Comparing the TGD view of color perception to the geometric models of color perception

Gary Ehlenberger sent an interesting link related to the theories of color perception (see this). It has been known for a long time that colors are not properties of the light or of any stimulus producing color sensation. The notion of color space tries to geometrize hue, lightness and saturation as attributes of color perception. Already Riemann suggested that the Riemannian geometry alone could explain the perceived color differences in terms of a distance in this color space. Schrödinger proposed later a geometric theory of color perception but it has its own problems.

Bujac et al (see this and this) have proposed a non-Riemannian geometric model for color perception claiming to solve the problems of the Riemannian models. TGD suggests a quantum theory of color perception (see this) involving in an essential way the new physics predicted by TGD.

1. Problems of models based on Riemannian geometry of color space

The theories based on Riemann geometry of color space have however problems.

1. Bezold- Brücke effect (see this) is a phenomenon in which a changing light intensity can make a color appear to shift in hue. Neuroscientists would say the geometric model is simply unrealistic. This effect could be due to varying responses of light-receptors to the increasing light intensity. For instance, decreasing the intensity, the contribution of rods which is white or black increases and the hue changes.
2. The sum of small color differences along a geodesic line connecting two different colors is larger than the perceived color difference: this is known as the problem of diminishing returns and is not consistent with Riemnannian geometry. The principle of diminished returns (Weber-Fechner law) is actually a general principle of sensory perception.
3. There is also a problem with the identification of neutral direction along which the color dominated by white becomes dominated by black. A non-Riemannian variant of color space has been proposed. The link indeed describes a new theory (see this) claiming to solve a basic problem of Schrödinger’s geometric theory of color perception. This theory would not rely on Riemannian geometry and claims to solve the also the problem of diminished returns (see this). The assignment of Weber-Fechner law to the geometry of color space looks to me an implausible idea. However, it would seem to me that the notion of color space is too naive.

2. TGD view of color perception

In the TGD framework, the notion of color space is given up and instead a model of color perception as quantum measurement is developed. This model relies in an essential way to the new physics predicted by TGD but also involves geometrical considerations.

2.1 Color perception as measurement of color quantum numbers

Consider first the general observations motivating the proposal.

1. In TGD inspired theory of consciousness (see this and this), sensory perceptions can be identified either as the quantum numbers of outcomes of a quantum measurement or their differences for the states in the quantum jumps: for vision both options give 3+3 fundamental colors. It turned out that in zero energy ontology (ZEO) of TGD only the first option is correct: conscious self corresponds to a sequence of “small” state function reductions and sensory qualia correspond to the quantum numbers measured in SSFR and moment of consciousness has duration measured by the geometric time between two subsequent SSFRs.
2. The first option allows two variants. Fundamental colors could correspond to
   1. the color quantum numbers for quark triplet and their complementary colors to antiquark triplet.
   2. or to 3+3 pairs of gluons forming a color octet: the 2 states with vanishing color quantum numbers do not have color.

   At this moment I cannot distinguish between these options. Interestingly, the sum of the numbers of charged states in the fundamental representation and its conjugate and the number of charged states in adjoint representation are equal only for SU(3) and Sp₂ equivalent to SO(5) or B₂. Therefore it seems that for CP₂ there is an additional symmetry analogous to supersymmetry. The assignment of perceived colors to the quantum numbers characterizing quarks or gluons looks of course complete nonsense in the standard physics framework. The irony is that the properties of QCD color fit nicely with the properties of visual colors and it is this observation which must have motivated the terminology as a kind of joke.

