# Real scalars for complex scalar results

• earth2
In summary, real scalars for complex scalar results involve multiplying real numbers with complex numbers to produce complex results. This concept is useful in science for representing physical quantities that have both magnitude and direction, such as electric and magnetic fields. One example of using real scalars for complex scalar results is when calculating the impedance of an electrical circuit. The properties of real scalars and complex scalars include commutativity, associativity, distributivity, and the existence of multiplicative inverse. However, complex scalars also have the property of conjugation, which can add complexity to calculations and interpretations. One limitation of using real scalars for complex scalar results is that the resulting complex numbers may not always have a physical meaning and can complicate calculations.
earth2
Hi guys,

I'm a bit puzzled. I'm just reading some offline lecture notes where the Feynman rules of real (!) scalars coupled to gluons are given. However, with these rules the amplitude for phi g -> \bar{phi} g is considered. There are no further instructions. I'm just wondering how one can use Feynman rules for a real scalar-gluon theory to construct the result for its complex sibling. I mean, i know that a complex scalar can be written as two real ones but still there should be a mess in terms of numerical prefactors, shouldn't there? Moreover, how does one apply this to Feynman rules?

Well, I just don't see how to use the real rules to construct the complex result...

Any help is appreciated,
earth2

sky

Hi earth2sky,

Great question! It can definitely be confusing to apply Feynman rules for real scalar-gluon theories to complex scalar-gluon theories. The key here is to remember that the Feynman rules for real scalars are derived from the Lagrangian of the theory, which only includes real fields. So when we want to apply these rules to a complex scalar, we have to first rewrite the Lagrangian in terms of complex fields.

To do this, we can use the fact that a complex scalar field can be written as the sum of two real scalar fields, as you mentioned. This means that the Lagrangian for a complex scalar-gluon theory can be written as the sum of two Lagrangians for real scalar-gluon theories. Then, we can use the Feynman rules for real scalar-gluon theories to calculate the amplitudes for each of these Lagrangians separately, and then add them together to get the total amplitude for the complex scalar-gluon theory.

As for the numerical prefactors, these will depend on the specific theory and interactions involved. But the important thing to remember is that the Feynman rules themselves do not change, just the way we apply them to the complex scalar-gluon theory.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask.

## 1. What is the meaning of "real scalars for complex scalar results"?

Real scalars for complex scalar results refer to a mathematical concept where real numbers are multiplied with complex numbers to produce complex results.

## 2. How is this concept useful in science?

This concept is useful in science because it allows for the representation of physical quantities that have both magnitude and direction, such as electric and magnetic fields.

## 3. Can you provide an example of using real scalars for complex scalar results?

One example is when calculating the impedance of an electrical circuit, which involves multiplying a real resistance value with a complex reactance value to obtain a complex impedance value.

## 4. What are the properties of real scalars and complex scalars?

The properties of real scalars and complex scalars include commutativity, associativity, distributivity, and the existence of multiplicative inverse for non-zero values. Additionally, complex scalars have the property of conjugation, where the imaginary component is flipped in sign.

## 5. Are there any limitations to using real scalars for complex scalar results?

One limitation is that the resulting complex numbers may not have a physical meaning in certain cases, such as when dealing with negative physical quantities. Additionally, the use of complex numbers may add complexity to calculations and interpretations.

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