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[Phys-l] S Duality and Time Reversal Invariance




S Duality and Time Reversal Invariance


As outlined in a previous post a direct connection can be made between T
Duality and time reversal, per a proposal by Gasperini and Venziano in their
string theory cosmology. This raises the question, is there any connection
between S duality , the duality between Noether ( local) and Topological
charges? I am going to assert the answer is yes.

To begin this I need to introduce some unconventional notation. These are
the;

Time Operator T(op)


( What serves as the time operator is really a Hamiltonian with potential
energy terms but here we will ignore the bound from below issue.)

The Parity Operator , space reflection. P(op)


The charge conjugation operator C(op)

which reverses all Noether charges.


However, we need to introduce the reinterpretation versions of these
operators we shall call

Tr(op), Pr(op) and Cr(op)

The reinterpretation operator reveres the action of its associated non
reinterpreted operator.


The reason for this strange notation and the need for reinterpretation
operators is illustrated by the Feynman reinterpretation principle where anti
particles are defined as normal particles reverses in time. The local
charge reversal is the result of this principle. These operators form an Abelian
Z4 symmetry group where;

TP= C =PT

TC=P=CT

CP= T= PT

CC=PP=TT=I

CPT=I


This symmetry reflects the Feynman treatment of these operators in his
paper

" The Reason for Antiarticles"


Where he writes

" If we define P as the parity operator which changes the sign of the
three spatial dimensions , T as the time reversal operation which changes the
flow of time , and finally C as charge conjugation which changes particles
to antiparticles and visa versa , then operating on a state with P and T is
the same as operating on the State with C, that is PT=C

end quote.

All this will become important in connecting S duality and time reversal
invariance. The role of particle Helicity will also become of some interest
in this modified operator notation. It will be revealed that we should think
of particle helicity as a topological charge.

S Duality

The Duality between Noether and topological charges play a very important
role in particle physics , especially in QCD. This Duality draws a
connection between the unipole magnetic charge and its associated Noether charge. We
know that based on Maxwell's equation electric and magnetic charges are
related by the equation;


F= q_e*{E+v/c X B}+ q_m*{B-v/c XE}

And we can Re Write Maxwell's equations as


DivE =4*pi*rho_e

DivB =4*pi*rho_m

GradE = - (1/c)*pd(B)/pd(t)+4*piJ_m/c

GradB= (1/c)*pd(E)/pd(t)+4*pi*J_e/c

That is moving electric charge created a magnetic field and moving
magnetic charge creates a electric field.

Since we have never observed a monopole we normally write this equation
as;


F= q_e*{E+v/c X B}


However, in S duality monopoles are valid solution in QFT. There we get
the Polyakov- t'Hooft QCD monopoles for example. Using the Polyakov-t'Hooft
proposal we can model quarks as composite monopoles , that is the result a
tangle of monopole fields. Based on Duality we might also model the
Polyakov-t'Hooft monopole as a tangle of quark fields.

So in essence a Noether charge as a result of moving magnetic charge is
dual to a topological charge as the result of moving electric charge. This has
interesting implications for time reversal invariance.

( Note one might think that you can't get unipole solutions this way but in
fact the Polyakov -t' Hooft model and other S dual monopole models do
produce Unipole solutions based on relativistic motion. (See Scientific
American January 1996 " Everything Explained" by Madhusree Mukerjwee. )

Given an electric charge , we know with certainty that;

T[+>= [+>

Electric charge does reverse when acted upon by the time operator.

But if we model this charge as the result of moving magnetic charge then we
must assert that

T[N>= [S>

Time reversal must flip the polarity of the magnetic monopole.

Conversely if this is true when we model a magnetic monopole as the
result of moving electric charge then the electric charge must not flip polarity
under time reversal. We see that the S dual description remains
consistent.

(We now see why when Carrigan and Trower in their Scientific American
article assert that the existence of magnetic monopoles violate time reversal
invariance they are wrong. This article resulted in a very active discussion
on this list a few years ago. Their assertion assumes that the polarity is
invariant under time reversal.)

Therefore;


Tr(op)[+>= [-> Tr(op)[N>= [N>


And

Pr(op)[+>= {+} Pr[S>= [N>

Which allows for

C_r(op)[+>= {-> Cr(op)[S> =[N>

Note also that


T_r(op)[e(-)_L> = [e(+)_L >

and

Pr(op)[e(-)_L> [e(-)_R>


We see that helicity transforms as a topological charge.


This meshes well with the left right symmetry and charge conjugation in
SU(5) and SO(10) where


C_r(op)[F(+)_L>= [F(-)_R>


Therefore we must conclude that time reversal invariance requires that
Noether and topological charges operate in a complementary manner under space
and time reflection.

Bob Zannelli