Normalisation constants in physics?

[pivoxa15]

Are normalisation constants in physics always real valued? If yes, is that because the normalisation constanst only normalises measureable quantities which are always real so the constants are real also.

Any exceptions?

 

[Xezlec]

This isn't a complete answer, but I'd say it's all in how you write your equations. Any 2D vector quantity (or any pair of related quantities) can be phrased as a complex number, and then the formulas involved can be rewritten for complex numbers, and everything works out. I'm sure it's possible to write a lot of physics equations in such a way that the normalization constants are all complex. It's just that that isn't the "standard" way to write stuff.

I think the reason that real numbers and vectors are the standard representations of many physical laws is that real numbers are simpler, more familiar, have a longer history, and are easier to do math with. Also, if you need a vector, you might as well just use an actual vector instead of a complex number. Vectors are a more general concept anyway.

I'd appreciate any feedback on those statements, because this is another one of those things I think about a lot.

 

[Xezlec]

One little addendum: in electrical engineering we use complex numbers a lot, and many properties of materials are described in terms of complex numbers. I don't know if this matters to you.

 

[AlephZero]

This isn't a complete answer, but I'd say it's all in how you write your equations.

I agree. Just because the maths appears to model the real world, that doesn't mean the maths IS the real world. An extreme example of that is the quantum mechanics arguments about how to interpret the maths, which start from the (unjustified, and wrong, IMO) assumption that everything in the maths has to correspond to something "physically real" however bizarre the interpretation might be.

In EE and some parts of vibration engineering, the main point of using complex numbers and complex normalization is because it makes the maths simpler and computationally more efficient. You could reformulate the whole thing using two real variables instead of one complex, but (in the vib. eng. case which I'm more familiar with) the downside is needing 2x the computer memory and 4x the run time to do the equivalent calculations compared with the complex arithmetic formulation - which is a pretty good practical reason for using complex.

 

[Xezlec]

An extreme example of that is the quantum mechanics arguments about how to interpret the maths, which start from the (unjustified, and wrong, IMO) assumption that everything in the maths has to correspond to something "physically real" however bizarre the interpretation might be.

Interesting. I always thought that was just how they defined the term "physically real".

In EE and some parts of vibration engineering, the main point of using complex numbers and complex normalization is because it makes the maths simpler and computationally more efficient. You could reformulate the whole thing using two real variables instead of one complex, but (in the vib. eng. case which I'm more familiar with) the downside is needing 2x the computer memory and 4x the run time to do the equivalent calculations compared with the complex arithmetic formulation - which is a pretty good practical reason for using complex.

Wait... why would you need 2x the memory? A vector of two real numbers requires the same amount of memory as a single scalar that is complex. But the vector can be more easily generalized to more complicated cases.

And why does the vector take 4x the CPU time? I've never figured out why complex math hardware would be desirable over true vector hardware to some people. Anyway, today's machines probably handle vectors better than complex numbers in general. Maybe this is an artifact of some library?

 

[Claude Bile]

If yes, is that because the normalisation constants only normalises measurable quantities which are always real so the constants are real also.

I am inclined to agree with this. I can't think of a purely mathematical reason why a complex number would be forbidden in these circumstances.

Any 2D vector quantity (or any pair of related quantities) can be phrased as a complex number, and then the formulas involved can be rewritten for complex numbers, and everything works out.

Complex numbers share similarities with 2D vectors because they are both represented over an orthogonal 2D basis - but there are cases when the two are not directly interchangeable, for example when a complex number is itself a component of a vector or matrix.

I think the reason that real numbers and vectors are the standard representations of many physical laws is that real numbers are simpler, more familiar, have a longer history, and are easier to do math with. Also, if you need a vector, you might as well just use an actual vector instead of a complex number. Vectors are a more general concept anyway

I used to share the same thoughts, as I suspect many of those learning the ropes of complex numbers, however I have found the more I studied them and used them, there more I have understood that a) complex numbers have been around for over a century, and are used extensively in just about every field of physics and engineering b) there comes a point where you simply cannot avoid the use of complex numbers (just as there is some point in primary school where integers become insufficient and one must use fractions and so on) - it is a matter of necessity not choice and c) complex numbers ARE simple to use - as a simple example, it can turn a trig problem into a simple multiplication problem by allowing us to use exponential functions (which are easy to multiply and divide) to represent an oscillatory function.

At the end of the day, Complex numbers are simpler to use for the same reason problems in real space are easier if we permit ourselves to use negative numbers, fractions and irrational numbers.

Claude.

 

[Xezlec]

Complex numbers share similarities with 2D vectors because they are both represented over an orthogonal 2D basis - but there are cases when the two are not directly interchangeable, for example when a complex number is itself a component of a vector or matrix.

I'd like to hear a true case where they are not interchangeable. A vector of complex numbers is equivalent to a 2xN matrix or reals, and a matrix of complex numbers is equivalent to a 2xMxN tensor of reals.

I used to share the same thoughts, as I suspect many of those learning the ropes of complex numbers, however I have found the more I studied them and used them, there more I have understood that a) complex numbers have been around for over a century, and are used extensively in just about every field of physics and engineering

I agree and I mentioned that. Convention is a reason.

b) there comes a point where you simply cannot avoid the use of complex numbers (just as there is some point in primary school where integers become insufficient and one must use fractions and so on) - it is a matter of necessity not choice

I think you could use vectors in these same circumstances.

c) complex numbers ARE simple to use - as a simple example, it can turn a trig problem into a simple multiplication problem by allowing us to use exponential functions (which are easy to multiply and divide) to represent an oscillatory function.

Works the same way if you define e^<vector> using the right identity.

At the end of the day, Complex numbers are simpler to use for the same reason problems in real space are easier if we permit ourselves to use negative numbers, fractions and irrational numbers.

No, that wouldn't explain why complex numbers are better than vectors. It would explain why either complex numbers or vectors is better than having neither one at all. I'm not saying throw both out, I'm saying replace complex numbers with vectors.

 

[Chronos]

I agree with claudeB, there are no mathematical rules that exclude exotic numbers. Introducing vectors does not change things.

 

[Claude Bile]

Xezlec - Clearly your knowledge of Maths is far beyond what I had assumed from your previous post, I admit coming up with concrete counter-examples was not really my intention - my intention was to simply to give anecdotal reasons why complex numbers are useful.

As you rightly pointed out though, I have not proven that complex numbers cannot be substituted by vectors.

Claude.