java barcode generator library THE R SECTOR in Java

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THE R SECTOR
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Now we review the R sector in more detail. The left + and right modal expansions are
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+ ( , ) = d n e 2 in( + )
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( , ) = d n e 2 in( )
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(11.3)
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Quantization proceeds using the anticommutation relations:
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, d n = d m , d n = m+n ,0
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(11.4)
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These relations are augmented, of course, by the usual bosonic commutation relations. We also have the number operators: N ( d ) = nd n dn
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n =1
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N ( d ) = nd n dn
n =1
(11.5)
CHAPTER 11 Type II String Theories
The total number operator for the left-moving and right-moving sectors is given by adding the bosonic number operator N = n n
n =1
to Eq. (11.5). We obtain NL = N + N (d ) N R = N + N (d ) (11.6)
Taking n > 0,we can de ne creation and annihilation operators as follows:
d n acts as a creation operator by adding n to the eigenvalue of N ( d ). d n acts as an annihilation operator by subtracting n from the eigenvalue of N ( d ) .
The states are constructed in the usual way by using a fock space (or number states). Letting n > 0 , the ground state is annihilated by the bosonic and fermionic annihilation operators:
n 0
= dn 0
(11.7)
for the right-moving modes, and similarly for the left movers. We can construct an arbitrary state by tracing on the ground state multiple times: n
= ( ni ) pi ( d m j ) 0
qj i j
(11.8)
For states in the right-moving sector, the expression for the left-moving sector is similar. Now consider the special case of d 0 . The anticommutation relation is
, d 0 =
(11.9)
This is almost the same as the commutation relation obeyed by the Gamma matrices (i.e., the Dirac algebra ): { , } = 2 (11.10)
String Theory Demysti ed
This suggests that these operators are related to the Gamma matrices using
= i 2d 0
(11.11)
This tells us that the states in the R sector are space-time spinors. We can write the ground state as 0
where a is a spinor index that ranges over a = 1, , 32. As we have seen earlier, this is because a general Dirac spinor has 2D/2 components where D is the number of space-time dimensions. Since D = 10 for superstring theories, there are 32 a components. The state 0 R is a 32-component Majorana spinor. Now recall that the chirality operator 11 = 0 1 9 acts on states 0 R of de nite chirality according to 11 0 11 0
+ R R
=+ 0 = 0
+ R R
(11.12)
States with de nite chirality are Majorana-Weyl spinors, which have half the a number of components, (16 in this case). We can write the state 0 R as a direct sum of positive and negative chirality states: 0
(11.13)
This gives the state [Eq. (11.13)] 16 16 = 32 components. The states 0 are R + space-time fermions. However, they have the bizarre property that 0 R is bosonic and 0 R is fermionic on the worldsheet.
THE NS SECTOR
In the NS sector, we have the modal expansions of the left- and right-moving fermionic states given by
+ ( , ) = ( , ) =
r Z +1/ 2
br e 2 ir( + ) br e 2 ir( ) (11.14)
r Z +1/ 2
CHAPTER 11 Type II String Theories
The expansion coef cients satisfy the anticommutation relations:
{b , b } = {b , b } =
r s r s
r +s ,0
(11.15)
The number operators are N (b) =
r =1/ 2
rb r br
N (b) =
r =1/ 2
rb
br
(11.16)
This allows us to de ne number operators for right- and left-moving modes: N R = N + N (b) NL = N + N (b) Letting n > 0 : bm acts as a creation operator, increasing the eigenvalue of N(b) by r. -r bm acts as an annihilation operator, decreasing the eigenvalue of N(b) by r. r Again we construct the states using a fock space. The ground state is annihilated in the following way:
n 0
(11.17)
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