Lecture Notes in Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
Series: Mathematisches Institut der Universit~t Bonn • Adviser: F. Hirzebruch
52
III
D. J. Simms
Trinity College, Dublin
Lie Groups
and Quantum Mechanics
1968
Springer-Verlag- Berlin-Heidelberg-New York
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All rights reserved. N o part of this b o o k may be translated or reproduced in any form without written p e r m i s s i o n from
Springer Verlag. â by Springer-Verlag Berlin ã Heidelberg 1968
Library of C o n g r e s s Catalog Card N u m b e r 68 - 24468 Printed in Germany. Title No. 7372
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Preface
These notes are based on a series of twelve lectures given at the
University of Bonn in the Wintersemester 1966-67 to a mixed group
of mathematicians and theoretical physicists. I am grateful to
Professors H~rzebruch, Bleuler and Klingenberg for giving me the
opportunity of s p e a k ~ a t
their seminar.
These notes are written primarily for the mathematicianwho has an
elementary acquaintance with Lie groups and Lie algebras and who
would like an account of the ideas which arise from the concept of
relativistic invariance in ouantum mechanics. They may also be of
interest to the theoretical physicist who wants to see familiar
material presented in a f o ~ w h i c h
uses standard concepts from other
I
areas of mathematics.
The presentation owes very much to the lecture notes of Mackey [24]
and [25] and of Hermann [15] . Many of the original ideas are due to
Wigner and Bargmann [37] , [1] and [39] . A useful collection of
reprints is contained in Dyson [11] .
I should like to thank Professor F. Hirzebruch for his help and
stimulation, Dr. D. Arlt for useful criticisms, and the Mathematisches
Institut Bonn for support during this time and for the typing of the
manuscript.
Dublin, April 1967
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D.J.
Simms
Contents
I
Relativistic invariance
i
2
Lifting projective representations
9
3
The relativistic free particle
i7
4
Lie algebras and physical observables
2o
5
Universal enveloping algebra and invariants
26
6
Induced representations
42
7
Representations of semi-direct products
z~8
8
Classification of the relativistic free particles
56
9
The Dirac equation
72
SU(3): charge and isospin
76
io
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Section 1.
Relativistic Invariance.
Causality.
Let
M
be the set of all space-time events. Any choice of
observer defines a bijective map
x 1, x2, x 3
event
x
M * R 4 , x~-~ (Xl,X2,X3,X4)
are space coordinates and
x4
as seen by the chosen observer.
where
is the time coordinate of the
This gives
M
the structure of
a real 4-dimensionsl vector sp~co with indefinite Lorentz scalar
product
<x,y> = - xlY 1 - x2Y 2 - x3Y 3 + x4Y 4 ,
called the Minkowski structure of
scalar product
~
<x,y>
- y, x - y>
measured
M
relative to the given observer. The
may be given the following physical interpretation:
is the time interval between events
and
y
as
by a clock which moves with uniform velocity relative to our
observer and is present at both events. Let us write
x
x
is able to influence the event
means that
y
x < y
if the event
y , in the eyes of our observer. This
occurs later in time then
x
and that a physical body such
as a clock is able to be present at both events. Thus we define:
x < y
if and only if
This partial order on
M
chosen observer. An event
<x,y> > 0
x4 < Y4
and
<x - y , x - y> > 0 .
expresses the idea of causalit2, as seen by our
x
is time-like if
<x,x> > 0 . The relation
is an equivalence relation on the set of all time-like events,
with two equivalence classes: the future and past events. Moreover
if and only if
y - x
is a future time-like event.
A change of observer determines a bijective map
where
f(x)
as the event
x < y
f : M
~ M ,
is the event which appears to the new observer to be the same
x
does to the old observer.
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-2-
The diagram
f
M
old~
commutes. Event
f(x)
~M
/
4
ne~
will influence event
observer, if and only if
x
influences
f(y)
y
, according to the new
according to the old observe
The two observers will have the same idea of causality provided that
x < y :
for all
> f(x) < f(y)
x, y ~ M . In this case we call
f
a causal automorphism rela±
to our observer. If the idea of causality is to be preserved, we must li
ourselves to observers which are related to our chosen observer by a caw
automorphism.
