Kinematic and Dynam ic Subgroups of the Point Form

For the case of the point form it is very easy to isolate the dynamic and kinematic sub­ algebras. From formula 0.17 page 6 [47] we know that a general in fin itesim al Lorentz transformation can be w ritten

where Suppose we choose a transformation where the only non zero e’s are and of course then

-1-

= g° - (2.41)

’See Appendix 7 for finite forms

and

gl gl 4- glt'g^

= gl _ gOigO.

(2.42)

We can also express this transformation in terms of light cone coordinates. Since r = ] £ 1 if we put a = we have

r —> t H- aq^ + ((g^ + a t? + g^^ + g^^)2 = i + ag H- (I g 1^ 4-2o:gf)i ~ t + ag+ | g |

— t + aq+ I g j +0! r ^ = r -f oig ( 1 + Y~~t ) .

I l l V ig ly

For points on the light cone w ith apex at the origin we have f = — | g [ so that r —> r (= 0).

We also have that = g^ so

—> g^ + CKf = 4- at.

For a particle on the light cone through the origin this becomes

y i y i - û: i g 1= y i - q: j y I •

F inally

y^ -» y^ and y^ y^

[62]. The fact that r —> r shows that this type of transformation leaves the light cone w ith apex at the origin invariant. We can also show this just using Minkowski coordinates. Supose (t, g) lies on the light cone i.e. t = — j g |. P ut a ~ — t hen from ( 2.41) and

( 2.42) t' = gO 4- aq^ and 1 q' 1~ {{q^ 4- aq^? F g^ 4- g^ )^

= (I 9 +2ag°g^)i =| g | ("l +

=| g j 4-ay~

|

= —(g^ + aq^) 115

since | g |. Therefore t' — — \ ^ \ and the light cone is preserved. This effectively demonstrates the invariance of the light cone under boosts. We shall denote the generators of the special Lorentz transformations Therefore we have shown that the belong to the kinematic subalgebra of the point form.

Suppose now we consider a transformation where the only non zero e’s are and In this case

4- ~

and

I f we interpret the transformation passively we might put

t ' ~ t , q ^ '= q ^ , q ^ '- q ^ + ctq^^ q ^ '~ q^ - aq^ (2.43) where a — It is also possible to express this Lorentz transformation in terms of light cone coordinates. Since

T = t- h \q \ and y = q

we have

r ' = t + [(gi - + g^^ + (g^ + aq^)‘^]^ t + (g4^ — 2aq^q^ + a^g^^ + g^^ + 2oi(fq^ 4-

= f+ I g 1= T. Sim ilarly ^ 1/ 1 3 1 3 yi. _ — gj- _ Ciq^ = — ay y 2 ' = g2' _ ^ 2 _ y 2 y^' = q^' = g^ + aq^ - 2/ ^ 4- a y ^.

Taking the active point of view we would express this as

T - > T , y ^ ~ ^ y ^ ~ - > y ^ y ^ + a y ^

[62]. Notice that we have r r showing that this transformation preserves the light cone. We can also see this just working w ith the Minkowski coordinate description ( 2.43). Suppose t lies on the light cone w ith apex at r then t = r — ] g |. Now

1 q' !=

+ {q^ + W Ÿ + (g^ - «g^)^)^

116

= (g^ +g^ + 2org^g^ + a^g^ +g^ - 2og^g^ + cx^g^ )% =1 4 I .

Also

t' ~ t

so that

t' = r - I g' I .

Therefore (i, g) is moved to a new point on the light cone. This démontrâtes that the light cone is invariant under spatial rotations. The generators of spatial rotations, denoted ~T^, also belong to the kinematic subalgebra of the point form.

I t is easy to show tha t none of the translations leave the light cone invariant. For example under p i

t “ >t, g ^-> g ^ + o, g^ g^ g^ -+ g^.

g^a Clearly t' — t = ~ \ q \ whereas

I i 1= ((gi + Oif+ q^^ + g3^)s = (1 £ 1^ +2g^a)^ =1 g 1 ^ ^ ^

The last term in the above spoils the invariance of the light cone since w ith it ^ | g' |. Summarising we see that the kinematic and dynamic subgroups of the point form are the homogeneous subgroup and the group of translations respectively.

In a sim ilar way we can show that the generators of kinematic subgroup of the instant form are ~P^ and and the Hamiltonians are and IT°.

