Are all quadratic terms in gauge fields necessarily mass terms?

In summary, a mass term in a Lagrangian is a term that describes the mass of a particle in a specific field, and it can be determined by applying a Lorentz transformation.
  • #1
QuantumSkippy
18
1
Can someone please help me out with mass terms in the general case for a lagrangian?

It is known that for n scalar fields, any quadratic in these fields will be a mass term.
For classical fields [tex]\varphi_{j}[/tex] with the most general possible expression being [tex]M^{jk}\varphi_{j}\varphi_{k}[/tex] , the matrix [tex]M^{jk}[/tex] is guaranteed to be symmetric and so can be diagonalised with an orthogonal similarity transformation. So there is no argument there - we get a sum of squares after diagonalisation of the form [tex]\sum_{j} M^{jj}\varphi_{j}\varphi_{j}[/tex]

For the case of gauge fields, however, it does not seem (? help me out here!) that just any quadratic at all will necessarily be a mass term. Here is the reasoning as I see it:

No one would disagree that a term like [tex] M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu} [/tex] is definitely a mass term. Again, for classical fields the mass matrix [tex]M^{jk}[/tex] is guaranteed symmetric by the sum over symmetric terms and so is once more diagonalisable with an orthogonal similarity transformation.

Things seem different for the most general case. For example with a sum like[tex] M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}[/tex] , one would expect that a Lorentz transformation can reduce the term[tex] M_{\mu\nu}^{jk}[/tex] to something of the form [tex]g_{\mu\nu}m^{jk}[/tex].

In this way, the 'mass' term becomes [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] after the Lorentz transformation has been applied.

Observe however, that for the Lorentz transformation [tex]L_{\alpha}^{\mu}[/tex] which achieves this change we have

[tex]M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}[/tex] .

Multiplying both sides of this equation by [tex]{(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}[/tex], we obtain the result that

[tex]M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}[/tex].

This follows from the orthogonality of the Lorentz transformations with respect to the metric [tex]g_{\alpha\beta}[/tex].

So the upshot of this appears to be that unless terms are of the form [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] they cannot represent mass terms and are in fact, self interaction terms.

Please help me out here, as this is the only way I can interpret quadratic terms in the gauge fields at present.
 
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  • #2


I can definitely help you out with mass terms in the general case for a Lagrangian. Let's start with the basics - a mass term in a Lagrangian is a term that describes the mass of a particle in a specific field. In other words, it is the term that determines the strength of the interaction between the particle and the field.

In the case of n scalar fields, any quadratic term in these fields will indeed be a mass term. This is because the mass matrix M^{jk} is guaranteed to be symmetric and can be diagonalized with an orthogonal similarity transformation, resulting in a sum of squares.

However, for gauge fields, things are a bit different. A term like M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu} is definitely a mass term, as it describes the interaction between the gauge fields and the particle. But for the most general case, a term like M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k} may not necessarily be a mass term. This is because it is possible for a Lorentz transformation to reduce this term to something of the form g_{\mu\nu}m^{jk}, where m^{jk} is the mass matrix after the transformation.

In other words, not all quadratic terms in gauge fields represent mass terms. Some may be self-interaction terms instead. To determine whether a term is a mass term or not, we need to apply a Lorentz transformation and see if it can be reduced to the form m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}. If it can, then it is a mass term. Otherwise, it is a self-interaction term.

I hope this helps to clarify the concept of mass terms in the general case for a Lagrangian. If you have any further questions, please do not hesitate to ask.
 

1. How do gauge fields relate to quadratic terms in physics?

Gauge fields are mathematical constructs that are used to describe fundamental interactions in physics, such as electromagnetism and the strong and weak nuclear forces. Quadratic terms in gauge fields refer to mathematical expressions that involve the square of these fields, which have important physical implications.

2. Are all quadratic terms in gauge fields related to mass?

No, not all quadratic terms in gauge fields necessarily correspond to mass. In some cases, they may describe other physical properties, such as coupling strengths or self-interactions of the field. However, quadratic terms in gauge fields do play a crucial role in the generation of mass for certain particles through the Higgs mechanism.

3. Can quadratic terms in gauge fields be zero?

Yes, in some cases, quadratic terms in gauge fields can be zero. This occurs when the gauge symmetry of the system is unbroken, meaning that the field has no net value and does not contribute to the mass of particles. However, if the gauge symmetry is spontaneously broken, the quadratic terms can become non-zero and contribute to particle masses.

4. How do quadratic terms in gauge fields affect the behavior of particles?

Quadratic terms in gauge fields have a significant impact on the behavior of particles. They determine the mass of particles and can also influence their interactions with other particles. In theories with spontaneously broken gauge symmetry, the quadratic terms can also give rise to the Higgs field, which is responsible for giving mass to the W and Z bosons.

5. Are there any exceptions to the rule that all quadratic terms in gauge fields are not mass terms?

Yes, there are some exceptions to this rule. In certain theories, such as supersymmetry, quadratic terms in gauge fields can be directly related to mass terms. This is due to the presence of additional symmetries that relate different types of particles, leading to a more intricate relationship between gauge fields and particle masses.

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