Let
IR*
A dilatation of
latio.___~nof
M
M
denote the multiplicative
is a map
is a map
formation of
M
x, y E M . Here
M
of the form
x~-*~X,
A ¢ IR ~ • A
x~--~x + a , a c M . A homogeneous Lorentz tran~
is a linear map
M
~M
group of non-zero real nux
A : M
-M
is given its N i n k o w s k i
with
<Ax,Ay> :
foz
structure defined by a chosez
observer.
The group of translatiorsmay be identified with the additive
The group ~
oE homogeneous Lorentz transformations
homomorphisms
w(A) = ~ I
~ , W :~--*IR ~ , where
according to whether
A
~(A)
admits two continuou
is the determinant of
A
leaves the equivalence classes of f
and past events fixed or interchanges them~ The orthochron0us homogeneou
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-3-
If a group
G
acts on a group
as a group of automorphisms,
H
H × G
h,--) h g , g ( G , then the cartesian product
with the group ope-
ration
( h l , g l ) ( h 2 , g 2) = (hlhgl , glg2 )
is called a semi-direct product and denoted by
H ~ g . We note that
\
h~'~ (h,1)
and
subgroup of
HOG
g~-~ (1,g)
embed
H as a normal
subgroup and
G
as a
. Moreover
h g = ghg -I
in
H~)G
.
Any product of dilatations,
Lorentz transformations
translations,
is of the form
x~-~AA-x
where
(a,A,A) ¢ M x ~
x~@B #.x + b
is
x~AB
and orthochronous
+ a
x IR• . The composition of
x~-~ A A-x + a
with
A.#.x + A A.b + a , which corresponds to a group
operation
(a,A,A).(b,B,#)
= (a + A k,b , AB , A-~ ) .
This shows that the group generated by such transformations
product
)
M®
relative to the natural action of ~ i
x ]Re
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on
M .
is a semi-direct
-4-
Zeeman
[4o] has shown that
group of causal automorphisms of
M@(~
t x I~ )
is the complete
M .
Causal invariance in quantum mechanics.
The pure states of a quantum mechanical system are represented
by the projective space
Hilbert space
~ of l-dimensional subspaces of a separable complex
H . For each non-zero vector
be the l-dimensional subspace containing
The inner product
<~,~>
~
in
H , let
~ .
on the Hilbert space
H
defines a real
valued function
2
TIC!I'2 II~112
on
~ x ~ . The physical interpretation is that
<~,$~
is the transition
probabi!itY, the probability of finding the system to be in the state
when it is in the state
A bijective map
$ .
T: ~ - * ~ is an automorphis~ if it preserves the
transition probability:
for all
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-5-
If
A
is a unitary or anti-unitary transformation of
an automorphism
~ of
~
then it defines
by:
By an anti-unitary transformation
A
we mean one such that
A( @ + ~ ) - A ~
< A ¢,~p >
for all
H
+ A~
= < @,~o >
~, ~ • H , A ( @ . The product of two anti-unitary transformations
is unitary, so that the group
U(H)
of unitary transformations is a sub-
group of index two of the group
~(H)
mations. The map
embeds
Let
w:~(H) * Aut(H)
ei~
~ ei0-1
of unitary or anti-unitary transforU(1)
as a subgroup of
Aut(~) be the group of automorphisms of
be the map
Theorem of Wi~ner.
U(H) .
~ , and let
w(A) = ~ . We have the fundamental result:
The sequence
zr
(1)
is exact.
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-6-
This means that every automorphism of
where
if
~
is of the form
A
is a unitary or anti-unitary transformation of H . Moreover,
i8
~ = ~ then A = ei~B with e ¢ U(1) . For Wigner's proof see [38] ,
appendix to chapter 20. See also [2] .
If
f: M ...~.. M
is a bijective map arising from a change of ob-
server, it determines a bijective map
Tf: ~--~ ~ , where
Tf ~
is the
quantum mechanical state which appears to the new observer to be the same
as the state
~
does to the old observer. If
g: M - @
M
is another change
of observer we shall suppose that
Tfg = TfTg .
A physical justification of this assumption could be based upon the relation between states and assemblies of events in space-time, and on the
definition of
f
and
g
in terms of events in space-time.
