duality for the electromagnetic fieldjalfaro/supergravity/v-supergravity90-110.pdf · duality for...
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Duality for the electromagnetic field
Interacting theory with one ableian gauge field and a complex scalar
Bianchi identity and equation of motion
Define the tensor
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Duality for the electromagnetic field
These equations are invariant under the transformation
Where
is an SL(2,R) transformation
If Which is the transformation of the scalar
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Duality for the electromagnetic field
Magnetic and electric charges appear as sources for the Bianchi identityand generalized Maxwell equation
Transforms like
Schwinger-Zwanziger quantization condition for dyons
We have
S-duality
group
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Duality for the electromagnetic field
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Duality for the electromagnetic field
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Duality for the electromagnetic field
Dimension of the symplectic group m(2m+1)
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Duality for the electromagnetic field
• Duality transformations-symmetries of one theory (S-duality)-transformations from theory to another theory (M-
theory applications)
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Non-abelian Gauge FieldAn element of the gauge group in the fundamental representation
Gauge potential
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Non-abelian Gauge Field
Covarint derivatives
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Non-abelian Gauge Field
Bianchi identity
Where
Action
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Non-abelian Chern Simons
Equation of motion
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Internal Symmetry for Majorana Spinors
real and therefore compatible with the Majorana condition
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Internal Symmetry for Majorana Spinors
• D=4 Complex representation, we have the highest rank element
The chiral projectiosn transform
Variation of the mass term
If G=SU(n), the mass term is preserved by SO(n)
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The free Rarita-Schwinger field
Consider now a free spinor abelian gauge fieldwe omit the spinor indexes
Gauge transformation
This is fine for a free theory, but interacting supergravity theories are more restrictive .We will need to use Majorana and/or Weyl spinors
Field strenght gauge invariant
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The free Rarita-Schwinger field
• ActionProperties: a) Lorentz invariant, b) first order in space-time derivativesc) gauge invariant, d) hermitean
The lagrangian is invariant up to a total derivative
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The free Rarita-Schwinger field
• Equation of motion
Noether identities
Using
We can write the equations of motion as
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The free Rarita-Schwinger field
• Massless particles
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The free Rarita-Schwinger field
• Initial value problemThe gauge fixes completely the gauge
the equations of motion in components
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The free Rarita-Schwinger field
• Initial value problem
We have also
The restrictions on the initial conditions are
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The free Rarita-Schwinger field
• Initial value problemThe on-sheel degrees freedom are half of
In D=4, with Majorana conditions, we frind two states expected for aa masslees particle for any s>0. The helicities are +3/2 and -3/2
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Degrees of freedom