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Spin on a 4D Feynman Checkerboard

Abstract

We discretize the Weyl equation for a massless, spin-1/2 particle on a time-diagonal, hypercubic spacetime lattice with null faces. The amplitude for a step of right-handed chirality is proportional to the spin projection operator in the step direction, while for left-handed it is the orthogonal projector. Iteration yields a path integral for the retarded propagator, with matrix path amplitude proportional to the product of projection operators. This assigns the amplitude i ±T 3B/2 2N to a path with N steps, B bends, and T right-handed minus left-handed bends, where the sign corresponds to the chirality. Fermion doubling does not occur in this discrete scheme. A Dirac mass m introduces the amplitude i 𝜖 m to flip chirality in any given time step 𝜖, and a Majorana mass similarly introduces a charge conjugation amplitude.

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Notes

  1. See [14] for a general investigation of spacetime polytopes with null faces.

  2. The eigenvalues of A(𝜃) have modulus less than or equal to unity if and only if the step speed α is ≥3. Our choice α=3 is thus the marginal value for a convergent finite difference scheme.

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Acknowledgments

We are grateful to E. Hawkins for mathematical aid, and to Yigal Shamir and Paulo Bedaque for instruction about lattice fermions. This work was supported in part by the NSF under grants PHY-9800967, PHY-0300710, PHY-0601800, PHY-0903572, PHY-1407744 at the University of Maryland, in part by the CNRS at the Insitut d’Astrophysique de Paris, and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.

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Correspondence to Brendan Z. Foster.

Appendix A: Norm of the Amplification Matrix

Appendix A: Norm of the Amplification Matrix

In this appendix we prove that the norm of the matrix A(𝜃) defined in (28) is less than unity unless at least three of the 𝜃 i coincide, in which case the norm is unity. The proof is due to Eli Hawkins.

Let |ν〉 be any unit spinor. The squared norm of A|ν〉 is ∥A|ν〉∥2=〈ν|A A|ν〉, whose maximum is the larger eigenvalue of A A. This value defines the squared norm ∥A2.

Using the definition of the spin projection operators (8) and the inner products of the unit vectors \(\hat {n}_{i}\) (1 if i = j and \(-\frac 13\) if ij) we find

$$ \text{tr} (A^{\dagger} A) = 1 + \tfrac16 \sum\limits_{(ij)} \cos(\theta_i-\theta_j) $$
(44)

where the sum is over the 6 choices of {i,j}⊂{1,2,3,4}. This trace is at most 2, and therefore the smaller eigenvalue of A A is less than 1 unless A A=1. Hence

$$ {\Phi} := \det \left( A^{\dag} A -1\right) $$
(45)

has the same sign as 1−∥A∥.

The matrix A A is linear in terms of \(e^{i(\theta _{i}-\theta _{j})}\), therefore Φ is quadratic. Because Φ is invariant under all permutations of the 𝜃’s, it can be written as a quadratic function of the cosines cos(𝜃 i 𝜃 j ). Because Φ vanishes when the 𝜃’s are all equal, it is convenient to write it in terms of the cosines minus 1. It thus takes the form,

$$\begin{array}{@{}rcl@{}} {\Phi} &=& a \sum\limits_{(ij)} \left( 1 - \cos[\theta_i-\theta_j]\right) + b \sum\limits_{(ij)} \left( 1 - \cos[\theta_i-\theta_j]\right)^2 \\ &&+ c\!\!\sum\limits_{(ij)(kl)} \left( 1 - \cos[\theta_i-\theta_j]\right) \left( 1 - \cos[\theta_k-\theta_l]\right) . \end{array} $$
(46)

The last sum is over the 3 partitions of {1,2,3,4} into pairs. When 𝜃 2 = 𝜃 3 = 𝜃 4, A takes the form (16), which obviously has an eigenvalue of unit modulus. Therefore Φ=0 in this case, which shows that a = b=0. To determine the value of c, consider the case that 𝜃 1 = 𝜃 2=0 and 𝜃 3 = 𝜃 4 = π. Then \(A=\frac 12(\hat {n}_{1}+\hat {n}_{2})\cdot \vec {\sigma }\), so A A=1/3, hence Φ=4/9. The last sum in (46) is 8, so c=1/18.

The determinant Φ is thus given by

$$ {\Phi} = \tfrac1{18}\sum\limits_{(ij)(kl)} \left( 1 - \cos[\theta_i-\theta_j]\right) \left( 1 - \cos[\theta_k-\theta_l]\right). $$
(47)

This satisfies Φ≥0, therefore ∥A∥≤1. Each term is non-negative, therefore Φ=0 only if every term vanishes. This occurs only if at least three of the 𝜃’s are equal.

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Foster, B.Z., Jacobson, T. Spin on a 4D Feynman Checkerboard. Int J Theor Phys 56, 129–144 (2017). https://doi.org/10.1007/s10773-016-3170-0

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Keywords

  • Lattice
  • Path integral