Not sure which part of QM you’re referring to, but arguably QM hasn’t really been “found out” yet, so we shouldn’t be surprised that it’s not easy to understand. I mean seriously, what the hell are complex numbers doing in the Dirac equation?
QM is a solid theory that reliably predicts every known experiment dependent on it.
And it seems you need to brush up on your arithmetic theory.
There is a progression in the usual number fields, Naturals (w/ or w/o 0), Integers, Rationals, Reals, Complex, Quarternions, Octonions, Sedenions, etc.
Naturals have a starting point, countability, no negatives, no inverse elements and no algebraic closure.
Integers sacrifice a starting points to gain negative elements.
Rationals sacrifice finiteness of subsets to gain inverse elements.
Reals sacrifice uniqueness of representation to gain uncountability.
Complex numbers sacrifice absolute order to gain algebraic closure.
Then it gets a bit hazy in memory, but I know Quarternions sacrifice commutativity of multiplication and Octonions aren’t associative but I can’t remember what neat tricks you gain there. The Sedenions have zero divisors but I can’t remember what they loose.
Now the point is that complex numbers are the most interesting because they have algebraic closure; you cannot construct an equation with multiplication and addition or almost any other operation in which the solution isn’t a Complex number. Not so with the Reals (sqrt −1). Thus Complex numbers are completely logical to be physics rather than Reals.
Reals sacrifice uniqueness of representation to gain uncountability.
Not what I would have said. Instead, I think it would be better to say that the reals sacrifice countability in order to gain completeness.
(“Uniqueness of representation” isn’t a big deal at all. In fact, it doesn’t even hold for the natural numbers, which is why there is such a thing as “arithmetic”.)
Well, the simplest way to represent a real number is with infinite decimal expansion. Every rational number with a finite digit expansion has two infinite digit expansions. Every Natural has one corresponding string of digits in any positional number system with natural base.
I think at the time of writing I considered ‘completeness’ to be ill defined, since the real numbers don’t have algebraic closure under the exponential operator with negative base and fractional exponent, while with ordinary arithmetic it is impossible to shoot outside of the Complex numbers.
(EDIT: I cant arithmetic field theory today) The best I can come up with is uniqueness of representation, since it implies infinite representations and thus loss of countability. (insofar as I remember my ZFC Sets correctly, a set of all infinite strings with a finite alphabet is uncountable and isomorphic at least to the interval [0,1] of the reals)
Technically limits work with Rational Numbers too, so that isn’t an unique property either
No, they don’t; that’s precisely the point. There are Cauchy sequences of rational numbers which don’t converge to any rational number. For an example, simply take the sequence whose nth term is the decimal expansion of pi (or your favorite irrational number) carried out to n digits.
what the hell are complex numbers doing in the Dirac equation?
As I’ve pointed out to you before, if you have a problem with physical applications of complex numbers, you should be equally offended by physical applications of matrices, because matrices of the form [[a,-b],[b,a]] are isomorphic to complex numbers. In fact, your problem isn’t just with quantum mechanics; if you can’t stand complex numbers, you should also have a problem with (for just one example) simple harmonic motion.
In detail: we model a mass attached to a spring with the equation F=-kx: the force F on the mass is proportional to a constant -k times the displacement from the equilibrium position x. But because force is mass times acceleration, and acceleration is the second time derivative of position, this is actually the differential equation x″(t) + (k/m)x(t) = 0, which has the solution x(t) = ae^(i*sqrt(k/m)t) + be^(-i*sqrt(k/m)t) where a and b are arbitrary constants.
It’s true that people tend to write this as ccos(sqrt(k/m)t)+dsin(sqrt(k/m)t), but the fact that we use a notation that makes the complex numbers less visible, doesn’t change the underlying math. Trig functions aresums of complex exponentials.
Complex numbers are perfectly well-behaved, non-mysterious mathematical entities (consider also MagnetoHydroDynamics’s point about algebraic closure); why shouldn’t they appear in the Dirac equation?