3. Also black and white would be complementary colors and color black is a sensation produced by dark current, which gives rise to a genuine stimulus in the brain. In its absence there is no sensation of darkness. This would suggest that the notions of lightness and saturation could be reduced to the contribution of black and white colors in the total visual input. The contribution of cones (color sensitive in ordinary sense) and rods involved with night vision would be in the same role. Lightness would be determined by the relative contributions of white and black colors and saturation by the ratio of ordinary colors as compared to that of black and white.
4. The sums of two color quantum numbers vanish for both triplet and antitriplet: this explains why the third fundamental color can be produced by superposing the other 2 colors.
5. For the physical states total color quantum numbers vanish by color confinement (see this, this and this). A region of a given color is surrounded by a narrow frame with a complementary color. Could this be related to color confinement?
6. This model requires that one can speak of color at least in cellular scales. TGD indeed allows colored states in arbitrarily long length scales. The predicted hierarchy of Planck constants makes color symmetries, in particular scaled variants of color symmetries possible in arbitrarily long scales and in living matter the length scale range 10 nm-2.5 μm contains the p-adic length scales for 4 Gaussian Mersenne primes. This number theoretical miracle makes also scaled down possible copies of hadron physics possible in these scales (see this, this, this and this) .

2.2 The findings of Barbara Shipman as support for the TGD view of color perception

In the standard physics framework, the presence of QCD color in the physics of color perception is impossible since the length scale for hadron physics is hadron scale. Topologist Barbara Shipman (see this) has found direct evidence for the appearance of the mathematics of color symmetries of strong interactions in a mathematical model for the dance of honeybee.

1. The mathematics used by Barbara Shipman involves so-called momentum maps associated with the symplectic action of a group G, now SU(3) in a space X with a symplectic structure. SU(3) itself, SU(3)/U(1)× U(1) and CP₂ are examples of the space X. Symplectic action means that the elements of the Lie algebra g of G correspond to Hamiltonians as functions in X. The Lie algebra commutator in g is represented as a Poisson bracket.

The points of Lie algebra g of G can be mapped by exponential map to the points of X and the duality between g and g^* a map of the points of X to the adjoint g^* of g. What this means physically is that n commuting coordinates and the n values of their conjugate coordinates representing conserved charges as Hamiltonians determine the orbit of a point in X.

The space F=SU(3)/U(1)× U(1) is a 6-D space, which can be identified as the twistor space of CP₂ that is a a bundle having CP₂ as a base and 2-sphere as a fiber. F has also an interpretation as the space for the choices of color quantization axes. F plays a key role in the construction of quantum TGD (see this) and the existence of the twistor space with Kähler structure makes M⁴ and CP₂ unique (see this) so that TGD is unique.

A comparison with M⁴ helps here. For the rotation group SO(3) acting in CP₁=S², the space of quantization axis of angular momentum is SO(3)/SO(2)=SU(2)/U(1)=CP₁= S². For Minkowski space M⁴ the twistor space is a bundle having M⁴ as a base and the 2-D sphere of light-like rays from a given point of M⁴ as a fiber.

Also the double coset space F₁= CP₂/U(1)× U(1)= U(2)\ SU(3)/U(1)× U(1) is 2-dimensional. Eguchi-Hanson coordinates are standard coordinates for CP₂ (see this). Since the phase-like angle coordinates of CP₂ associated with U(1)× U(1) have as conjugates the coordinates r and θ of CP₂, this space does not allow a symplectic structure.

The lower-dimensional case of CP₁ helps to get some idea of the topology of F₁. U(1)\ CP₁/U(1) can be identified as a half geodesic from the North Pole to the South pole and is not a symplectic space. The identification as a half-geodesic is not unique since it is determined only apart from local U(1) rotation of the half-geodesic. F₁ has a sphere with 3 circles S¹ as boundaries and also now the identification is unique only modulo local U(1)× U(1) rotations. The full space CP₁ of quantization axes is needed also at S² and also at CP₂ so that F₁ cannot be identified as a kind of reduced space for the quantization axes.

The honeybee dance represents geometric information about the direction and distance of the food source so that only 2-D data are involved. Could the 2-D F₁ code for this information? Perhaps a more plausible option is that since F is a sphere bundle, the points of the fiber sphere S² code for the 2-D data, essentially the direction of color quantization axes. The path of the honeybee to the food source would be mapped to a path in S² and would represent the history of color perceptions of the honeybee along the path.

