What does it mean to say that a composite quantum particle containing red, green and blue quarks is “colourless” and does not feel the strong nuclear force?
Every quark has a wave function with 3 components, which we will call red, green and blue. But like the 3 directions in ordinary space, if you add up 3 equal-sized vectors in each of these directions, the sum certainly isn't zero!
But that’s not how colour charge works. When we have more than one particle, we need to form a larger vector space, with 9 directions for 2 particles, and 27 for 3 particles. Each of these 27 directions corresponds to a choice of colour for each of the particles: RRR, RRG, RRB, etc.
When a composite particle contains quarks of all 3 colours, the colourless state is not just one of the six permutations of RGB; rather, it is a sum of all 6, with a sign that depends on whether the permutation is even or odd:
w = (RGB - RBG + GBR - GRB + BRG - BGR)/√6
Why is this “colourless”?
The strong force depends on what happens to a wave function when you multiply its colour vector with a matrix U that belongs to a group called SU(3), which means it is a 3×3 matrix of complex numbers, with a determinant of 1, and its inverse is the matrix you get by taking its transpose and complex conjugate. If a wave function φ is unchanged by this, if Uφ = φ for every U, then φ will not feel the strong force. But the only 3-dimensional vector for which this is true for every U in SU(3) is φ=0.
However, if you have a composite particle with a 27-dimensional wave function, you need to multiply each of the 3 particle’s wave functions with U.
What happens if we do this for our vector w?
Greg Egan
npub12c…gd38p
2025-02-10 13:54:18 UTC
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"content": "What does it mean to say that a composite quantum particle containing red, green and blue quarks is “colourless” and does not feel the strong nuclear force?\n\nEvery quark has a wave function with 3 components, which we will call red, green and blue. But like the 3 directions in ordinary space, if you add up 3 equal-sized vectors in each of these directions, the sum certainly isn't zero!\n\nBut that’s not how colour charge works. When we have more than one particle, we need to form a larger vector space, with 9 directions for 2 particles, and 27 for 3 particles. Each of these 27 directions corresponds to a choice of colour for each of the particles: RRR, RRG, RRB, etc.\n\nWhen a composite particle contains quarks of all 3 colours, the colourless state is not just one of the six permutations of RGB; rather, it is a sum of all 6, with a sign that depends on whether the permutation is even or odd:\n\nw = (RGB - RBG + GBR - GRB + BRG - BGR)/√6\n\nWhy is this “colourless”?\n\nThe strong force depends on what happens to a wave function when you multiply its colour vector with a matrix U that belongs to a group called SU(3), which means it is a 3×3 matrix of complex numbers, with a determinant of 1, and its inverse is the matrix you get by taking its transpose and complex conjugate. If a wave function φ is unchanged by this, if Uφ = φ for every U, then φ will not feel the strong force. But the only 3-dimensional vector for which this is true for every U in SU(3) is φ=0.\n\nHowever, if you have a composite particle with a 27-dimensional wave function, you need to multiply each of the 3 particle’s wave functions with U.\n\nWhat happens if we do this for our vector w?",
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