The new observer will have the same idea of transition probability
as the old observer if and only if
of all transformations of
M
Tf
is an automorphism of
~ . The group
which are associated v,dth observers which have
the same idea of causality as our chosen observer is
M~(~t
x IR*)
by the
result of Zeeman. If all the observers also have the same idea of transition
probability, then we have a homomorphism
T: M ( ~ ( ~
× I ~ ) --@ Aut(~t) .
In this case we say that we have causal invariance. If the weaker condition
holds that all observers which are related to our chosen observer by transformations in the restricted i~_homg~eneousLorentz group
M(~
~
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have the
U
-7-
same idea of transition probability then we have a homomorphism
T: M ~ O
~ ~ Aut(~) .
In this case we have rel~tivistic inv~iance. From now on we will confine
ourselves to relativistic invariance.
The exact sequence (I) gives an exact sequence
where
two in
If
@
U(~)
is the image of
U(H)
under
~
and is a subgroup of index
Aut(~) .
is a connected Lie group, then the image of any homomorphlsm
T: G --~ Aut(~)
is contained in
in
G
U(~) . To see this we note that the exponential map
shows that there is a neighbourhood
V
of the identity in
which each element is a square and is therefore mapped by
Since
M~O
is a connected hie group, we have
in the case of relativistic invariance.
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T
a
into
G
in
U(~)
homomorphism
-8-
Let the Hilbert space
and let the projective space
to the surjection
the state
Tf ~
@-~
~
H
be given its usual norm topology,
be given the quotient topology relative
@ . We assume that, for each fixed stste
~ ,
will depend continuously on the change of observer.
This means that the map
M®A o
H
A
f~-~ Tf ~
is continuous for each
~ ~ H . This is equivalent to the
c o n t i n u i t y of
where
U(H)
~ ~
is given the weakest topology such that all maps
, are continuous,
obtained by giving
U(H)
~ ¢ ~ . The same topology on
....). ~ ,
may be
the strong operator topology, which is the
weakest topology such that all maps
tinuous,
U(~)
U(~)
U(H)
~ H , A~--~A~
, are con-
~ ~ H , and then taking the quotient topology relative to the
^ . The equality of these two topologies on U (I) follows
from [1] theorem 1.1.
surjection
A~¢A
By a pro~ective representation
T
of a topological group
we shall mean a continuous homomorphism
T: G
given above. By a representation
G
morphism
T: G---~U(H)
where
T
U(H)
of
~U(H)
G
with the topology
we mean a continuous homo-
is given the strong operator topology
defined above.
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-9-
Section 2.
T
If
U(H)
then
Lifting Pro jectiye Representations. '
is a representation of a topological group
~-T
is a projective representation of
G
in
G :
T
G
Conversely, if
T: G
admits a liftin6
such that
~
)U(~)
>
.
is a projective representation then
if there exists a representation
T: G
T
~ U(H)
T = ~.T .
Let
group
T
U(H)
G
be a connected Lie group with simply connected covering
and covering map
representation of
G
in
p
with kernel
K . Let
T
be a projective
U(~) . Using the theorem of Wigner we have the
diagram
P
1
>K
')~
)G
)1
1t'
1
> u(1)
) u(H)
)
)
with both rows exact.
T-p
is a projective representation of
T,p (K) = I . Suppose
Top
admits a lifting
o(K) C U(1) . Conversely any representation
that
G
in
a(K) C U(1)
U(~)
a: ~
a
of
~U(H)
~
in
~
and
; then
U(H)
such
will define a unique projective representation
such that
T
of
Top = ~oa .
These considerations show that, if the simply connected Lie
,group
G
has the property that every projective representation of
G
a~mlts a lifting, then the determination of all the projective representations of
G
is equivalent to the determination of the representations of G
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-
which map
K
into
lo-
U(1) . Bargmann in [I] Theorem 3.2, Lemma 4.9,
and § 2, proved that the possibility of lifting projective representations
of
G
depends on the cohomology of the Lie algebra of
G . The remainder
of this section will be devoted to giving a topological proof of Bargmann's
result. We shall prove four theorems, the fourth being the theorem of
Bargmann.
Theorem I.
U(H)
and fibre
Proof.
is a principal bundle with base space
U(1) .
