The discussion revolves around the nature of normalization constants in physics, specifically whether they are always real-valued or if complex normalization constants can also be valid. Participants explore the implications of using complex numbers in equations and their relevance in various fields such as electrical engineering and vibration engineering.
Participants do not reach a consensus on whether normalization constants must be real-valued or if complex values can be valid. Multiple competing views remain regarding the use of complex numbers versus vectors in various contexts.
Participants highlight limitations in their arguments, such as the dependence on how equations are written and the potential computational efficiency of different formulations. There is also mention of unresolved mathematical steps regarding the interchangeability of complex numbers and vectors.
This discussion may be of interest to those studying normalization in physics, electrical engineering, and mathematics, particularly in relation to the use of complex numbers and their applications in various fields.
Xezlec said:This isn't a complete answer, but I'd say it's all in how you write your equations.
AlephZero said: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.
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.pivoxa15 said:If yes, is that because the normalisation constants only normalises measurable quantities which are always real so the constants are real also.
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.Xezlec said: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 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.Xezlec said: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
Claude Bile said: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 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
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.