How the mass term of the Hamiltonian for a scalar fields transform?

In summary, the mass term of the Hamiltonian for scalar fields transforms under Lorentz transformations according to the properties of the scalar field. Specifically, the mass term remains invariant since it is a scalar quantity, ensuring that the physics described by the Hamiltonian is consistent across different inertial frames. The transformation properties reflect the underlying symmetries of the theory, preserving the form of the mass term in the Hamiltonian formulation while facilitating the analysis of particle dynamics and interactions.
  • #1
PRB147
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TL;DR Summary
The mass term under Lorentz transformation
The Hamiltonian for a scalar field contains the term
$$\int d^3x m^2 \phi(x) \phi(x)$$, does it changs to the following form?
$$\int d^3x' {m'}^2 \phi'(x') \phi'(x')=\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$? As it is well known for a scalar field: $$\phi'(x')=\phi(x)$$ .
 
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  • #2
The mass is Lorentz invariant AFAIK.
 

FAQ: How the mass term of the Hamiltonian for a scalar fields transform?

What is the mass term in the Hamiltonian for a scalar field?

The mass term in the Hamiltonian for a scalar field is a component that represents the energy associated with the mass of the field's quanta. For a real scalar field, the mass term typically appears in the Hamiltonian density as \( \frac{1}{2} m^2 \phi^2 \), where \( m \) is the mass of the field and \( \phi \) is the scalar field itself.

How does the mass term transform under Lorentz transformations?

The mass term in the Hamiltonian for a scalar field is invariant under Lorentz transformations. This means that while the field \( \phi \) itself transforms according to the representation of the Lorentz group, the mass \( m \) remains constant, ensuring that the physical predictions of the theory do not depend on the observer's inertial frame.

What is the significance of the mass term in quantum field theory?

The mass term is crucial in quantum field theory as it determines the behavior of particles associated with the scalar field. It affects the dispersion relation, influencing how the particles propagate and their stability. The presence of a mass term leads to the existence of massive excitations, which correspond to physical particles with rest mass.

How does the mass term affect the energy-momentum tensor?

The mass term contributes to the energy-momentum tensor, which describes the distribution of energy and momentum in spacetime. Specifically, it contributes to the energy density and pressure of the field, influencing how the field interacts with gravity and other fields through the Einstein field equations.

Does the mass term change under a change of field variables?

The form of the mass term can change under a change of field variables, such as a field redefinition or a transformation to a different representation. However, the physical implications, such as the mass of the particles and their interactions, should remain consistent across different formulations, provided the transformation is done correctly and maintains the underlying symmetries of the theory.

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