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Minimalising Quantum Mechanics

Minimalising Quantum Mechanics. Peter Rowlands. Creating a nilpotent structure. This is derived most simply by first taking the classical: E 2 – p 2 – m 2 = 0

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Minimalising Quantum Mechanics

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  1. Minimalising Quantum Mechanics Peter Rowlands

  2. Creating a nilpotent structure This is derived most simply by first taking the classical: E2 – p2 – m2 = 0 and factorizing using noncommuting algebraic operators (multivariate 4-vector quaternions or complex double quaternions): (± ikE±ip +jm) (± ikE±ip +jm) = 0

  3. ijk quaternion units ijk vector units 1 scalar i pseudoscalar The multivariate vector units are effectively a complexified quaternion system, which is commutative to i, j, k. Multivariate vectors a and b follow the product rule: ab = a.b + i ab Quaternions Multivariate Vectors The algebra is a tensor product of quaternions and vectors

  4. (±1, ± i) 4 units (±1, ± i) × (i,j,k) 12 units (±1, ± i) × (i, j,k) 12 units (±1, ± i) × (i,j,k ) × (i, j,k) 36 units The units applied to E, p and m can be seen to be isomorphic to the g matrices and generate the entire algebra: ikii ji ki 1j There are 64 possible products of the 8 units

  5. The nilpotent Dirac equation Now apply canonical quantization to the left-hand bracket of (± ikE±ip +jm) (± ikE±ip +jm) = 0 with an appropriate phase factor, e–i(Et – p.r), for a free particle, to give: (k / t±ii+jm)(± ikE±ip +jm) e–i(Et– p.r) = 0 This now becomes equivalent to the Dirac equation for a free fermion.

  6. The Dirac 4-spinor The E and p here represent the four simultaneous possibilities: fermion / antifermion E spin up / down p which are conventionally incorporated into a 4-component spinor y, and it is convenient to represent the four possible sign conventions by the components of a row (or column) vector.

  7. The Dirac 4-spinor Also, for a multivariate p, pp = (s.p) (s.p) = pp = p2 So we can also use s.p for p (or s.Ñ for Ñ) in the Dirac equation, where s.p is helicity, and s is a pseudovector of magnitude –1.

  8. The Dirac 4-spinor Significantly, all four terms have the same phase factor. The sign variation applies only to operator and amplitude.

  9. The nilpotent amplitude The important property is that the amplitude (± ikE±ip +jm) is a nilpotent, or square root of zero. If we assume this is always true, whether the particle is free or not, then we can get rid of the equation altogether.

  10. The nilpotent operator So we can use covariant derivatives or include field terms to represent the operators ikE and ip. For example, in the simplest case: Once the nature of the ikE and ip operators is decided, then the phase factor is uniquely determined, because it must be such that the amplitude that results squares to zero. Finding the phase factor which does this is what we mean by finding a ‘solution’. We never need to refer to an equation.

  11. The nilpotent operator and vacuum If we take (± ikE±ip +jm) as an operator, we may ask: what is it operating on? The answer has to be vacuum or the rest of the universe. To specify the exact state of a fermion, the ikE and ip terms will have to include its interactions with all other fermionic states. The fermion is an open system because (± ikE±ip +jm) (± ikE±ip +jm) = 0 conserves energy only over the entire universe.

  12. Pauli exclusion and zero totality Clearly, the expression (±ikE±ip +jm) (± ikE±ip +jm) = 0 is a statement of Pauli exclusion, if we take both terms as amplitudes. It can also be taken as an expression of a zero totality universe. If we extract (±ikE±ip +jm) from nothing, we are left with –(±ikE±ip +jm) and again –(±ikE±ip +jm) (± ikE±ip +jm) = 0

  13. The quantum field Because of the way they are defined, nilpotent operators are specified with respect to the entire quantum field. Formal second quantization is unnecessary. We can consider the nilpotency as defining the interaction between the localized fermionic state and the unlocalized vacuum, with which it is uniquely self-dual. The phase is the mechanism through which this is accomplished.

  14. Multiple meanings We can now see that the expression (± ikE±ip +jm) (±ikE±ip +jm)  0 has at least five independent meanings. classical special relativity operator  operator Klein-Gordon equation operator  wavefunction Dirac equation wavefunction  wavefunction Pauli exclusion fermion  vacuum nonequilibrium thermodynamics

  15. Creation operators It is possible to define the entire structure of quantum mechanics by defining the creation operator for a single fermion (±ikE±ip +jm) as a nilpotent. Of course, there are four creation (or annihilation) operators here (or two of each): (ikE+ip +jm) fermion spin up (ikE–ip +jm) fermion spin down (–ikE+ip +jm) antifermion spin down (–ikE–ip +jm) antifermion spin up But only the first term defines the real fermionic state. The others are vacuum ‘reflections’, or the states into which it could transform.

