M Theory Lesson 27

The planar algebras of Vaughan Jones arise from a coloured operad of empty discs in a larger disc, with string pieces (open or closed) in the surface, such that there are an even number of boundary points on each disc. The theory of Jones’ subfactors is a kind of Galois theory for $II_{1}$ factors. Matti Pitkanen has considered the relevance of this to physics.

In M theory we would also like to consider higher dimensional operads. For quaternionic number theory it is appropriate to start with 3-sphere discs, plus further structure. Rather than marked boundary points, for instance, a 3-sphere can contain knots and links and also surface boundary components. If the surface pieces are punctured spheres they can be made to look like 2-disc diagrams. We could pack enormous amounts of algebraic information into such an operadic structure. Batanin’s 2-level tree composition is a guide to horizontal and vertical compositions in the 2-operad case.

4 Responses so far »

  1. 1

    CarlBrannen said,

    The paper by Matti Pitkanen was interesting. He talks about replacing the complex numbers with octonions or quaternions.

    This is similar to what I am proposing. But instead of replacing complexes with another field, I’m suggesting arrays of primitive idempotents in the Clifford algebra. This can always be put into a matrix representation, but it is far more restrictive than an arbitrary matrix.

    I would feel better about the physics if they were out looking at resonances and building relationships with observation.

  2. 2

    Kea said,

    I would feel better about the physics if …

    I agree, Carl. We’ll get there soon.

  3. 3

    Doug said,

    Planar Algebra [one_2D_image] is very useful, but truly lacks the power of more realistic [array] 3D imaging [three_2D_iamges].

    One should note that anytime magnetism or electricity is involved, there may be unseen orthogonal “imaginary” [or invisible] axes to each of the three “real” views.

    I just thought of a better way to demonstrate architectural [or medicinal] geometry when I read that CarlB used the term “resonance “. The references have images and explanations.

    “Magnetic resonance imaging (MRI) often uses Three Planar Views [sums to 3D]:
    1 – Sagittal View – side-side, usually sliced from left to right
    2 – Axial View – top-down, usually sliced from bottom to top
    3 – Coronal View – front-back, usually sliced from face to occiput

    If functional MRI, adds time.

    “Magnetic resonance imaging (MRI) is a non-invasive way to take pictures of the body.
    Unlike x-rays and computed tomographic (CT) scans, which use radiation, MRI uses powerful magnets and radio waves. The MRI scanner contains the magnet. The magnetic field produced by an MRI is about 10 thousand times greater than the earth’s.
    The magnetic field forces hydrogen atoms in the body to line up in a certain way (similar to how the needle on a compass moves when you hold it near a magnet). When radio waves are sent toward the lined-up hydrogen atoms, they bounce back, and a computer records the signal. Different types of tissues send back different signals. For example, healthy tissue sends back a slightly different signal than cancerous tissue.”
    “MRI is superior to computed tomography (CT) in most cases where differentiation of soft tissues is necessary. It can view organs without obstruction by bone and foreign bodies. It is capable of showing the tissues from multiple viewpoints and is a noninvasive way to evaluate blood flow.”

    Consider the ‘Mathematics and Physics of Emerging Biomedical Imaging’.
    within Chapter 4 Magnetic Resonance Imaging
    Copyright © 2007. National Academy of Sciences

  4. 4

    Doug said,

    Kea, your diagram in this thread reminds me of one MRI planar view of the abdomen.

Comment RSS · TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: