Coupling constants with fractional dimensions

In summary, the renormalizability of an interacting theory requires coupling constants to have correct dimensions, limiting the interest in scalar fields with higher order interactions. It is unclear if similar limitations apply to vector and spinor field interactions. However, introducing fractional powers of the scalar field, such as ##\phi^{7/2}##, may lead to unphysical behavior due to the inability to define constants with fractional units in physics. Additionally, constructing a Feynman vertex with 3 or 5 scalar neutral particles, such as Higgs bosons, would be challenging and may cause issues similar to taking the square root of the Klein-Gordon operator. It is uncertain if there is a way to incorporate square-root creation/annihilation operators
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hilbert2
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Most QFT texts, such as Peskin&Schroeder and D. Tong's lecture notes, contain a mention that the renormalizability of an interacting theory requires the coupling constants to have correct dimensions, making scalar fields with ##\phi^5 , \phi^6, \dots## interactions uninteresting. Maybe there are similar limitations for vector and spinor field interactions, but someone more familiar with QFT must answer that.

Now, if instead of an integer power of the scalar field, I make a field equation that has something like ##\phi^{7/2}## or similar in it, does this lead to some kind of unphysical behavior too? In theories of physics, you don't really see constants of nature that have fractional powers of kilogram or second in their units (not sure why). Is this kind of a fractional interaction equivalent to a non-renormalizable one because the term with fractional order can be expanded to a power series with arbitrarily high integer powers of the field variable?
 
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Can you build a Feynman vertex out of 3,5 scalar neutral particles, such as Higgs bosons? How would you do it, draw 3 full lines and the 4th reduced to half length?
 
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A perturbation expansion containing products of three creation/annihilation operators with one square-root creation/annihilation operator would definitely be a bit difficult, and maybe it would cause problems similar to when you try to take the square root of the Klein-Gordon operator. It's just not immediately obvious to me that there's no way to do that.
 

1. What are coupling constants with fractional dimensions?

Coupling constants with fractional dimensions are mathematical constants used in theoretical physics to describe the interaction between different particles or fields. They are often represented as numbers with non-integer values, such as 1.5 or 2.7.

2. How are coupling constants with fractional dimensions calculated?

The exact method for calculating coupling constants with fractional dimensions varies depending on the specific theory or model being used. However, in general, they are calculated using mathematical equations and experimental data.

3. What is the significance of coupling constants with fractional dimensions?

Coupling constants with fractional dimensions play a crucial role in understanding the behavior of particles and fields in theoretical physics. They provide insight into the strength and nature of interactions between different components of a system.

4. Can coupling constants with fractional dimensions change over time?

Yes, in some theories, coupling constants with fractional dimensions are considered to be dynamic and can change over time. This is often seen in theories that attempt to unify different fundamental forces, such as the strong and weak nuclear forces.

5. How do coupling constants with fractional dimensions affect our understanding of the universe?

Coupling constants with fractional dimensions are an essential part of many theoretical models used to describe the fundamental laws of the universe. By studying these constants, scientists can gain a better understanding of the underlying principles that govern the behavior of matter and energy in the universe.

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