  16. P, T, C transformations And, strictly, the spinor structure of the operator is redundant, along with the phase factor, the amplitude, and the QM equation, as it is an automatic consequence, by sign variation, from the choice of the lead term, or by respective P, T or C transformations: Pi (ikE+ip +jm) i = (ikE–ip +jm) Tk(ikE+ip +jm) k = (–ikE+ip +jm) C –j (ikE+ip +jm) j = (–ikE–ip +jm)

  17. Vacuum operators There is another way to look at this. If we take (ikE+ip +jm) and postmultiply it by k(ikE+ip +jm) or i(ikE+ip +jm) or j(ikE+ip +jm) the result is (ikE+ip +jm), multiplied by a scalar. This can be done an indefinite number of times. So these three idempotent terms behave as a vacuum operators.

  18. Vacuum operators We can even suggest specific identifications on the basis of the pseudoscalar, vector and scalar characteristics of the associated terms. k(ikE + ip + jm) weak vacuum fermion creation i (ikE + ip + jm) strong vacuum gluon plasma j (ikE + ip + jm) electric vacuum SU(2) The 3 additional terms in the fermion spinor then become strong, weak and electric vacuum ‘reflections’ of the state defined by the lead term.

  19. Vacuum operators We can see the three vacuum coefficients k, i, j as originating in (or being responsible for) the concept of discrete (point-like) charge. The operators act as a discrete partitioning of the continuous vacuum responsible for zero-point energy. In this sense, they are related to weak, strong and electric localized charges, though they are delocalized.

  20. Bosonic states The transformations here also lead to the production of the three types of bosonic state: Spin 1 boson: (ikE+ip +jm) (–ikE+ip +jm) T Spin 0 boson: (ikE+ip +jm) (–ikE–ip +jm) C Bose-Einstein condensate / Berry phase, etc.: (ikE+ip +jm) (ikE–ip +jm) P

  21. Bosonic states and vacua We can see how the 3 bosonic states are related to vacua produced by the 3 charge operators: weak spin 1 (ikE + ip + jm) k(ikE + ip + jm) k(ikE + ip + jm) k(ikE + ip + jm) … (ikE + ip + jm) (–ikE + ip + jm) (ikE + ip + jm) (–ikE + ip + jm) … electric spin 0 (ikE + ip + jm) j(ikE + ip + jm) j(ikE + ip + jm) j(ikE + ip + jm) … (ikE + ip + jm) (– ikE –ip + jm) (ikE + ip + jm) (– ikE – ip + jm) … strong B-E condensate (ikE + ip + jm) i(ikE + ip + jm) i(ikE + ip + jm) i(ikE + ip + jm) … (ikE + ip + jm) (ikE – ip + jm) (ikE + ip + jm) (ikE – ip + jm) …

  22. Supersymmetry and renormalization Ultimately, this means that we don’t need renormalization. Every fermion is its own supersymmetric bosonic partner in the vacuum, and vice versa, with the same E, p, m. The fermion and boson loops should cancel automatically. There is no hierarchy problem, and we don’t need any supersymmetric particles.

  23. Fermionic spin Fermionic spin is a routine derivation from the p component of the nilpotent structure. If we mathematically define a quantity s= –1, then [s, H] = [–1, i (ip1 + jp2 + kp3) + ijm] = [–1, i (ip1 + jp2 + kp3)] = –2i (ijp2 + ikp3 + jip1+ jkp3 + kip1 + kjp2) = –2ii (k(p2 ­– p1) + j(p1 ­– p3) + i(p3 ­– p2)) = –2ii1p

  24. Fermionic spin If L is the orbital angular momentum rp, then [L, H] = [rp, i (ip1 + jp2 + kp3) + ikm] = [rp, i (ip1 + jp2 + kp3)] = i [r, (ip1 + jp2 + kp3)] p But [r, (ip1 + jp2 + kp3)]y = i1y . Hence [L, H] = ii1p , and L + s / 2 is a constant of the motion, because [L + s / 2, H] = 0.

  25. Helicity Helicity (s.p) is another constant of the motion because [s.p, H] = [–p, i (ip1 + jp2 + kp3) + ijm] = 0 For fermion / antifermion with zero mass, (kE + iis.p + ijm)  (kE – iip) (–kE + iis.p + ijm)  (–kE – iip) Each of these is associated with a single sign of helicity, (kE + iip) and (– kE + iip) being excluded, if we choose the same sign conventions for p.

