# Wave function - Dirac Notation

Based on that notes (equation 54): https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/stuff/dirac.pdf

I was reading about the wave functions and I have a question about the notation. You can define the wave function like:

$\psi(p) = e^{-ix_{\mu}p^{\mu}}u(p)$

My question is: Why there is no dependence in the $\mu$ on the left hand side? The dependence in the wave function is only taking care $u(p)$.

UPDATE:

My question is because I am doing something more advanced and I have this same question, but for the wave function related to the matrix elements and fermions representation:

I have the follow definition for the pseudoscalar current:

$P^{ij}_5 (\vec{x}, t) = \sum_{y} \bar{\psi}^{i} (\vec{x}+\vec{y}, t) \gamma^5 \psi^{j} (\vec{y}, t)$

and then the wave function is defined as:

$\Psi_{P}(\vec{x}) = <\Omega|P^{ij}_5(\vec{x}, 0)| P>$

i and j are flavour indices.

then the same question: why there is no dependence in the LHS of the equation? Should I apply some theory or some convention?

Paper with the information: http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/22/036/22036311.pdf

• The IJ in the bra/ket are free indices. They should be carried in the PSI in the LHS, too. Jan 31, 2018 at 16:09
• @DanielC thanks. Also the index of the gamma matrix (dirac matrix) in this case \gamma_5 or for the vector case \gamma_\mu should also appear in the LHS? I mean do you think people hide it to be more easy to write or there is another explanation? Also last question, I don't know if you will be able to answer but for this wave function they say it is "gauge fixed". Do you know what it means? Thanks a lot Jan 31, 2018 at 16:48
• Might be useful to include context of the $\Psi_P(x)$ equation (author, publication or text title, etc) Feb 1, 2018 at 0:39
• @KyleKanos this is the link for the paper: iaea.org/inis/collection/NCLCollectionStore/_Public/22/036/… Feb 1, 2018 at 10:12

It seems that the notes you are using have used Einstein's summation convention - this means that when you have an index appearing both as a subscript and a superscript, it is summed over, i.e. $$x_\mu p^\mu := \Sigma_\mu x_\mu p^\mu$$ Thus since the index is summed over, there is no dependence on the LHS.

• Thanks for your answer, but I have update the question, considering my goal. Jan 31, 2018 at 14:42
• It seems to me that the Pij are matrix elements, so the bra should provide (say) the other i term over which to sum and the ket should provide the j. So again since you are summing over then there is no dependence on the LHS. Feb 1, 2018 at 1:37

The equation above makes use of the Einstein's summation convention. According to this convention, all the repeated indices get summed over. Eq: $$x^\mu y_\mu \equiv \sum_{\mu=0} x^0y_0 + x^1y_1 + ...$$ Therefore, there is no dependence on $\mu$ on the LHS. The index $\mu$ is called a dummy index.

Generally, Greek symbols are used to denote the space-time variables ($\mu = 0,1,2,3$) with appropriate metric definition and Latin symbols for space variables ($i=1,2,3$). However, many people use it differently.

The upper index $x^{\mu}$ refers to the contravariant quantity and a lower index $x_{\nu}$ refers to a covariant quantity with a metric connecting the two as $$x^{\mu}=g^{\mu\nu}x_{\nu}$$

• thanks for your answer. I have update the question. But thanks a lot, because I think I can get some ideas for the next step. Jan 31, 2018 at 14:42
• I guess, you refer again to the indices $i$ and $j$. In this (second) example Einstein's summation convention does not apply, because there are no repeated indices. Actually from your description it is not clear what $i$ and $j$ mean. So please tell us what they mean, respectively to what they refer. Jan 31, 2018 at 15:14
• @FredericThomas we have i and j different. And they are the flavour indices. Jan 31, 2018 at 16:17