- #1
Simon_Tyler
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The notation below, is consistent with Wess and Bagger's https://www.amazon.com/dp/0691025304/?tag=pfamazon01-20.
Given a Majorana spinor field in 4D, written in 2-component notation as
\Psi(x) = \begin{pmatrix} \psi(x) \\\\ \bar\psi(x) \end{pmatrix} ,<br /> \quad (\psi_\alpha)^* = \bar\psi_{\dot\alpha} \ ,<br />
how many linearly independent Lorentz invariants can be formed using 2n spinors and n space-time derivatives, tied together with the sigma/Pauli matrices and various metrics?
And, more importantly for my application, how many are there modulo total derivatives?
Is there a general (eg representation theory) approach to this type of problem?
Notes:
--------------------
For example with n=1 the only invariant is the standard kinetic term
\psi \sigma^a \partial_a \bar\psi <br /> = \psi^\alpha \sigma^a_{\alpha\dot\alpha} \partial_a \bar\psi^{\dot\alpha}<br /> = - (\partial_a\psi) \sigma^a \bar\psi + \text{total derivative} \ .<br />
For n=2, I believe (and want to prove) that there are only 6 invariants up to total derivatives.
Defining the matrix
v_a{}^b = i \psi\sigma^b\partial_a\bar\psi
and its complex conjugate
\bar v_a{}^b = -i (\partial_a\psi)\sigma^b\bar\psi,
I chose the basis(?)
(\partial^a\psi^2)(\partial_a\bar\psi^2) \ ,\; tr(v)tr(\bar v) \ , \;<br /> tr(v)^2\ ,\; tr(v^2)\ ,\; tr(\bar v)^2 \,\; tr(\bar v^2)\ .<br />
Other terms being related by (for example)
tr(v\bar v) = tr(v)tr(\bar v)<br /> +\tfrac12\Big(tr(v^2)-tr(v)^2+tr(\bar v^2)-tr(\bar v)^2\Big)<br /> + \text{total derivative}<br />
Given a Majorana spinor field in 4D, written in 2-component notation as
\Psi(x) = \begin{pmatrix} \psi(x) \\\\ \bar\psi(x) \end{pmatrix} ,<br /> \quad (\psi_\alpha)^* = \bar\psi_{\dot\alpha} \ ,<br />
how many linearly independent Lorentz invariants can be formed using 2n spinors and n space-time derivatives, tied together with the sigma/Pauli matrices and various metrics?
And, more importantly for my application, how many are there modulo total derivatives?
Is there a general (eg representation theory) approach to this type of problem?
Notes:
- I am mainly (at the moment) concerned with the n=2 case.
- Due to anticommutativity, there are no such terms with n>4.
- This can obviously be rewritten using 4-component Majorana spinors and Dirac matrices.
--------------------
For example with n=1 the only invariant is the standard kinetic term
\psi \sigma^a \partial_a \bar\psi <br /> = \psi^\alpha \sigma^a_{\alpha\dot\alpha} \partial_a \bar\psi^{\dot\alpha}<br /> = - (\partial_a\psi) \sigma^a \bar\psi + \text{total derivative} \ .<br />
For n=2, I believe (and want to prove) that there are only 6 invariants up to total derivatives.
Defining the matrix
v_a{}^b = i \psi\sigma^b\partial_a\bar\psi
and its complex conjugate
\bar v_a{}^b = -i (\partial_a\psi)\sigma^b\bar\psi,
I chose the basis(?)
(\partial^a\psi^2)(\partial_a\bar\psi^2) \ ,\; tr(v)tr(\bar v) \ , \;<br /> tr(v)^2\ ,\; tr(v^2)\ ,\; tr(\bar v)^2 \,\; tr(\bar v^2)\ .<br />
Other terms being related by (for example)
tr(v\bar v) = tr(v)tr(\bar v)<br /> +\tfrac12\Big(tr(v^2)-tr(v)^2+tr(\bar v^2)-tr(\bar v)^2\Big)<br /> + \text{total derivative}<br />
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