Paul Bird's Scientific Research

Cosmological Constant Prediction Here is a paper that shows how to derive the Cosmological Constant f rom first principles and also shows why gravity curves differ from GR at precisely the places they do.

This is an article I wrote on the intrinsic Riemann Curvature of Implicit Surfaces (Algebraic Varieties) . If you have any thoughts on this please comment below.

We shall define the  Generalised Complex Contour Integral as: \[  \oint_X \Delta[\phi]D\phi \equiv i \int\int \delta[X]\left( \overline{\Delta}[\overline{\phi}]\det\left[\frac{\delta X}{\delta \phi}\right] + \Delta[\phi] \det\left[\frac{\delta X}{\delta \overline{\phi}}\right] \right) D\phi D\overline{\phi} \] The function \[ \Delta[\phi] \]  is for \[ S_0 \] : \[ \Delta[\phi]  = \frac{1}{2\pi i (\phi_1 - \phi_2)}\] For \[S_n\] where n>0 \[ \Delta[\phi]  = \frac{1}{\int \varepsilon_{abc..}\phi^a(x) \partial_1 \phi^b(x) \partial_2 \phi^c(x)...     dx^n}\]  

 Type Theory My understanding of Type theory. Defining natural numbers: zero: Nat succ(n: Nat ): Nat So for example defining the number 2: succ(succ(zero)): Nat same as??? isNat(zero) ForAll(n) :  isNat(n)=>isNat(succ(n)) We can define operations like plus and times and equals //define proofs that  n==n equals(zero,zero): Equals< zero,zero > equals(succ(n: Nat ),succ(n: Nat )): Equals< succ ( n :Nat ) , succ ( n :Nat ) > Proofs and Dependent types Define a dependent type Even <n: Nat > which is a proof that n is even.  {a: Nat, p: Equals <times(2,a),n>}: Even <n: Nat > So that for example the following is a proof that 6 is even: {3, equals(times(2,3),6)}: Even <6> which becomes {3: Nat ,equals(6,6): Equals <6,6>} : Even <6> {a: Nat ,  p: Equals <times(2,a),n>}: Even <n: Nat > isEven(n) = n =>    (a => (isNat(a) & equals(times(2,a),n))) isEven(6) (3)  In C syntax typedef Eve...

We can define a distance invariant under superspace transformations: \[ R^2 = |x-y|^2 + \theta \gamma_\mu(x-y)^\mu \zeta \] For D space-time dimensions and E Grassman dimensions, does this satisfy a Cayley-Menger type equation? In 2 dimensions we know the volume of a tetrahedron is zero. Or: \[ \det \begin{bmatrix} 0 & R_{12} & R_{13} & R_{14} & 1\\ R_{12} & 0 & R_{23} & R_{24} & 1\\ R_{13} & R_{23} & 0 & R_{34} &1\\ R_{14} & R_{24} & R_{34} & 0 &1 \\ 1&1&1&1&0 \end{bmatrix} =0 \] What about when D=11 and E=32 ? Simplest is D=1, E=2 with (?): \[ R^2 = (x-y)^2 + (x-y)(\theta \zeta + \zeta^* \theta^* ) \] with for example: \[ R^4 = (x-y)^4 + 2 (x-y)^3(\theta \zeta + \zeta^* \theta^* ) + 2(x-y)^2( \theta \theta^* \zeta \zeta^* ) \] \[ R^6 = (x-y)^6 + 3 (x-y)^5(\theta \zeta + \zeta^* \theta^* ) + 6(x-y)^4( \theta \theta^* \zeta \zeta^* ) \] Then the area of the triangle is 0 (or an...

A network of belts would be equivalent to a network of directed graphs. A half twist in a belt corresponds to a directed arrow. In particular a sub structure rotating 360 degrees would reverse the arrows going into the substructure:   Can we be more clear on graph equivalences? Therefor such a patch as fermionic structure. But what are fermions? (Perhaps all nodes are fermions.)

