y - x) ==

°

and xo"; Yo' We introduce furthe rmore the sets:

C(x) =

{y!

Q(y - x) '-=

Or,

S(x)

={y!

Q(y - x)

<Or,

l(x)={y! Q(y -x) >o},

rex)

={y

!x« y},

r(x)={y!y«x}.

C is the group of bijections of M, preserving the rela-tion« .

Zeeman3 _{proved that C and G' coinCide. Zeeman's} theorem has been generalized in several ways, cf. Refs. 4-9.

2. THE SEPARATING TOPOLOGY FOR MINKOWSKI SPACE M

Usually 1\;1 is endowed with the Euclidean topology, but one can argue (Zeeman3_{,10) that this is objectionable for} physical reasons. On the other hand, it is impossible to define a topology for M by means of the indefinite quadratic form Q in a way similar to the Euclidean topology by means of the definite quadratic form. In Copyright © 1975 American Institute of Physics 1210

Ref. 10 Zeeman has proposed several non-Euclidean topologies for M related to the Lorentz group L. Nandall_-13

investigated them and added some more of this kind of topologies. All these topologies have the property that the corresponding group of autohomeomor-phisms of M coincides with G and for that reason they seem to be physically significant. Unfortunately, they are very complicated from a topological point of view; for instance, they fail to satisfy the normal property and hence they are not metrizable. In this section we shall deal with that one of the topologies, proposed by

Zeeman, that seems to be the most suitable for phySics, cf. Ref. 9. We call it the separating topology. Similar

topologies are also proposed by Cole14 _{and Cel'nik.} 15 Let d(x, y) denote the Euclidean metric

Given x EO M and E > 0, let Nl(x) denote the Euclidean

E-neighbourhood of x, given by

N;C,,)={Yld(x,y) <E}.

We introduce

N:(x)=Nl(x)n(c(x)\{x})*, XEOM

(by v* we mean the complement of a set V).

Definition: The separating topology for M is the

topo-logy, given by the basis of open sets N;(x), x EO M. We use the notations Ms for M with the separating topology and ME for M with the Euclidean topology.

Remark: It is also possible to define our topology by using only the relations «, <, and <'. That offers the possibility of introducing the separating topology in more general causal spaces, cf. Refs. 9, 16.

Let x, y, Z E M; ~,« x« Z and let us write

OJv,?) =r(y) II r(?) n (C(x)\{x})*.

Clearly the topology for

M

with basic open sets

OrCv,

z)

is equivalent with the topology with basic open sets N,'(x). Notice that Ms is a Hausdorff space; it satisfies the first axiom of countability and it is a separable space but it does not have a countable basis. However Ms

is locally connected and path wise connected it is not locally compact. From a physical point of view it seems to be interesting that on lightlike lines the discrete topo-logy is induced and that on timelike lines and spacelike hyperplanes the Euclidean topology is induced, cf. Ref. 10.

Comparing ME and Ms we still note the following properties:

(1) The set 0 is open in Ms and not in ME if and only

if for all x EO there is an E > 0 such that Ns'(x) e 0

and there is an x EO 0 with the property (C(x) \{x}) n N;(x) n

0*"*

~ for all E > O.

(2) The subset X of Ms is compact in Ms if and only if

X is compact in ME and all x E X are isolated in

xn

C(x) (with respect to ME)'

(3) The group of autohomeomorphisms of Ms is G. For details we refer to Ref. 9.

1211 J. Math. Phys., Vol. 16, No.6, June 1975

3. THE SEPARATING TOPOLOGY FOR LORENTZ GROUP L

This is the main part of our paper; we shall investi-gate the topology for L induced by the separating topol-ogy for M. As is to be expected, Ms induces a topology for L, deviating from the usual Lie group topology, such as we obtain by considering L as a six-dimensional manifold in R. 9 There are several ways to topologize a set of maps. In this section we shall deal with the topol-ogy of pointwise convergence. See e. g., Ref. 17.

A. Introduction

For each x EO Ms and for every open set 0 eM, we define

(x, 0) = {I EOL:

I

Ix EO}.

Let L, denoteL:, endowed with the topology that has the family of all sets (x, 0) as a subbasis, and let LE denote

L:,

endowed with the topology, defined in a similar way as for L" but coming from ME instead of Ms' The family of intersections of sets of the form (x, 0) is a basis for the topological space L" each number of this basis having the form

n7:l

(xj , OJ), where Xi E Ms and OJ is open in Ms' Notice that Ls is finer than LE , for Ms is

finer than ME' As we shall show below, L, is strictly finer. Notice furthermore that L, is a Hausdorff space, for M, has that property.

It is also possible to describe our topology by means of convergence of nets (see, e. g., Ref. 17, p. 77). To that end one can define: The net of Lorentz transforma-tions

(lJ

converges to I in Ls if and only if (lvx) con-verges to lx for all XE Ms' We shall say that a set

oeL,

is open if and only if every net (1

J,

converging to an element 1 E 0, is eventually in O. Remark that, if the net (Z

vl

does not converge to I in L E, it does not

con-verge to l in L,. As we shall show below, the converse is also true if we restrict ourselves to time like vectors.

B. Properties of Ls

L, is strictly finer than L_E• Example.

. h 1 SIll - 0 '1 1 cosh- 0 I -_{n -} '1 O 0 1 ' X == 0 0 0

l is the unit element of L:. In L E we find that (In)

con-verges to l if 11_00, but lnX=e1/nx, and therefore (In)

does not converge in L" for In(x)<i N;(x), even for all n.

Also in the case of space like vectors, there are nets converging in L_{E ,} but not in L,. The same sequence (In) as above, but applied to the spacelike vector y

=(1,1,0,1), gives us lny)£N:Cv) for all 11.

Theorem 1: L sand L E induce the same topology on

the subgroup

0;.

Proof: It suffices to prove that a net of pure rotations (rJ, converging in LE , also converges in Ls (with the

same limit). Suppose that (rJ converges to r in L_E•

Then we have for all x that eventually {(rvx)}c N;(rx). P.G. Vroegindeweij 1211

On the other hand, we know that all rvx are situated in the same space like hyperplane through rx and therefore

{(rvx)}n (C(rx)\ {rx}) = ¢

and, consequently,

{(rvx)}n N:(rx) = {(rvx)}

n

N;( rx),

This means that (rvx) eventually belongs to N;(rx). In other words, (r) converges to r in L,.

_o

Corollary: Ls induces the same topology as LE on

every compact subgroup of

L:,

because 0; is a maximal compact subgroup of LE and consequently of Ls'

A semitopological group G is a topological space,

provided with a group structure such that the product

map GXG-G, given by (a,b)-ab, (a,bEG), is

sepa-rately continuous. See, e. g., Ref. 18.

Theorem 2: L s is a semitopological group.

Proof: Suppose that (l) converges to I, L e., (lvx)

converges to lx, xEMs' In particular, if we consider

l'x instead of x, then (lvl'x) converges to ll'x. There-fore, for all neighborhoods Oil' of ll' there is a neigh-borhood 01 of l such that 0, .[, CO". 0 On the other hand,

we know that the elements of Ls are homeomorphisms of Ms and therefore it follows from (lvx) converges to

lx that (l'lvx) converges to I'lx for all I' EL, Le., for

all Oil' there is a neighborhood 01 of I such that I' 0 1

COl'/" 0

C. The main theorem

The definition of Ls uses the action of L: on M and the topology of Ms' Now we want to give an intrinsic defini-tion of Ls ' by comparing it with LE • In Sec, 1 we have

seen that L: is very close to SL(2, C).

Lemma 1: For timelike vectors x, (Ivx) converges to

Ix in ME if and only if (lvx) converges to Ix in Ms'

Proof: Obviously, convergence in Ms implies

con-vergence in ME' To prove the converse, we remark that the nets, converging in LE and not in L s' are exactly those having the property that there is an x such that eventually

(Zvx -Ix, Ivx -Ix) = 0 and Ivx,nx,

i. e. ,

(rllvx - x, rllvx - x) = 0 and rllv x* x.

It is sufficient to consider only one timelike vector. We choose x' = (a, 0, 0, 0) and note that it is possible to transform all timelike vectors, situated on the same hypersurface (x, x) = a2

, into (a, 0, 0, 0) by a suitable

Lorentz transformation (a* 0). The intersection of

{x 1 (x, x) = a2

} and the light cone C(x') consists only of the

vertex x' of the cone. Therefore, the relations

U-1lv x - x , r1lvx-x)=0

and Z-llv x'

*

x' do not hold together. In other words,

(lv x) converges to Ix in ME implies that (tv x) converges

to Ix in Ms' 0

Let <p denote the surjective homomorphism of SL(2, (:)

onto L: (as introduced in Sec. 1) and let B denote the image under <p of the set of upper triangular matrices of the form [~

:-d

with 10'1

*

1.

Lemma 2: Let x be an isotropic vector and let (t) be

1212 J. Math. Phys., Vol. 16, No.6, June 1975

a net of Lorentz transformations. Then (Iv x) converges to Ix in M, if and only if

(i) Iv ~ converges to Ix in ME'

(ii) no tEL exists such that eventually r-lrlzJEB.

