# Detecting a particle in QFT

#### smileii

Some friend asked me the following question:

For a real scalar field \phi, assume that H = H_free - \int d^3 x\ J \phi. J(x, t) is just some real number, source, or background field, without second quantization. Now, what is the amplitude \psi(x, t) for finding a particle at time t(before, during, or after source is on/off) at position x? The J(x,t) is nonzero only for finite period of time. And the initial state is vacuum, when t --> -\infty .

This question looks simple. However, I cannot find a solution which satisfies both causality and Lorentz invariance.

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#### meopemuk

Some friend asked me the following question:

For a real scalar field \phi, assume that H = H_free - \int d^3 x\ J \phi. J(x, t) is just some real number, source, or background field, without second quantization. Now, what is the amplitude \psi(x, t) for finding a particle at time t(before, during, or after source is on/off) at position x? The J(x,t) is nonzero only for finite period of time. And the initial state is vacuum, when t --> -\infty .

This question looks simple. However, I cannot find a solution which satisfies both causality and Lorentz invariance.
In my opinion, this problem is not correctly formulated. Your Hamiltonian has explicit time dependence, so it describes not an isolated system but a system in an external field. In this case, I am even not sure what is the correct definition of Lorentz invariance. (In the case of isolated systems, the Lorentz invariance means that generators of inertial transformations - the total momentum, angular momentum, Hamiltonian, and boost operators - satisfy commutation relations of the Poincare Lie algebra).

What is the physical situation which your friend tried to model with this Hamiltonian?

Eugene.

2018 Award

#### smileii

To meopemuk:

I think the external source makes sense in some cases. For example, if we consider the scattering of electron by some electroganetic field, usually, the electron field is quantized, but not the EM field. This half-second- quantization is for calculation simplicity.

#### meopemuk

To meopemuk:

I think the external source makes sense in some cases. For example, if we consider the scattering of electron by some electroganetic field, usually, the electron field is quantized, but not the EM field. This half-second- quantization is for calculation simplicity.
I agree that external field is a useful approximation. My question was: how one can guarantee the Lorentz (Poincare) invariance in this approximation? External field defines a preferred frame of reference, then the original question "find a solution which satisfies ... Lorentz invariance." becomes ill-posed. In my opinion.

Eugene.

#### smileii

I agree that external field is a useful approximation. My question was: how one can guarantee the Lorentz (Poincare) invariance in this approximation? External field defines a preferred frame of reference, then the original question "find a solution which satisfies ... Lorentz invariance." becomes ill-posed. In my opinion.

Eugene.
OK, the Lagragian should be Lorentz invariant if $phi$ and $J$ are scalr classical fields. Now, when only \phi is quantized, \phi's Lorentz transformation rule is still the same. So, to keep the Lorent invariance of this "new" Lagragian, I require that $J$ transforms in the same way as $\phi$. Same for translation symmetry.... I understand that you cannot find a set of P^{\mu}, J^{\mu\nu} operators to satisfy those commutation relations, since J is not quantized. But in the sense of classical field theory, the Lagragian is still Lorent invariant.

In the exterme case, when $J(x') = \delta^4(x' - y)$, and you detect particle at x, you will expect your answer should be the for of f(x-y) due to translation symmetry. And the amplitude of detection should be still the same when x-> x'. y -> y' in another reference frame. For example, $G(x-y) \theta(x^0 - y^0)$ clearly doesn't satisfy Lorentz invariance. This is how I check "Lorentz symmetry" here.

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