Idea is that the state of the Universe is represented by a knot. The complex amplitude is given by a knot invariant. This is the sum of histories. A history is a series of slicings which return the knot to the unknot. Knot states related by Reidmeister moves are equivalent. The cosmological time of a knot is the 'unknotting' number. One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1. However, many of these may have very low amplitudes. The amplitudes might be given by the knot polynomials , e.g,for some constant,k,q: \[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]

First is General Relativity . The action can be put in this form: \[ S_{EH'}=\frac{1}{8\kappa}\int \sqrt{-g}\left( g^{ab}g^{de}g^{cf} +2 g^{ac}g^{bf}g^{de} + 3g^{ad}g^{be}g^{cf} -6 g^{ad}g^{bf}g^{ce} \right)\partial_c g_{ab}\partial_f g_{de} dx^4 \] Where \(g^{ab} = \frac{1}{4!}\varepsilon^{apqr}\varepsilon^{auvw}g_{pu}g_{qv}g_{rw} / \det(g) \) We can get Maxwell's equations from the above by a Kaluza-Klein compactification on (D+n) dimensions. They are:  \[S_{YM} = \int\sqrt{-g}\frac{1}{2}(g^{ab}g^{cd} - g^{ac}g^{bd})(\partial_a A_b + i A_a A_b) (\partial_c A_d + i A_c A_d) dx^4 \] Where the A fields are matrices which generate the gauge group. Then Dirac's equations are: \[S_D = \int e \psi^\dagger e^a\sigma (\partial_a + i A_a.\tau + i\Omega_a)\psi + M (\psi \varepsilon\psi + \psi^\dagger \varepsilon\psi^\dagger )dx^4\] with: \[ \Omega_a = \varepsilon e_a e^b e^c \partial_b e_c \] and \(\tau\) is a generator of a gauge group, such as SU(n). M is a matrix which...

Why not expand LQG around a ground state? It makes more sense? psi(A) = Sum_loops exp(iA.dx) * Ground(A) In 2+1 dimensions Ground(A) might be for example the chern simons invariant. AdA+(2/3)A^A^A

Let us take a wave function of the metric at time T=0. Expanded around a flat space g  = n + h. Ψ[h] = a +  Ψ( x )h( x ,0) +  Ψ( x , y )h( x ,0)h( y ,0) + ..... The Wheeler-de-Witt equation should give a relation between the  Ψ s. Because g is a linear function of h, d/dh=d/dg. When expanded in terms of h, this has an infinite number of terms. This imposes very special constraints on  Ψ  to give it the right symmetries. (A similar thing happens in string theory). ((gg+gg-gg) d/dh( x ,0) d/dh( x ,0) + det(g)R[g] )   Ψ[h] = 0 This equation says how the wave function amplitudes are connected  be when 2 particles are in the same place.  If h^2 is approximately 0. Then h behaves like a gauge field. Part 2 ------- Lets take a different expansion g=exp(h) , det(g) = exp(Tr(h)) , delta h = da + db? Then this timer d/dg = g^-1 d/dh (  d/dh d/dh +  exp(Tr[h]+h+h-h)*(dh)^2 )   Ψ[h] = 0 ? Ψ(x) = exp(-x^...

Invention A pair of interlocking tori with a helical grooves may serve as a transmission gear???

Suppose we take the idea that particles can never be inside a black hole event horizon as an axiom. This means we can either set the 2-particle wave function for cases where particles are too close to zero. But this does not lead to smooth functions. Another option is to reflect this part of the wave function back out to infinity. For stationary masses m_1 and m_2. Ψ (x,y) =  Ψ ( (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2,    (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 ) u=  (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2 v= (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 L = (d/dy^2-m2^2)(d/dx^2-m1^2) Ψ (x,y)  + (d/dv^2-m1^2)(d/dv^2-m^2) Ψ (u,v) ?? invariant under x->u, y->v so  Ψ(0,r) =  Ψ(0,m/r) ? Ψ(0,m) =  Ψ(0,m) Ψ(m,0) =  Ψ(m,0) Ψ(m/2,m/2) =  Ψ(m/2,m/2) H=s(x,y)[ Ψ (x,y) -  Ψ ( (x + y)/2 +  (x-y) (m_1+m_2) /|x-y|/2,    (x + y)/2 - (m_1+m_2) (x-y) | x-y|/2 ) ] dxdy

One of the axioms of field theory is that the amplitude for a path is exp(i L ) where L is distance from x to y. L is an invariant of O(n) where n is the number of dimensions. One might ask, it possible to have quantum mechanics in space where the symmetry is a different group? Sp(n) and SU(n) have invariants of the form xy-yx, which unfortunately are identically zero for commuting coordinates. So really we are only interested in groups that can be defined in terms of symmetric invariants. An interesting group is F4. In 26 dimensions it has the symmetric invariant (S): So we could define a quantum theory in this space with an invariant based on these two invariants. For example we could define our amplitude as: exp( i L + j S)  i  X' X'  +    j X' X'X'  ?

