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

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SUMMARY

The discussion centers on the transformation of the mass term in the Hamiltonian for scalar fields. It confirms that the mass term, represented as $$\int d^3x m^2 \phi(x) \phi(x)$$, transforms to $$\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$ under Lorentz transformations. The scalar field itself remains invariant, as indicated by the relationship $$\phi'(x')=\phi(x)$$. This establishes that while the mass term appears modified by the Lorentz factor $$\gamma$$, the underlying scalar field retains its form.

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  • Understanding of Hamiltonian mechanics
  • Familiarity with scalar field theory
  • Knowledge of Lorentz transformations
  • Basic concepts of invariance in physics
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  • Study the implications of Lorentz invariance in quantum field theory
  • Explore Hamiltonian formulations for different types of fields
  • Learn about the role of the Lorentz factor $$\gamma$$ in relativistic physics
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The discussion is beneficial for theoretical physicists, graduate students in physics, and researchers focusing on quantum field theory and relativistic mechanics.

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TL;DR
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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The mass is Lorentz invariant AFAIK.
 

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