• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Real Majorana wavefunction / field: What is the big deal?

+ 4 like - 0 dislike

It is known that there is a set of gamma matrices that can be purely imaginary (called Majorana basis), thus one can solve the 1st quantized Majorana wave function in terms of real wave function.

However, I am confused by the implications for this real Majorana wave function.

  1. Isn't that the wave function should still be complex under time evolution? If so, what is the big deal to call this real Majorana wave function?

  2. What is the meaning for this set of real Majorana wave function, when we go from 1st quantized to 2nd quantization language?

p.s. One should clarify 1st quantized and 2nd quantization languages. The Ref below seems to mix two up.

Below from Wilczek on Majorana returns:

enter image description here

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
asked Jun 14, 2018 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]

1 Answer

+ 7 like - 0 dislike

Neat question. The short answer is no, the real spinor does not turn complex under time evolution. That's at the very heart of the "big deal".

To see this explicitly, go to the rest frame, as you may always Lorentz-transform yourself back. So, the Dirac equation reduces to just $$ (i\tilde \gamma ^0 \partial_t - m)\psi =0, $$ hence $$ (\partial_t +im\tilde \gamma^0)\psi=0 . $$ Note in this representation $i\tilde \gamma ^0$ is real and antisymmetric, and thus antihermitean, as it should be!

Since it does not mix up the real and imaginary parts of the Dirac spinor $\psi$, we may consistently take the imaginary part to vanish, so the spinor is real: A Majorana spinor in this, Majorana, representation. (In the somewhat off-mainstream basis you display, you may take $C=-i\tilde \gamma^0=-C^T=-C^\dagger=-C^{-1} $.)

The evident solution, then, is $$ \psi (t) = \exp (-itm\tilde \gamma ^0 ) ~~\psi(0)=\exp (-itm ~\tilde \sigma_2\otimes\sigma_1 ) ~~\psi(0)~~, $$ where, as seen, the exponential is real, so the time-evolved spinor is real, as well, forever and ever. Thus, $$ \psi (t) = (\cos (tm)-i\tilde \gamma^0 \sin (tm) ) ~\psi(0) . $$

Now, thruth be told, this is an "existence proof" of the consistency of the split. Few of my friends actually use the Majorana representation in their daily lives. It just reminds you that a Dirac spinor is resolvable to a Majorana spinor plus i times another Majorana spinor.

The properties of the Dirac equation are the same in both first and second quantization, so all the moves and points made also hold for field theory as well, without departure from the standard textbook transition of the Dirac equation.

  • Edit in response to @annie marie hart 's question.

In effect, the complex nature of Schroedinger's equation is replicated with real matrix quantities, given $\Gamma\equiv i\tilde \gamma^0 ~\leadsto ~\Gamma^2=-I$. As a result, all complex unitary propagation features of Schroedinger's wavefunctions are paralleled here by the real unitary matrices acting on spinor wave functions, normalized in the technically analogous sense.

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user Cosmas Zachos
answered Jul 23, 2019 by Cosmas Zachos (370 points) [ no revision ]
thanks very much for the nice answer. vote up

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
Another way of saying this could be to think of the free Hamiltonian as a matrix $\chi^T H \chi$, where $\chi$ are Majorana fields. If this is to be nonzero then by fermionic statistics it must be antisymmetric, and since $H$ is Hermitian we can conclude that the Hamiltonian for a collection of Majoranas must be purely imaginary. This means that the time evolution $e^{-iHt}$ acts as a real matrix.

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user user3521569
How do we appreciate better that the $\exp(-iEt)$ type of time evolution does not get involved? In the naive picture we have $H |\psi_E> = i \hbar \partial_t |\psi_E>$ with a textbook solution $ |\psi_E>= \exp(-iEt) |\psi_0>$ and $H |\psi_0>=E |\psi_E>$. But $\exp(-iEt)$ complexes the solution...

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
Also are all the real solutions, (such as $\psi (t) = (\cos (tm)-i\tilde \gamma^0 \sin (tm) ) ~\psi(0) $) normalizable as in the Hilbert space requires $\int |\psi (t)|^2 d^3x=1$?

This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights