Am I Hallucinating? (confusion in geometric algebra)

In summary, the conversation revolves around geometric algebra and a claim that abcdabcd equals -1. The experts say this is true, but the speaker has found evidence that contradicts it. They discuss the properties used in this claim, including the fact that aa=1 and ab=-ba only when a and b are orthogonal vectors. They also mention that the vectors used in this context are not arbitrary, but rather the basis vectors of a 4D space. It is pointed out that the square of the pseudoscalar depends on the metric of the space. Ultimately, it is determined that the speaker was working with the wrong space and the claim is in fact correct in the proper context.
  • #1
Hornbein
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I'm learning geometric algebra. There is a very simple statement which I think is wrong. But it must be right, because all the experts say so. Arrg!

The only properties used are

1a = a1 = a

aa = 1

if b<>a then ab = -baTheir claim is that abcdabcd = -1.

Let's see:

aa = 1

abab = -abba = -aa = -1

abcabc = -abacbc = ababcc = abab = -1

abcdabcd = -abcadbcd = abcabdcd = -abcabcdd = -abcabc = --1 = 1

But the experts say abcdabcd = -1. See for yourself at
http://geocalc.clas.asu.edu/pdf-preAdobe8/ZBW_I_QM.pdf, eqn 10 and 11

This can't be. To reduce the 4D case to the 3D case there have to be an odd number of swaps. They are of opposite signs and nonzero. They can't possibly be equal.

Elsewhere they say that abcabc and abab equal -1.

?
 
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  • #2
I think you are ignoring Eqs. (7)-(8) which basically tell you that your 4D space has signature (+,-,-,-). In other words, this means that you are working on ℝ1,3 and not on ℝ4, thus, using your notation, aa=1 and bb=cc=dd=-1.

Another thing worth noticing is that those vectors a,b,c,d are not arbitrary vectors: they are the basis vectors of your 4D space, and b,c,d by definition square to -1. In the context of that paper abcd represents the pseudoscalar of the space. Whether the square of the pseudoscalar is positive or negative (or zero) depends on the metric of the space.

It shall also be pointed out that ab=-ba is not true in general, but only when a,b are orthogonal vectors, which is expressed in Eq. (8).
 
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  • #3
mnb96 said:
I think you are ignoring Eqs. (7)-(8) which basically tell you that your 4D space has signature (+,-,-,-). In other words, this means that you are working on ℝ1,3 and not on ℝ4, thus, using your notation, aa=1 and bb=cc=dd=-1.

Another thing worth noticing is that those vectors a,b,c,d are not arbitrary vectors: they are the basis vectors of your 4D space, and b,c,d by definition square to -1. In the context of that paper abcd represents the pseudoscalar of the space. Whether the square of the pseudoscalar is positive or negative (or zero) depends on the metric of the space.

It shall also be pointed out that ab=-ba is not true in general, but only when a,b are orthogonal vectors, which is expressed in Eq. (8).

Aha, that's it. I had the wrong space.
 

1. What is geometric algebra and how does it relate to hallucinations?

Geometric algebra is a mathematical framework that extends traditional algebra to include geometric concepts such as vectors, planes, and rotations. It can be used to model physical systems and phenomena, including those that may be associated with hallucinations.

2. What causes hallucinations in the brain?

Hallucinations can be caused by a variety of factors, including drug use, mental illness, sleep deprivation, and neurological conditions. In some cases, they may also be a side effect of certain medications.

3. How can I tell if I am hallucinating or if something is real?

The best way to determine if you are experiencing a hallucination is to seek help from a medical or mental health professional. They can assess your symptoms and provide a diagnosis and treatment plan.

4. Can geometric algebra be used to explain or predict hallucinations?

There is ongoing research on the use of geometric algebra to model and understand hallucinations. However, it is important to note that hallucinations are complex phenomena and may have multiple causes, so a single mathematical framework may not be able to fully explain or predict them.

5. Are hallucinations always a sign of mental illness?

No, hallucinations can be caused by a variety of factors and may not necessarily indicate a mental illness. However, if you are experiencing hallucinations, it is important to seek help from a medical or mental health professional to determine the underlying cause and receive appropriate treatment.

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