111–120 of 166 posts
There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…
I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all. It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers. Which makes perfect sense, of course, because all of t…
https://academic.oup.com/book/26363
As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories.
Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers.
Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.
I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all. It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers. Which makes perfect sense, of course, because all of t…
Earlier quoted context omitted.
Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of fre…
I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?
For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers.
If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers.
Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…
In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space.
David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgo...
I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all. It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers. Which makes perfect sense, of course, because all of t…
Quantum mechanics is defined by three axioms:
1. States are unit vectors. Vectors whose 2-norm is 1.
2. Operations preserve the 2-norm. They are described by unitary matrices.
3. Systems are combined using the tensor product. If system A has state u and system B has state v, then the combined system (A, B) has state u⊗v. If you apply operation x to system A and operation y to system B, the operation on the combined system is x⊗y.
What this paper proved is that, if you use these axioms but limit yourself to unit vectors and unitary matrices with real entries, you cannot explain some experiments. This is surprising because a complex number can be thought of as just a pair of real numbers. For example, in numpy, you can turn any complex ndarray into a real ndarray by making an ndarray with one additional index of length 2, like this:
def complex_to_real(complex_ndarray):
real_ndarray = np.zeroes(shape=(*complex_ndarray.shape, 2), dtype=np.float64)
real_ndarray[..., 0] = np.real(complex_ndarray)
real_ndarray[..., 1] = np.imag(complex_ndarray)
return real_ndarray
The problem is axiom (3), where systems are combined using the tensor product. The issue is that complex_to_real(A ⊗ B) has one more index than A⊗B, but complex_to_real(A)⊗complex_to_real(B) has two additional indices because you gained one real-vs-imaginary index from A and also one from B. The tensor product doesn't understand that these indices should be merged, instead of concatenated. Using complex numbers tweaks the definition of the tensor product so that it does merge these indices. The experiment is basically a way of checking that you contracted those extra indices instead of keeping them both.
So really this result is not about complex numbers vs real numbers. It's about how the states of quantum systems are combined. If you want to use real numbers, you can't use the normal tensor product; you have to use a modified one that understands every system has a special real-vs-imaginary index and that when combining systems you must contract these indices together. Complex numbers just happen to have a tensor product that packages this contract-one-index functionality nicely.
There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…
[0] https://www.youtube.com/watch?v=_Sm7SNlNUOI&list=PLOFVFbzrQ4...
Earlier quoted context omitted.
By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.
Something I've always wondered is: what is the weakest algebra that could be used to model physics? E.g.: Are nationals sufficient? Integers? Finite integers?
Earlier quoted context omitted.
Because they are separate but interacting with real numbers? I don’t understand.
He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0] You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane.…
I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).