  26. Helicity Numerically, E = p, so we can express the allowed states as E(k– ii) Multiplication from the left by the projection operator (1 – ij) / 2  (1 – g5) / 2 leaves the allowed states unchanged while zeroing the excluded ones.

  27. Idempotent and nilpotent The conventional idempotent QM uses the same equation as nilpotent QM, but operator and wavefunction are differently defined. IDEMPOTENT [(ik /t + i + jm) j] [j(ikE + ip + jm) e–i(Et – p.r)] = 0. operator wavefunction NILPOTENT [(ik /t + i + jm)jj] [(ikE + ip + jm) e–i(Et – p.r)] = 0. operator wavefunction

  28. Discrete differentiation The nilpotent operator has three terms, compartmentalised using the quaternions k, i, j, in a similar way to real and imaginary parts. If we use discrete differentiation, we can reduce it to two. In discrete differentiation, as defined by Kauffman, to preserve the Leibniz rule, we take and The mass term disappears in the operator (though it has to be introduced in the amplitude).

  29. Discrete differentiation Suppose we define a nilpotent amplitude y= ikE + iiP1 + ijP2 + ikP3 + jm and anoperator D with and

  30. The discrete nilpotent Dirac equation With some straightforward algebraic manipulation, we find that Dy = iy(ikE + iiP1 + ijP2 + ikP3 + jm) + i(ikE + iiP1 + ijP2 + ikP3 + jm)y– 2 i(E2 – P12 – P22 – P32 – m2). When is y nilpotent, then Dy = This is a Dirac equation using discrete differentials.

  31. The Discrete nilpotent Dirac equation • There are 3 remarkable things about this equation. • It makes no difference whether we introduce i before the differentials. It is either classical or quantum. • (2) It contains no mass term in the operator, and so annihilation and creation operators are exact negatives of each other. • (3) We can convert the differentials to covariant derivatives, and so have an operator for a distorted space-time without a mass term.

  32. A Finsler connection Because there is no mass term, the opportunity arises to represent the appearance of mass directly in terms of an anisotropic space-time structure. For example, Bogoslovsky’s geometric phase transition, interpreted as a mass-creating spontaneous-symmetry breaking in the fermion-antifermion consendate (represented in nilpotent theory as a product of two nilpotent operators / amplitudes).

  33. Zitterbewegung A nilpotent Hamiltonian can be written in the form H = –ijcs.p – iiimc2 = –ijc1p – iiimc2 = acp – iiimc2. Taking a = –ij1as a dynamical variable, we define a velocity operator, which, for a free particle, becomes: [r, H]

  34. Zitterbewegung The equation of motion for this operator then becomes: [a, H] (cp – H a) The standard solution for this gives the zitterbewegung term for the free fermion, which is interpreted as a switching between the fermion’s four spin states.

  35. Antisymmetric wavefunctions We may note that nilpotent wavefunctions or amplitudes, which are Pauli exclusive because nilpotent, are also automatically antisymmetric (y1y2 – y2y1), and so Pauli exclusive in the conventional sense as well: (± ikE1±ip1+jm1) (± ikE2±ip2+jm2) – (± ikE2±ip2+jm2) (± ikE1±ip1+jm1) = 4p1p2 – 4p2p1 = 8 ip1p2

  36. Antisymmetric wavefunctions This is a remarkable result. It implies that, instantaneously, any nilpotent wavefunction must have a p vector in real space (a spin ‘phase’) at a different orientation to any other. The wavefunctions of all nilpotent fermions instantaneously correlate because the planes of their p vector directions must all intersect, and the intersections actually create the meaning of Euclidean space, with an intrinsic spherical symmetry generated by the fermions themselves.

  37. Antisymmetric wavefunctions At the same time, the equation could also be interpreted as suggesting that each nilpotent also has a unique direction in a quaternionic phase space, in which E, p and m values are arranged along orthogonal axes. We may suppose here that the mass shell or real particle condition requires the coincidence between the directions in these two spaces. In addition, the p vector carries all the information available to a fermionic state, its direction also determining its E and p values uniquely.

  38. Antisymmetric wavefunctions Three consequences of this are immediately apparent. (1) To avoid direction duplication, one at least of the three nilpotent terms (the mass term, in fact) must have only one algebraic sign (2) A hypothetical massless fermion and antifermion pair would require opposite helicities (say, ikE+ ip and – ikE+ ip) to avoid being on the diagonal (3) A massless fermion could not exist in practice because, since the magnitudes of E and p would always be equal in such cases, then the resultant angles would always be the same.

