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

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The Hamiltonian for a scalar field includes a mass term represented as the integral of the product of the field and its mass squared. The discussion questions whether this term transforms under Lorentz transformations, specifically if it changes to a form involving a modified mass and transformed field. It is noted that the scalar field itself remains invariant under Lorentz transformations, meaning that the field at the transformed coordinates is equivalent to the original field. The mass is asserted to be Lorentz invariant, suggesting that it does not change under such transformations. The conversation emphasizes the importance of understanding how mass terms behave in the context of scalar field theory.
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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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