# A Quantum Field theory profound insight antiparticles

1. Jan 16, 2017

### binbagsss

Hi,

I have recently began studying quatum field theory and have just seen how the quantization of the complex scalar field, noting that there is invariance of the action under a phase rotation shows the existence of antiparticles.

I just have a couple of questions, apologies in advance if they are stupid:

- I'm after some more physical intuition on the phase rotations? I can't find much in a google
- Probably a stupid question: are there any other field types with other in variances that demonstrate the existence of antiparticles in the same way?

Many thanks

2. Jan 16, 2017

### Staff: Mentor

Can you give a reference?

3. Jan 17, 2017

### binbagsss

is it incorrect?

4. Jan 17, 2017

### stevendaryl

Staff Emeritus
What you may thinking of is this:
• From the Lagrangian of the complex scalar field, you can use Noether's theorem to derive a conserved current.
• This current, unlike that of the Schrodinger equation, is not always positive.
• So it cannot be interpreted as a particle probability density.
• But it can be interpreted as a charge density, with contributions from both negatively charged and positively charged particles.

5. Jan 17, 2017

### vanhees71

Any useful QFT textbook is a reference.

The line of argument roughly goes like this: To define a Poincare covariant local (micro causal) QFT you need a combination of positive and negative-frequency solution of free-field equations of motion. Here local means that you have field operators that transform under Poincare transformations (represented by unitary representations of the covering group of the proper orthochronous Lorentz group) as their classical pendants, i.e., for a scalar field
$$\hat{U}(\Lambda) \hat{\phi}(x) \hat{U}^{\dagger}(\Lambda) = \hat{\phi}(\Lambda^{-1} x),$$
where $\Lambda$ is an arbitrary orthochronous Lorentz transformation (and analogously for space-time translations).

In order to have a Hamiltonian that is bounded from below, which we need to have a stable ground state, the negative-frequency modes have to enter as a term proportional to a creation operator and the positive-frequency modes as a term proportional to an annihilation operator (in non-relativistic QFT the field consists only of a superposition of annihilation operators!). In this way the negative-frequency solutions appear as positive-energy modes of either the same particles as represented by the annihilation operators associated with the positive-fequency modes (then for a scalar field you have an Hermitean scalar field) or a different kind of particles, which necessarily have the same mass as the particles but with the opposite charge of the conserved Noether current from the then possible U(1) symmetry (invariance under global phase rotations of the field). These are called the anti-particles associated with the particles.

This analysis goes through for particles with any spin $s \in \{0,1/2,1,3/2,\ldots \}$.

It also follows from the representation theory of the Poincare group that half-integer (integer) spin particles must be quantized as fermions (bosons), which is the famous spin-statistics theorem. Further, any local microcausal QFT with Hamiltonian bounded from below is also necessarily symmetric under the "grand reflection" CPT (charge conjugation, parity=spatial reflections, and time reversal), which is the famous Pauli-Lueders CPT theorem. There's no necessity for C, P, T, or CP to be conserved for such a QFT, and indeed the weak interaction violates all of these discrete symmetries, but of course not CPT.

6. Jan 17, 2017

### binbagsss

mmm I have noether's theorem roughly as: if S is invariant up to a surface term under some transformation this implies there is a conserved current.

The complex scalar field leave S invariance under a complex phase rotation. we find the conserved current.

We then operate with H and Q, H the hamiltionian and Q the conserved charge assoaiced to the conserved current described above, on both a+|0> and b+|0> , where, from the quantisation of the complex scalar field, a+ is the creation of particle type 1 and b+ is the creator of particle type 2, and the outcome is that operating with H returns the same energy(mass), whilst operating with Q returns the same charge but with a different sign, thus showing the existence of particles and antiparticles.

7. Jan 17, 2017

### stevendaryl

Staff Emeritus
Yes, what you're saying is right. I was saying that even at the level of a classical field theory, the current for a complex scalar field must involve charge densities of both signs.