  39. The fundamental interactions The fermionic nilpotent operator can be used to do QM in the conventional way by define a probability density, etc. It can also do QED, QCD and weak interaction theory, and define propagators. It solves the relativistic hydrogen atom in six lines and provides analytic solutions for spherically symmetric fields involving other potentials. More important than these, however, is the fact that its structure demands the existence of all the three fundamental particle physics interactions.

  40. Spherical symmetry: the point source If we assume that the constraint of spherical symmetry exists for a point particle, then we can express the momentum term of the operator in polar coordinates, using the Dirac prescription, with an explicit spin term: We need the spin term because the multivariate nature of the p term cannot be expressed in polar coordinates.

  41. The scalar or Coulomb term The nilpotent Dirac operator now becomes: Now, whatever phase we apply this to, we will find that we will not get a nilpotent solution unless the 1 / r term with coefficient iis matched by a similar 1 / r term with coefficient k. So, simply requiring spherical symmetry for a point particle, requires a Coulomb term of the form A / r to be added to E. It is a fundamental statement of the nilpotent condition.

  42. The scalar term Every nilpotent solution related to a point source requires a Coulomb term, with a U(1) group symmetry, as a fundamental aspect of nilpotent structure, and it is a fundamental component of all three interactions. But, while all three terms in the nilpotent have a scalar component or magnitude, one term alone (the m term) is a pure scalar, and the existence of a pure scalar term also suggests the existence of a pure scalar interaction (the electric interaction).

  43. The vector term (ikE +iipx + jm)(ikE + ijpy + jm) (ikE + ikpz + jm) +RGB (ikE –iipx + jm)(ikE – ijpy + jm) (ikE – ikpz + jm) –RBG (ikE + iipx + jm)(ikE +ijpy + jm) (ikE + ikpz + jm) +BRG (ikE – iipx + jm)(ikE – ijpy + jm) (ikE – ikpz + jm) –GRB (ikE + iipx + jm)(ikE + ijpy + jm) (ikE +ikpz + jm) +GBR (ikE – iipx + jm)(ikE – ijpy + jm) (ikE –ikpz + jm) –BGR

  44. The vector term The condition for making the transitions between these phases gauge invariant requires the SU(3) symmetry associated with the strong interaction. (It also requires the system to have mass because of the co-existence of right- and left-handed helicity states.) The fact that the state is completely entangled and that the interaction is nonlocal means that the rate of exchange of p in the process is independent of any physical separation that might exist between the three components.

  45. The vector term In effect, this requires a constant force and a potential that is linear with separation. Applying a combination of Coulomb and linear potential terms produces a structure whose nilpotent solutions require the characteristic infrared slavery and asymptotic freedom associated with the strong interaction.

  46. The pseudoscalar term The third aspect of the nilpotent structure which generates a physical effect is the spinor structure, and the accompanying zitterbewgung. Though this is only a vacuum process, it specifically requires the creation of bosonic structures of the form (ikE+ip +jm) (–ikE+ip +jm) and (ikE+ip +jm) (–ikE–ip +jm) via a harmonic oscillator mechanism, which is the characteristic defining process of the weak interaction, in real as well as vacuum states.

  47. The pseudoscalar term In effect, the zitterbewgung ensures that a fermion is always a weak dipole in relation to its vacuum states, and the single-handedness of the weak interaction can be regarded as the result of a weak dipole moment connected with fermionic ½-integral spin. Significantly, all weak interactions between real particles require sources that are in some senses dipoles (fermion-antifermion) and so can be expected to require a dipolar potential, in addition to the Coulomb term.

  48. The pseudoscalar term Any such potential combination applied to the nilpotent operator produces a series of energy levels characteristic of the harmonic oscillator. In this case, the interaction and its SU(2) symmetry appears to be generated by the duality of the pseudoscalar term ikE in generating antifermion, as well as fermion states.

  49. The structure of the nilpotent It seems to be that the structure of the nilpotent alone that produces the three fundamental interactions characteristic of particle physics, and that no external physical input is required. Simply by defining an operator which is a nilpotent 4-component spinor with vector properties, we necessarily imply that it is subject to electric (or other pure Coulomb), weak and strong interactions, and no other, because no other analytic nilpotent solution exists.

  50. The pseudoscalar term This structure is a product of the three types of quantity (pseudoscalar, multivariate vector and scalar) which it contains, and, ultimately, these are reflections of the need for a discrete (point) source to preserve spherical symmetry and hence to conserve angular momentum. We can, in fact, identify these and their associated symmetries as being connected with the three separately conserved aspects of angular momentum: magnitude (scalar, U(1), spherical symmetry does not depend on the length of the radius vector), direction (vector, SU(3), spherical symmetry does not depend on the choice of axes), and handedness (pseudoscalar, SU(2), spherical symmetry does not depend on whether the rotation is left- or right-handed).

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