For the relevant definitions we refer to [17] I. 3.2.a)
[19] I!I. 4.
We first note that although
group in general since the multiplication
continuous, see [28] Đ 33.2,
For each non-zero
V
U(H)
=
~ Â H
~,U(H)
the set
= IA I < A ~,~ • $ 0 I
"~ = ,(V )
therefore form an open cover of
>U(1)
U(~) , and
U(H) . The sets
,-I(T~) = ~
be defined by
< A~,@
;9~(~A) = A-w~(A)
is not
is an open map.
U(H) , and such sets give an open cover of
Up: V~
)U(H)
U(1) x U ( H )
is open in
Let
and also
is not a topological
U(H) x U(H)
the multiplication
is continuous. It follows that
Then
U(~) , pro~ection
for each
>
A c U(1) . Let
be defined by
hq~(A) = (=A , Wcp(A)) ã
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h : V
- w
ì
u(l)
.
-11
-
This is continuous with continuous inverse
(=A , eiS);
~ei~[w~(A)]-IA
and is therefore a homeomorphism. This proves the bundle space property
that
=-1(W'~)v is homeomorphic to the topological product
It remains to show that the structure group is
If
@,
are non-zero vectors in
H
and if
~
× U(1)
@
U(1) .
=A ~ W @ ~ ] %
and
iG
e
E U(1)
then
h @ % I (=A,e i0) : h@
:
Thus the fibre coordinate
e
i8
I ei6~(A) -IA I
(=A , e
i0
n~(A)
-1
is multiplied by
w@(A))
.
%(A) -I
on change of coordinate neighbourhood.
Moreover the function
=A ~
-~(A)-Iw@(A).
Theorem 2.
Any continuous map
connected Lie group
with
is continuous. This completes the proof.
G
T: G
> U(~)
of a connected simply
can be lifted to a continuous map
r: g
> U(H)
T = ==T .
G
Proof.
T
induces a
U(1)
bundle,
E
) G , with base space
to~al space
E = I(g,A) i T(g) = =(A) I C G x U(H)
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G ,
-
and projection
a(g,A) = g .
12-
See C17J I. 3.3. or [33] page 98 .
The diagram
E
) U(H)
,-p
co~nutes, where
=(g,A) = A . By a generalisation [3J 5., 17., and 18.,
of a result of Cartan, the second homotopy group of
Since
G
G
is zero.
is also simply connected, it follows from a theorem of Hurewicz
[19J II . Corollary 9.2., and by the universal coefficient theorem [12] V.
Exercise G. 3., that the singular cohomology group
Since the first homotopy group of the fibre
from obstruction theory [35] 35.5 and 29.8
s: G
ME
exists with
U(1)
~(G,~
is ~
)
is zero.
, it follows
that a continuous map
aos = 1 .
The continuous map
T = ¢=s
has the property required since
#oT = #==os = T=acs = T . This completes the proof.
Extensions and factor sets.
Let
the pair
(G,K)
G
and
K
be Lie groups with
is a continuous map
=: G x
such that
G
...~..K
~(1,1) = I , and
z) -
for all
x, y, z ¢
G .
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K
abelian. A factor set for
-
Let
is the product
E~
be the topological group which as a topological space
G x K
and has group operation
(xl'kl)(x2'k2)
The properties
13-
of
~
: (:'1:'2'
ensure that
imbedded in the centre of
E~
E~
~(x1'x2)k~k2) •
is a topological
by the map
k,
~(1,k)
group, with
. Since
E~
K
is
locally Euclidean it is a Lie group by the well known result [27] 2.15
of Montgomery,
Zippin and Gleason, and we have an exact sequence of Lie
groups
I
where
~K
~E ~
~G
~1
a(x,k) = x .
Let
LG
and
symmetric bilinear map
the pair
(LG,LK)
LK
be the Lie algebras of
0: LG × L ~
) LK
G
and
K . A skew-
is called a factor set for
if
0(x,[y,,1) + 0(y,[z,x]) + e(z,[x,yl) = 0
for all
x, y, z ¢ LG . The factor set
a linear map
~: LG
~ LK
is trivial if there exists
with
e(x,y)
for all
0
~( [x,yl )
x, y ¢ LG . The quotient of the additive group of factor sets
by the subgroup of trivial factor sets is denoted by
called the 2 nd cohomology ~roup of
space
LK (with trivial action of
LG
LG
H2(LG,LK)
and
with coefficients in the vector
on
LK ) . See [20] III. lo.
and [6] XIII. 8. for more details.
