Posted on 12/21/2021 8:16:07 PM PST by algore
Isn’t -1 x +1 = -1
Maybe that expresses the idea that there can be an electron in two separate spaces? The reason we know they are in separate spaces is the positive and negative attached to them. The positive indicates one world (a space) and the negative is another world (a space). But they are essentially the same electron.
If this theory doesn’t fly, I blame 3 Dog Night.
“imaginary” is just a word. This set of numbers is as “real” as the rest of mathematics.
imho.
Will they ever figure out whether ‘i’ or ‘j’ is the correct representation of sqrt(-1)?
I do this daily with the bank account. The experts can call me.
The ancient Greeks were scandalized by the existence of numbers that couldn’t be written as the ratio of two whole numbers, and labelled them “irrational” numbers. Pythagoras is said to have sentenced the discoverer to death.
Even zero was overlooked for the longest time... What? A number that has no value?? Preposterous!
Didn’t realize people were still resisting imaginary numbers. Holy cow, they were only discovered several centuries ago!
What is the square root of minus 4?;-)
Like eleventeen and thirty-twelve?
During some of my EE classes we negotiated that hurdle using i or j as a coefficient of the root.
If this theory doesn’t fly, I blame 3 Dog Night.
—
I blame a bunch of Monkeys.
2i, -2i. A more interesting question is what is the square root of i? (i+1)/√ 2
I blame the Bee Gees. You know, i started a joke...
Depends if you’re an EE or not ;-)
“… if quantum mechanics is correct, imaginary numbers are a necessary part of the mathematics of our universe.…”
Maybe quantum mechanics isn’t correct. Whose that guy with a razor?
Very Hobbesian of you...
To the photon particle, the wave function can only be imaginary. The particle exists in linear space only, so it lacks the variability to sense something in planar space. The wave ‘function’ lacks a third variable of space so when it arrives at a surface in a volume universe, it collapses into two of the three variables of volume.
Here is an example. The guy combines the argument from silence fallacy [philosophical square root _/¯ ] with the straw argument fallacy [philosophical -1 ] for 2 synergystic fallacies that are more effective together.
https://freerepublic.com/focus/chat/4022814/posts?page=141#141
That’s comforting as I always made up numbers to answer algebra questions in 10th grade. 😆
Does this answer the question?
Smiles
————-
On the Role of Complex Numbers in Quantum Mechanics
Motivation
It has been claimed that one of the defining characteristics that separate the quantum world from the classical one is the use of complex numbers. It’s dogma, and there’s some truth to it, but it’s not the whole story:
While complex numbers necessarily turn up as first-class citizen of the quantum world, I’ll argue that our old friend the reals shouldn’t be underestimated.
A bird’s eye view of quantum mechanics
In the algebraic formulation, we have a set of observables of a quantum system that comes with the structure of a real vector space. The states of our system can be realized as normalized positive (thus necessarily real) linear functionals on that space.
In the wave-function formulation, the Schrödinger equation is manifestly complex and acts on complex-valued functions. However, it is written in terms of ordinary partial derivatives of real variables and separates into two coupled real equations - the continuity equation for the probability amplitude and a Hamilton-Jacobi-type equation for the phase angle.
The manifestly real model of 2-state quantum systems is well known.
Complex and Real Algebraic Formulation
Let’s take a look at how we end up with complex numbers in the algebraic formulation:
We complexify the space of observables and make it into a 𝐶∗
C
∗
-algebra. We then go ahead and represent it by linear operators on a complex Hilbert space (GNS construction).
Pure states end up as complex rays, mixed ones as density operators.
However, that’s not the only way to do it:
We can let the real space be real and endow it with the structure of a Lie-Jordan-Algebra. We then go ahead and represent it by linear operators on a real Hilbert space (Hilbert-Schmidt construction).
Both pure and mixed states will end up as real rays. While the pure ones are necessarily unique, the mixed ones in general are not.
The Reason for Complexity
Even in manifestly real formulations, the complex structure is still there, but in disguise:
There’s a 2-out-of-3 property connecting the unitary group 𝑈(𝑛)
U
(
n
)
with the orthogonal group 𝑂(2𝑛)
O
(
2
n
)
, the symplectic group 𝑆𝑝(2𝑛,ℝ)
S
p
(
2
n
,
R
)
and the complex general linear group 𝐺𝐿(𝑛,ℂ)
G
L
(
n
,
C
)
: If two of the last three are present and compatible, you’ll get the third one for free.
An example for this is the Lie-bracket and Jordan product: Together with a compatibility condition, these are enough to reconstruct the associative product of the 𝐶∗
C
∗
-algebra.
Another instance of this is the Kähler structure of the projective complex Hilbert space taken as a real manifold, which is what you end up with when you remove the gauge freedom from your representation of pure states:
It comes with a symplectic product which specifies the dynamics via Hamiltonian vector fields, and a Riemannian metric that gives you probabilities. Make them compatible and you’ll get an implicitly-defined almost-complex structure.
Quantum mechanics is unitary, with the symplectic structure being responsible for the dynamics, the orthogonal structure being responsible for probabilities and the complex structure connecting these two. It can be realized on both real and complex spaces in reasonably natural ways, but all structure is necessarily present, even if not manifestly so.
Conclusion
Is the preference for complex spaces just a historical accident? Not really. The complex formulation is a simplification as structure gets pushed down into the scalars of our theory, and there’s a certain elegance to unifying two real structures into a single complex one.
On the other hand, one could argue that it doesn’t make sense to mix structures responsible for distinct features of our theory (dynamics and probabilities), or that introducing un-observables to our algebra is a design smell as preferably we should only use interior operations.
While we’ll probably keep doing quantum mechanics in terms of complex realizations, one should keep in mind that the theory can be made manifestly real. This fact shouldn’t really surprise anyone who has taken the bird’s eye view instead of just looking throught the blinders of specific formalisms.
(Unsigned web post)
I think it’s fairly obvious we are living in a computer simulation program.
In Daniel 5 we see a real being, existing in a real spacetime not sensed by the partiers in palace party central, Babylon. That real being ‘reached’ from their real spacetime ‘into’ the real space and time reality (albeit having less variable), to write upon the wall. And there are many more examples of a different spacetime reality intersecting our 4D realm.
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