Q charge in Supersymmetry

[BJC]

Does anyone know how to expand the last term. My understanding is

sigma refers to the 3 Pauli matrices with the addition of the identity matrix for time

theta the Grassmann coordinates

mu space time Einstein summation index

 

My assumption is this expands to 2 equations (Q, theta are 2x1 matrices) and the theta sub-index on the sigma matrices means to select the alpha row, beta column for each of the sigma matrices.

 

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Also my first post - i would appreciate hints on formatting, especially how to insert HTML code symbols for Greek

α θ

etc.

[A Tripolation]

I can't help with the math, but I can direct you to the LaTeX guide. This forum is nice in that it allows us to make our math pretty.

[ajb]

You could pick a basis for the Pauli matrices and then write it all out. Can I ask why you want to so this?

[BJC]

You could pick a basis for the Pauli matrices and then write it all out. Can I ask why you want to so this?

 

Can I ask why you want to so this?

Curious question? Part of a web-based course on Super symmetry.

 

In the algebra of super symmetry the 4 Q generators form two 2-spinor elements that relates the 4 Grassmann coordinates with the 4 space-time coordinates. The Q generators can be thought of as rotations or translations in super space.

 

The 4 D generators are required to insure invariance in the Lagrangian. (The 4 D generators are similar, in concept, to the derivative operator for Gamma corrections in general relativity.)

 

 

You could pick a basis for the Pauli matrices and then write it all out.

what is "a basis for the Pauli matrices" ?

 

How to "and then write it all out" is my question.

If you notice the last differentiation term there is the Einstein summation terms ( [latex]\sigma^{\mu} \partial_{\mu}[/latex] ) but i assume that only a single variable is required so i am assuming the Grassmann coordinates alpha, beta indexes on sigma represent a row, column in the sigma matrix

 

Let alpha=1, beta=2 in this example

[latex]\sigma^0_{12} \partial_t + \sigma^1_{12} \partial_x + \sigma^2_{12} \partial_y + \sigma^3_{12} \partial_z[/latex]

 

which inserting (row 1, column 2) element of the sigma matrices (identity of t, time) gives

[latex](0) \partial_t + (1) \partial_x + (- \imath) \partial_y + (0) \partial_z[/latex]

 

or the entire expression as

[latex]Q_1 = \partial_{\theta_1} - \imath \partial_{\theta^*_2} (\sum_0^3 \sigma^{\mu}_{12} \partial_{x_{\mu}})[/latex]

 

which seems okay but i am not sure ????

[ajb]

In the algebra of super symmetry the 4 Q generators form two 2-spinor elements that relates the 4 Grassmann coordinates with the 4 space-time coordinates. The Q generators can be thought of as rotations or translations in super space.

 

Yes, that is the idea. The supersymmetry transformations should be like a translation in superspace.

 

The 4 D generators are required to insure invariance in the Lagrangian. (The 4 D generators are similar, in concept, to the derivative operator for Gamma corrections in general relativity.)

 

Yes, you need to introduce a covariant derivative. Interestingly, if the correct operators were just the derivatives with respect to the odd coordinates then supersymmetry would not mix even and odd coordinates.

 

 

what is "a basis for the Pauli matrices" ?

 

How to "and then write it all out" is my question.

 

Being slack with the indices you should have defined something like

 

[math](\sigma^{\mu})_{a\dot{b}} = (1 , \sigma^{i})[/math],

 

so all you have to do is pick a representation of the Pauli matrices. I don't know what one would be convenient to you.

 

I am not sure if there is much to gain writing it out fully. It has been a while since I looked into supersymmetry.

Edited April 1, 2011 by ajb

[BJC]

Yes, that is the idea. The supersymmetry transformations should be like a translation in superspace.

 

Yes, you need to introduce a covariant derivative. Interestingly, if the correct operators were just the derivatives with respect to the odd coordinates then supersymmetry would not mix even and odd coordinates.

 

 

Being slack with the indices you should have defined something like

 

[math](\sigma^{\mu})_{a\dot{b}} = (1 , \sigma^{i})[/math],

 

so all you have to do is pick a representation of the Pauli matrices. I don't know what one would be convenient to you.

 

I am not sure if there is much to gain writing it out fully. It has been a while since I looked into supersymmetry.

 

Thank you - i just started this physics stuff after retiring (50 years since university) - good to see some of my intuition is still functioning.

 

I have encountered this

[math](\sigma^{\mu})_{a\dot{b}} = (1 , \sigma^{i})[/math],

but was not able to find what the shorthand means.

No - there is not much physics in writing it out fully but the process helps my understanding

 

First time on this forum. The editor, especially the preview, is quite good. LaTex is cool - very logical

 

The biggest problem i am encountering is all the implied shorthand notations in physics and remembering all the "by convention, this symbol means .."

[ajb]

supersymmetry and supermathematics generally is full of conventions that differ person to person. The biggest trouble is keeping track of the minus signs.

 

I think the shorthand

 

[math](\sigma^{\mu}\overline{\theta})_{a}[/math] means

 

[math](\sigma^{\mu})_{a\dot{b}} \overline{\theta}^{\dot{b}}[/math]

 

where you have raised the indices by using [math]\epsilon^{\dot{a}\dot{b}}[/math]. There may be an extra sign in the above, you would have to check that carefully with the conventions you are using. (I think I would reorder that myself and include any further signs needed.)

 

I don't know if this helps at all, but I think of supersymmetry very generally as being described by particular Grassmann odd vector fields on supermanifolds. Very loosely there are two kinds of odd vector field

 

1. Homological
   
2. Super
   

 

Homological satisfy

 

[math][Q,Q] = 0[/math].

 

 

Note this is nontrivial as the Lie bracket is graded.

 

Super is "everything else",

 

[math][X,X] = Y[/math].

 

 

Both kinds of odd vector field turn out to be very important in physics.

Edited April 1, 2011 by ajb