Not sure which part of QM you’re referring to, but arguably QM hasn’t really been “found out” yet, so we shouldn’t be surprised that it’s not easy to understand. I mean seriously, what the hell are complex numbers doing in the Dirac equation?
It’s hard to engage with someone whose readiness to opine so vastly exceeds their readiness to meaningfully opine.
Eh?
QM is a solid theory that reliably predicts every known experiment dependent on it.
And it seems you need to brush up on your arithmetic theory. There is a progression in the usual number fields, Naturals (w/ or w/o 0), Integers, Rationals, Reals, Complex, Quarternions, Octonions, Sedenions, etc.
Naturals have a starting point, countability, no negatives, no inverse elements and no algebraic closure.
Integers sacrifice a starting points to gain negative elements.
Rationals sacrifice finiteness of subsets to gain inverse elements.
Reals sacrifice uniqueness of representation to gain uncountability.
Complex numbers sacrifice absolute order to gain algebraic closure.
Then it gets a bit hazy in memory, but I know Quarternions sacrifice commutativity of multiplication and Octonions aren’t associative but I can’t remember what neat tricks you gain there. The Sedenions have zero divisors but I can’t remember what they loose.
Now the point is that complex numbers are the most interesting because they have algebraic closure; you cannot construct an equation with multiplication and addition or almost any other operation in which the solution isn’t a Complex number. Not so with the Reals (sqrt −1). Thus Complex numbers are completely logical to be physics rather than Reals.
Not what I would have said. Instead, I think it would be better to say that the reals sacrifice countability in order to gain completeness.
(“Uniqueness of representation” isn’t a big deal at all. In fact, it doesn’t even hold for the natural numbers, which is why there is such a thing as “arithmetic”.)
Well, the simplest way to represent a real number is with infinite decimal expansion. Every rational number with a finite digit expansion has two infinite digit expansions. Every Natural has one corresponding string of digits in any positional number system with natural base.
I think at the time of writing I considered ‘completeness’ to be ill defined, since the real numbers don’t have algebraic closure under the exponential operator with negative base and fractional exponent, while with ordinary arithmetic it is impossible to shoot outside of the Complex numbers.
(EDIT: I cant arithmetic field theory today) The best I can come up with is uniqueness of representation, since it implies infinite representations and thus loss of countability. (insofar as I remember my ZFC Sets correctly, a set of all infinite strings with a finite alphabet is uncountable and isomorphic at least to the interval [0,1] of the reals)
EDITED to fix elementary error.
No. Every terminating number has two infinite decimal expansions, one ending with all zeros, the other with all nines.
1⁄3, for instance is only representable as 0.333… , while 1/8th is representable as 0.124999… and 0.125.
Oh right, thanks for catching that.
No, they don’t; that’s precisely the point. There are Cauchy sequences of rational numbers which don’t converge to any rational number. For an example, simply take the sequence whose nth term is the decimal expansion of pi (or your favorite irrational number) carried out to n digits.
Noted and corrected.
As I’ve pointed out to you before, if you have a problem with physical applications of complex numbers, you should be equally offended by physical applications of matrices, because matrices of the form [[a,-b],[b,a]] are isomorphic to complex numbers. In fact, your problem isn’t just with quantum mechanics; if you can’t stand complex numbers, you should also have a problem with (for just one example) simple harmonic motion.
In detail: we model a mass attached to a spring with the equation F=-kx: the force F on the mass is proportional to a constant -k times the displacement from the equilibrium position x. But because force is mass times acceleration, and acceleration is the second time derivative of position, this is actually the differential equation x″(t) + (k/m)x(t) = 0, which has the solution x(t) = ae^(i*sqrt(k/m)t) + be^(-i*sqrt(k/m)t) where a and b are arbitrary constants.
It’s true that people tend to write this as ccos(sqrt(k/m)t)+dsin(sqrt(k/m)t), but the fact that we use a notation that makes the complex numbers less visible, doesn’t change the underlying math. Trig functions are sums of complex exponentials.
Complex numbers are perfectly well-behaved, non-mysterious mathematical entities (consider also MagnetoHydroDynamics’s point about algebraic closure); why shouldn’t they appear in the Dirac equation?