There are interesting relations between the lepton masses (see Koide formula). Also, we can see that the 3 masses come approximately from the equation: x 3 - 41 x 2 + 90 x -1 = 0 Where the answers (|x0|, |x1|, |x2|) * 45.76MeV Is this significant? Probably not. Most equations seem to have integer values.

An Exceptional Supersymmetry It is proposed in this article a way to construct an exceptional supersymmetry distinct from the family of usual supersymmetries usually labelled by N. (N=1, N=2, N=4 and N=8). It relies on the special properties of the root system of E 8 . It is a different kind of super-symmetry in that the superpartners don't have the same charge but together the charges from all the particles in the representation form a group. In a nut-shell we are assigning spins to elements of the E8 algebra and promoting them to super-symmetry operators. The special properties of E8 allows us to do this. Introduction The similarities between  N=8 Supersymmetry with 256 generators and the group E8 with 248 generators is suggestive that there might exist a combined structure with properties of both. Noting that E8 can be split into a bosonic part (a representation of O(16) ) and a fermionic part a spinor representation of O(16). We can assign a "spin" to each...

These unit take the mass of the Top Quark (175GeV) equal to 1. Then we have approximately: m(top) = 1 m(H) = 1/ √2 m(W), m(Z)= 1/2 m(b) = 1/42 = 1/(6.7)    m(strange) = 1/1842 = 1/42^2 m(up) = 1/72000??? = 1/42^3 m(charm) = 1/141     (close to fine structure) m(down) <  1/182^2 m(tau) = 1/100   = 1/10^2 m(mu) = 1/1673  = 1/41^2  m(e)=1/345000 = 1/587^2   t(1)              b(A) c(B)             s(A^2) d(B^2)        u(A^3) A=1/42 B=1/135 m(v)=?=exp(-9) h=1 = 6.6[-34]m^2 kg/s c=1  = 3[8]m/s m(H)=2.22[-25]kg 1 s = 3*10^(-17) If we consider only mass^2 as important then we see: m(H)=1/sqrt(2), m(W or Z)=1/2, m(top)=1, m(bottom)=1/42 1 Length Unit = 1.0[-17]metres  /(1.414) 1 Time Unit = 3.3[-26] seconds   /(1.414) 1 Mass Unit = 2.22[-25]kg   (*1.414) G  = 6.67[-11]m^3/ kg/s^2 = 1.6[-35] Units = ex...

Classical Temporal Logic In classical theory any two events satisfy a time ordering. We shall call this classical temporal logic CTL . The rules of classical temporal logic are very simple. Since they are just the rules of inequalities of real numbers: A>B    →   B<A !(A>B)   →  A<B or A=B A>B & B>C  →  A>C Or equivalently given that x  →  y is equivalent to saying that (!x or y) is true, these rules can be given by saying that in CTL these statements are always true: !(A>B) or B<A A>B or A=B or A<B !(A>B) or !(B>C) or A>C With these rules we can dispense with the underlying geometric idea of events as points in space-time and we have all we need to know to compare the temporal ordering of events. Relativity In relativity there are four possibilities for events. Either event A is is in the future light-cone of event B in which case we write A>B or A is i...

The Fourier transform the continuous variable "position" is the continuous variable "momentum". The Fourier transform of the interval around a finite dimension is an integer charge. Therefore particles of different charges are simply the different modes around a finite dimension. In position space we should also talk about position in compact space. Everything becomes geometrical. In momentum space we should talk about charge. Can we extend this to spin also? Is spin part of the position or momentum space picture? If X (a,b) = ((x+t,y+iz),(y-iz,x-t)) then X (a,b) X (-a,-b) = x^2+y^2+z^2-t^2. Perhaps this is the better momentum picture? What do a and b mean?  ++ -- +- -+ Ψ *(-a)( D (a,b)+i A (a,b)) Ψ (b) + ( D (a,b) A (c,d)- D (c,d) A (a,b)).( D (-a,-b) A (-c,-d)- D (-c,-d) A (-a,-b)) +m Ψ L(a) Ψ R(-a)                  = D(a,b)A(c,d)D(a,b)A(c,d) - D(a,b)A(c,d)D(-c,-d)A(-a,-b) X*(a,b) = X(b,a)

If we have a particle (X,Y,Z,T) = (a(T),a(T),a(T),T) in comoving coordinates, in Milne coordinates this is: t=T cosh( a(T)*sqrt(3) ) x=T/sqrt(3) * sinh( sqrt(3)*a(T) ) which gives a parametric equation in T. We can plot these curves. e.g. for linear expansion we have the lines a(T) = v.T for various constants v.