Proof: Similarly, as in the proof of Lemma 1, it

suffices to consider only one isotropic vector. We choose x'=(l,l,O,O) and (compare Sec. 1) the relation

Zx=sxs*,

written out and applied to our Situation, becomes

Again, we have to exclude nets (I) with the property that eventually U-llvx-x, rIlvx-x)=O and rlzvx*x.

The intersection of {xl (x,x)=O} and the light cone C(x')

only consists of the line i\ (1,1,0,0), i\ E R, and therefore we must look for Iv with rllvx'=i\vX' (i\v* 1).

Let

then for such Iv we have

IOIvI

2

*1, OI_{vYv=O, avyv=O,}

l)lvI

2=0;

in other words,

1 0' v 12

*

1 and Yv = O.

Consequently, the 2 x2 matrices in question correspond with elements bEB; i.e., rIZv=b or lv=lb. The Lorentz transformations, leaving invariant the other one-dimensional isotropic subspaces, have the form

lbZ-

I

, where

7

is a suitable Lorentz transformation.

Summarizing, we have to exclude lv, eventually satisfy-ing the relation ril v = [bZ-l or

"i

-IZ-ll/ = b. Now the proof

is complete. 0

Corollary: Ls induces the discrete topology on the

sub-group Z and on its conjugates.

Proof: As is known the elements of <p-I(z) have the

form

[6

to-il with t E

R:,

being a subset of B. 0 Notice that in the case of Z there are two isotropic eigenvectors, viz., (1, ± 1,0,0) but in the case of B there is only the isotropic eigenvector (1,1,0,0).

Let C be the image under <p of the matrices [~ ~J of

SL (2, C) with properties:

(i)

1

0'

12 -

1

f312 - 1

Y

12

+ 1 0 12 = 2,

(ii)

10I!2_1f31 2

*1.

Lemma 3: Let x be a spacelike vector and let (l

J

be

a net of Lorentz transformations. Then (lvx) converges to lx in M" if and only if:

(i) (tv x) converges to Ix in ME'

(ii) no ; E L exists such that eventually

r

-lrll

J

E C.

Proof: Again we only need one space like vector to

start with and we choose x' = (0, a, 0, 0), situated on the

hypersurface (x, x) = - a2

(a

*

0). Similarly, as for Lemma 2, we find

Now the intersection of {xl (x, x) = - a2

} and C(x') is

situated in the hyperplane Xl

=a

and therefore we have to look for the elements of L:, transforming (O,a,O,O) into

(v, a, vcosu, vsinu), where v*O.

or

This means that

(

_ve

1I+a

lU

ve-iU)=a(I~12- ~1312

v -a

\

0'1' - 136

v

+a=a(10'12_1131 2),

v - a

=

a(

1

I'

12 - 161 2),

veiu =a(ay-i30),

and these relations are equivalent with the conditions: (i)

10'12_11312_11'12+1612=2,

(ii)

10'1 2_1,91 2*1

(v*O),

(iii)

(I

0'1 2 - 1 131 2 - 1)2

=

1

iiI' -

1'36 12;

but condition (iii) is superfluous for it is implied by (i) and

0'6 -

fYy

= 1.

Similarly, as for Lemma

2,

it turns out that in this case we must exclude the nets (l) with the propertythateventuallyf-lrllJ=c, withcEC. 0

Now we are able to state:

Theorem: Let (l

J

be a net of Lorentz transformations. Then (I v) converges to I in L, if and only if:

(i) (l

J

converges to , in L E'

(ii) no IE L exists such that eventually f-ll-l'J E B U C (B and C as defined above).

Proof: The theorem follows immediately from the

Lemmas 1, 2, and 3. 0

Remarks:

1. The theorem gives us an intrinSic definition of the topology of Ls by means of convergence of nets. 2. The topology of Ms can be recovered from Ls'

1213 J. Math. Phys., Vol. 16, No.6, June 1975

3. The Lorentz transformations that we have excluded for convergence, are exactly those leaving not only (x, x) invariant but also the intersections of the hypersurfaces (x,x)=p and the light cones in the points of contact. In Lemma 1 this intersection only consists of one point and in Lemma 2 we found the one-dimensional subvarieties.

4. Probably the condition

10'1 2 - 11312 - 11'12 +

I

61 2 =2

has to do with the roots of the equation s xs

*

= AX.

5. The set B U C has the property that 1 E B U C im-plies 1-1 E B U C and therefore L s is a T I-group (see

Ref. 19, p. 27). As is known (cf. Ref. 1) L, has no proper invariant subgroups and hence L, is con-nected (see Ref. 19, p. 28).

6. Probably L, has any representations that are not representations of LE ; these representations might

lead to new invariants of phySiCS. I did not succeed in finding examples of these new representations until now.

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