The Front Form Operator Representation of the Poincare Algebra in Basis Adapted to the Point and Instant Forms

The Poincare Algebra is given by

== (2^14)

( F ’.p ''] = 0 [50) page 150. Suppose we put

p '‘ = p f, J “‘ = pV . 7 ’ = pV - ? V (2.45)

where p° = (p ^+ m ^)2. These classical observables obey the commutation relations ( 2.44) under Poisson bracket { , } which is defined in the usual way from the symplectic 2-form as f f q l == ’ d q ^ d p s d q ^ d p s ' For example p ° ‘ , F ] = n«P” - r ) “ F = - P ° and = _pO _ _ p O as required. Also whereas c Ps i ~rit = PoOis— ~ P i = —p ~ —P . Po As a final example consider

[T^,P ^] = = F where of course we have assumed i ^ j. We have

{ F , F } - { ~ p i ( ^ + g 'P j, - P i }

— Pj^is^is *b Pi^js^is'

Since i ^ j this becomes

as required.

As is well known these classical observables can be quantised geometrically in Tirp

to obtain an operator representation of the Poincare algebra. We can show that when expressed in terms of front form variables the classical generators are in C °°(to M p ,n . 1) and in fact can be quantised to give self-adjoint operators in W hat is more the pairing maps effect a unitary transformation between the representations.

In terms of front form variables we have

~t32 _3 ___

j - y tt2-

- g - , : so l f + { m c Ÿ \ ^ i f

-rlO

-f3\J — 5— / U * \ __ J = - y 7Ti - I = ■■■ — --- } 7T J = - ÿ '^1 p O _ / 7r^ + (m c )2\ 1_ /7r f - 7r | - 7r | - ( m c )2\ ^ - ( 2.1 ) ' ^ 2Ï I ---) ' ^

In Appendix 8 we show that these classical observables generate complete vector fields and can be quantised to give the following self adjoint operators

^ y2 ^ dy^J

5if-a(»‘|r + |)

—1 f - ÿ 2^ - (mc)2\ —

^ \ 2y i j ^ ^ 2y i ^ ^

We can show that these operators are u nita rily related, via the pairing maps, to the usual representation of the Poincare algebra in H p . For example

== -z)*g3 ==

Clearly € C °°{toM i, P, 1) and in fact generates a complete vector field so that it can be quantised in Tip to obtain the self-adjoint operator

® Now we shall show that

^np'-^nn‘^n ^ n n ^ p n W ell

C % n 7f t^nn = ( - D ^ n +

Ur,TsUr;r,J^Ur,r,U-^r, = J ? .

’This can also be written

-t3 0 _{Jp ~ —iri—p=.pQ—}1 d _—ypo. ---

VPo ap3

This makes it particularly easy to see that our representation of the Poincare generators is the same aa appears in [94) remembering that we use a different metric on the cartesian momentum space.

= n M + ® + H ) ! ) y „ 3 + , , ^ SO üpf'cui'/S'z/niiffpri -- (-;;*)(--;%) (^gârq:^;r2;=ï -H ÿ ( ÿ ) + ÿ ) _ _0/ ..\ t

d

f \ iîi ÿ ^

\ d t

2gO(ÿ + g i) j 2 gi+gO a , g V ^ ÿ

(gi + ÿ ) agi 2 f ( ÿ 4- gi) ^ 2 (ÿ + ÿ )

2 f { f ‘ + q'-) { f + ' t ) ) '

*

+ 1®)= - * (î’o é + # )

'!

as required. As a further example we shall show that

^ n p ^ n n '^ n ^ n n ^ p n “ "^P- We have

so

^nn*^n ^ n n ~ (“ ^ n )^ n “

^ n p ^ n n '^ n ^ n n ^ p n ~ + flp Y —

ag2 \ g i 4-^ y agi 2ÿ(gO + gi)

dq^ \ ÿ 4 - ÿ y agi 2 ÿ ( ÿ 4- gi) dq2 gi 4- ^ agi 2 ÿ ( ÿ 4- gi) ;^2 ^ .* g ^ ÿ <9 , .. ÿ ÿ 4~î7ig Tczô — fwT ”h ag3 agi 2 ^ (g ^ 4 -g i)

=

-

^ w

)

' ' %)

whereas according to [94] or using geometric quantisation

as required. In this way we demonstrate that the classical Poincare generators can be quantised in the instant or front form pictures and lead to operator representations of the Poincare algebra that are unitarily related by the pairing maps.

In document Geometric quantisation and quantum mechanics in Dirac's front form (Page 125-132)