The path to the food source ends up with detection involving pattern recognition. The color pattern of the food source must play a key role in the recognition. Does the dance also represent this information?

2.3 The role of the twistor space of CP₂ in color perception

The space F is of obvious interest concerning the understanding of color perception.

1. From a given state characterized by color isospin and hypercharge, one can obtain all possible colored states by making SU(3) color rotations. The action of the elements of the Cartan group U(1)× U(1) multiplies the state with a mere phase factor so that the physical state is not changed.

Therefore the 6-D space F=SU(3)/U(1)× U(1) describes the space of all color perceptions for a given irreducible representation of the color group when the choice of quantization axes is allowed to vary? Is the perceived color independent of the choice of the quantization axes but that the probability for a given color quantum numbers in the measurement of color quantum numbers is determined by the state as in quantum measurement theory? This kind of view is suggested by the analogy of the Relativity Principle for color symmetries. For a fixed choice of quantization axes, one has only a discrete set of colors: 3+3 for both quark and gluon states.

For the tensor products of the simplest color states, color representations with a much larger repertoire of color quantum numbers are possible and the number of different hues becomes large. The number of the tensor factors contributing to the observed color, determining the possible dimensions of the irreducible color representations, should correspond to the intensity of light. This should correspond to the physiological situation and large quantum number limit behaving classically.

An interesting challenge is to understand the change of the hue with the increase of the intensity (Bezold- Brücke effect, see this) and therefore with the number of the contributing tensor factors. The increase of the intensity could increase the number of the active tensor factors involved and therefore extend the color palette. The probabilities of the receptors to respond to the incoming light depend on its intensity.

It is also possible to construct color multiplets in F consisting of color eigenstates. Two degrees of freedom correspond to 2 angle variables related to color quantum numbers and the remaining 4 to additional degrees of freedom which should geometrically relate to CP₂. Are these multiplets realized physically as the twistor lift of TGD suggests (see this)? A note about the role of CP₂ coordinates is in order. Induced classical electroweak gauge fields depend on gradients of CP₂ coordinates. The CP₂ part of the trace of the second fundamental form, involving second derivatives of CP₂ coordinates, is vanishing except at the 3-D edges of the space-time surface at which minimal surface property fails: it has quantum numbers of the Higgs field. The 2 complex CP₂ coordinates appear as such in the spinor harmonics associated with CP₂.

2.4 What could happen in color perception?

What could happen in color perception identified as quantum measurement.

1. Does the perceived color correspond to the sum of the colors quantum numbers assignable to the color sensitive receptors reducing to the quark or gluon level? How are the common color quantization axes determined?
2. Is the choice of the quantization axes for a given individual determined somehow? In TGD, electroweak interactions are color interactions in the spin degrees of freedom for CP₂ and color isospin and hypercharge correspond to electroweak isospin and hypercharge. The directions of the latter quantization axes are fixed. This would mean that the directions of color quantization axes are always the same. If so, the perceived color would be determined by the sums of the discrete color quantum numbers for receptors.
3. If the individual states in the tensor product have different color quantization axes, their states are not color eigenstates for the choice of quantization axes made in the tensor product. The state would decompose to a direct sum of irreducible color representations. State function reduction would occur and give rise to a state with definite color quantum numbers. One might hope that the outcome is highly unique at large color quantum number limit. One should however understand what determines the choice of the quantization axis for the entire tensor product.
4. Is it possible to test which option is correct? After images change color could this be due to different choices of the quantization axes or is it a genuine change of the analog of sensory input determining the after image?

See the article Comparing the TGD view of color perception to the geometric models of color perception or the chapter General Theory of Qualia.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2026/06/comparing-tgd-view-of-color-perception.html