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-
14-
C o n s i d e r now the exact sequence of Lie algebras
&
0
Choose a l i n e a r map
for each
If
8
• LK
B: LG
;LE W ~ *
• LE ~
~ 0 .
such that
&.# = I .
x, y ¢ LG . This is a factor set for the pair
is a trivial factor set then
~: L G
LG
~ LK
is linear,
defines a h o m o m o r p h i s m
~ = ~
and
(LG,LK)
.
where
so that
#: L G - - - ~ L E ~
connected and simply connected,
such that
8(x,y) = ~([x,y])
Put
with
~@~ = I . If
G
then there is a h o m o m o r p h i s m
~o~ = I . Thus
~
is
¥: G
must be of the f o r m
~(~) -- (x,~(x))
where
A
is a continuous map
(xy,~(xy))
G
~ K . Since
_- (~,~(~))(y,~(y))
=
(~y,~(x,y)~(x)~(y))
so that
A(xy)
for all
¥
: ~(x,~)~(x)~(y)
x, y c G .
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is a homomorphism,
E~
- 15 -
We have now proved the following;
Theorem ~.
Let
connected, and
for
G
K
(G, K)
and
K
be Lie groups t
abelian. Let
G
connected and simply
H2(LG,LK) = 0 . Then for any factor set
there is a continuous map
A: G
)K
with
(xy) :
for all
x, y ~ G .
We are now ready to prove the main result on lifting projective
representations.
Theorem 4 (Bar,mann).
group with
T: G
~U(H)
Proof.
?Toa
=
Let
G
be a connected and simply connected Lie
H2(LG,IR) = 0 . Then any pro~ective representation
admits a liftin~
r: g
)U(H)
which is a representation.
By theorem 2 there exists a continuous map
T
.
G
i
We c a n
(and will)
)u(1)
choose
~
>u(H)
so that
,I
o(1)
= I
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.
a: G
) U(H)
with
- 16-
For each
~(=(x)o(y))
x, y ~ G
we have
: ~oo(x).~oo(y)
: T(~). T(y) : T(~y) : ,oo(xy)
.
Therefore
o(x)o(y)
where
~(x,y)
¢ U(1)
. The map
Indeed for any unit vector
[~(x,,y,)
= ~(~,y)[o(~y)
~
= ~(x,y)o(xy)
~: G × G
in
H
>U(1)
is continuous.
we have
- ~(x,y)]o(x'y,)~
- o(x'y')]o
+ o(~,)[o(y,)
- =(y)]~
+ [o(x')
- o(x)]o(y)~
so that taking norms
@
The continuity of
(x,y)~
~
+ll o(y,)~
- o(y)~
+if o(~,)¢
=(x)¢
U(1)
~ a(xy)~ , e(y)~ , a(x)~ . Moreover
is
~: G--->U(1)
Now put
11.
follows from the continuity of the maps
tions for a factor set for the pair
of
Jt
IR
~
satiesfies the condi-
(G,U(1)). Since the Lie algebra
we can apply theorem 3 to obtain a continuous map
with
T(x) = ~ ( x ) ' ~ x )
.
Twww.pdfgrip.com
is the required representation lifting
T .
-
Section 3.
17
-
The Relativistic Free Particle.
We have seen in section I that the relativistic invariance
of a quantum mechanical system requires a projective representation
T:
of the restricted inhomogeneous Lorentz group. A permissible choice of
observer defines a bijective map
M
e ~IR 4
which is a vector space
isomorphism preserving the Lorentz scalar products. The group
of homogeneous Lorentz transformations of
IR4
0(3,1)
relative to its scalar
product
<x,y> = - xlY I - x2Y 2 - x3Y 3 + x4y 4 = x'Ax
is the group of
4 × 4 real matrices
A
with
<Ax,Ay> =
0(3,1) = I A i A'.A.A = A
where
A
is the diagonal matrix with entries
of translations of
IR 4
-I, -I, -I, I . The group
is naturally isomorphic to
of inhomogen~ous Lorentz transformations of
IR4
is the semi-direct
product
m @0(3,1)
relative to the natural action of 0(3,1)
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on
IR 4 , and the group
IR 4 .
-
18-
An inhomogeneous Lorentz transformation of
will appear to the observer as a transformation of
+ a(a)=
M , x:
.~Ax + a
IR 4 , a(x),
) ~(Ax) +
(aAa-1)a(x) + ~(a) . The map
> (~(a),aAa -1 )
(a,A),
is an isomorphism
a,
of
M@~
onto
~R4~0(3,1)
the connected component of the identity in
mapped isomorphically onto
~R4~S0(3,1)
0(3,1) , then
2 x 2
S0(3,1)
M ~
0
be
is
.
S0(3,1)
The simply connected covering group of
the group of
. Let
is
SL(2,¢) ,
complex matrices with determinant I . The covering
map
sT,(2,¢)
> so(3,1)
can be defined as follows. Let
TI =
[:II
O
Identify each
,
T2 =
[oiI
i
,
T3 =
O
x = (x 1,x2,x3,x4) ¢ IR 4
[I:]
,
O
I
with the Hermitian matrix
x I - ix21
x : x1~ I + x 2 ~ 2 + x3~ 3 + x4T 4 : (i I + x3
+ ix 2
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x4
x3 /
:]
-19-
so that det ~ = <x,x> . For each
rl(A)
where
A*
A e SL(2,¢)
........
:
;A#,
is the .,,,i,.m..v.e..Pse-transpose of
let
,
A . Since
det(A~A~) = det
it follows that
w(A) ¢ 0 ( 3 , 1 )
connected group
SL(2,¢)
. Since
W
is continuous it maps the
into the connected group
S0(3,1) . Finally,
is surjective [13] Part II section I § 5 , with discrete kernel
The simpl~
IR4@SL(2,¢)
, relative to the action
Let ~ o
M@%
connected cover of
ym4@so(3,1)
a~: M ® , , ~ °
>A S ~)SL(2,~)
Bargmann [1]
° , M ~
of
is therefore
SL(2,¢)
on
m 4 .
be the simply connected covering group of ~ o ' so that
~,: M ~ o -
M ~
~---~A~A ~
is the simply connected cover of
~(LG,IR) = 0
IR4~S0(3,1)
11,-1} .
where
M~%
. The isomorphism
induces an isomorphism
.
(6.17) has shown that the cohomology group
LG
is any one of the isomorphic Lie algebras of
, m 4~S0(3,1)
, IR 4 @ S L ( 2 , C )
. By the results of
section 2 we can conclude that the projective representation
T
is induced
by a representation
such that
T(I1,-ll) C U(1) . If
T
is irreducible then we call the
quantum mechanical system an elementary relativistic free p~ticle.
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Section 4.
Let
T : G
a Hilbert space
X ¢ LG
20
-
Lie Algebras and Ph2sical Observables.
~U(H)
be a representation of a Lie group
H , and let
LG
be the Lie algebra of
the l-parameter subgroup
continuous) l-parameter group
t~-* exp tX
t ~-~ To exp tX
T(X)
on
H
in
G . For each
is mapped into a (strongly
of unitary operators on
By the fundamental theorem of Stone [34] Theorem B
adjoint operator
G
H .
there is a unique skew-
such that
T@ exp tX = exp tT(X)
for all
t ¢ ~
. For
@ ¢ H , T(X)@
is the derivative at
t = 0
of the
map
]R
~ H , t:
The domain of the operator
T(X)
~(T.exp tX)~ .
is the set of
~
for which this derivative
exists.
There exists a dense set D T
and essentially skew-adjoint for all
determined by its restriction to
skew-symmetric operators having
in
on which
X ¢ LG ; each
D T . Let
DT
H
S(DT)
T(X)
T(X)
is defined
is therefore
be the Lie algebra of
as common invariant domain. Then
T)
is a Lie algebra homomorphism. This implies for instance that if
X,Y ¢ LG
then
T(X)T(Y) - ~(Y)T(X)
with unique self adjoint extension
is esBentially self adjoint on
DT
T[X,Y] . For these results see for
instance [31 1 , Theorem 3. I.
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