# Particle production in 1+1D QFT

• A
• QFT1995
In summary, the Massive Thirring Model has a Hamiltonian$$\int \mathrm{d}x \imath \Psi^\dagger \sigma_z \partial_x \Psi + m_0 \Psi^\dagger \Psi + 2g \Psi^\dagger_1 \Psi^\dagger_2 \Psi_2\Psi_1\\$$and conservation laws$$N = \int \mathrm{d}x \Psi^\dagger \Psi$$ is commutes the HamiltonianParticle production is not present in the theory, because all self-energy
QFT1995
I am currently studying the Massive Thirring Model (MTM) with the Lagrangian
$$\mathcal{L} = \imath {\bar{\Psi}} (\gamma^\mu {\partial}_\mu - m_0 )\Psi - \frac{1}{2}g: \left( \bar{\Psi} \gamma_\mu \Psi \right)\left( \bar{\Psi} \gamma^\mu \Psi \right): .$$
and Hamiltonian
$$\int \mathrm{d}x \imath \Psi^\dagger \sigma_z \partial_x \Psi + m_0 \Psi^\dagger \Psi + 2g \Psi^\dagger_1 \Psi^\dagger_2 \Psi_2\Psi_1\\$$
Due to the infinite set of conservation laws, particle production is said to be absent from this theory. However why isn't it sufficient to show that particle production is absent if the number operator $$N=\int \mathrm{d}x \Psi^\dagger \Psi$$ commutes the the Hamiltonian? Also, by particle production being absent, is that just a statement that all Feynman diagrams with self energy insertions evaluate to 0 but all other Feynman diagrams are possible?

Particle production is a nonperturbative phenomenon, and cannot be explained in terms of Feynman diagrams! The number operator is usually undefinable in an interacting theory - it doesn's survive renormalization; it is certainly not given by your expression.

A. Neumaier said:
Particle production is a nonperturbative phenomenon, and cannot be explained in terms of Feynman diagrams!
What? In the Standard Model, particle production is definitely a perturbative phenomenon described by Feynman diagrams.

Delta2 and vanhees71
Why isn't something like the following classed as particle production?

QFT1995 said:
Why isn't something like the following classed as particle production?

Because only one line comes in and only one line goes out. "Particle production" would mean more lines go out than come in.

PeterDonis said:
Because only one line comes in and only one line goes out. "Particle production" would mean more lines go out than come in.
Okay thank you. Would you be able to explain to me (or provide a reference) of how particle production arrises in non-perturbative QFT?

QFT1995 said:
Okay thank you. Would you be able to explain to me (or provide a reference) of how particle production arrises in non-perturbative QFT?
A nice class of examples are particle production from classical sources. See e.g. Itzykson and Zuber, Quantum Field Theory, Sec. 4-1-1.

Demystifier said:
What? In the Standard Model, particle production is definitely a perturbative phenomenon described by Feynman diagrams.
No. Feynman diagrams describe (usually infinite) contributions to S-matrix elements. Only the S-matrix as a whole describes scattering processes - among others those involving particle production.

Demystifier
QFT1995 said:
Okay thank you. Would you be able to explain to me (or provide a reference) of how particle production arrises in non-perturbative QFT?

A. Neumaier said:
No. Feynman diagrams describe (usually infinite) contributions to S-matrix elements. Only the S-matrix as a whole describes scattering processes - among others those involving particle production.
Are you saying that only exact calculations describe physical processes, while approximations don't?

Demystifier said:
Are you saying that only exact calculations describe physical processes, while approximations don't?
No. But approximations should be at least finite. The only Feynman diagrams leading to finite contributions are the tree diagrams, and these give poor approximations.

Already to get masses for the particles one needs to resum and renormalize infinite series of diagrams with loops. This resummation is already regarded as a nonperturbative step.

A. Neumaier said:
No. But approximations should be at least finite. The only Feynman diagrams leading to finite contributions are the tree diagrams, and these give poor approximations.
In some cases those are good approximations, and are certainly perturbative.

A. Neumaier said:
Already to get masses for the particles one needs to resum and renormalize infinite series of diagrams with loops. This resummation is already regarded as a nonperturbative step.
But in many cases one can renormalize a finite number of diagrams, typically a one-loop integral. This is a perturbative operation.

Demystifier said:
In some cases those are good approximations, and are certainly perturbative.But in many cases one can renormalize a finite number of diagrams, typically a one-loop integral. This is a perturbative operation.
But these cases do not determine particle masses and particle production - which is the topic of the thread.

QFT1995 said:
Why isn't something like the following classed as particle production?
View attachment 248324
Simply, because no new particle is produced. You start with one particle and end with one particle.

This can be two cases: If the in and out state refer to the same particle species, it's called a self-energy diagram. If they refer to different species, it's describing "particle oscillation". It's an interesting phenomenon (CP violation was discovered first in ##K_0 \bar{K}_0## mixing, resolving the "##\vartheta##-##\tau## puzzle"; neutrino oscillations, demonstrating that at least 2 neutrino sorts must have a non-zero mass and solving the "solar-neutrino puzzle").

weirdoguy

## 1. What is a 1+1D QFT?

A 1+1D QFT (Quantum Field Theory) is a mathematical framework used to describe the behavior of particles and fields in a two-dimensional space and time. It is a fundamental theory in physics that combines the principles of quantum mechanics and special relativity.

## 2. How is particle production studied in 1+1D QFT?

In 1+1D QFT, particle production is studied by using mathematical models and calculations to understand how particles are created and interact with each other in a two-dimensional space and time. This involves analyzing the behavior of quantum fields and their interactions.

## 3. What are the applications of studying particle production in 1+1D QFT?

Studying particle production in 1+1D QFT has various applications in physics, including understanding the behavior of particles in high-energy collisions, modeling the early universe, and developing new technologies such as quantum computing.

## 4. What are some challenges in studying particle production in 1+1D QFT?

One of the main challenges in studying particle production in 1+1D QFT is the complexity of the mathematical models and calculations involved. Additionally, there may be discrepancies between theoretical predictions and experimental results, which require further study and refinement of the theory.

## 5. How does particle production in 1+1D QFT differ from other QFTs?

Particle production in 1+1D QFT differs from other QFTs in that it is a simplified, two-dimensional version of the theory. This allows for easier mathematical calculations and a more manageable system to study, but it may not accurately reflect the behavior of particles in our three-dimensional world.

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