Abstract
The instant form and the front form of relativistic dynamics proposed by Dirac in 1949 can be linked by an interpolation angle parameter $\delta$ spanning between the instant form dynamics (IFD) at $\delta =0$ and the front form dynamics which is now known as the light-front dynamics (LFD) at $\delta =\pi/4$. We present the formal derivation of the interpolating quantum electrodynamics (QED) in the canonical field theory approach and discuss the constraint fermion degree of freedom which appears uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of relativistic fermion degrees of freedom. As discussed in our previous work, the helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. To exemplify the characteristic difference of the fermion propagator between IFD and LFD, we compute the helicity amplitudes of typical QED processes such as $e^+ e^- \to \gamma \gamma$ and $e \gamma \to e \gamma$ and present the whole landscape of the scattering amplitudes in terms of the frame dependence or the scattering angle dependence with respect to the interpolating angle dependence. Our analysis clarifies any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame approach and the LFD.
Highlights
For the study of relativistic particle systems, Dirac [1]proposed three different forms of the relativisticHfroanmtil(txoþni1⁄4anðxd0yþnaxm3Þic=spiffi2nffi1949: i.e., the 1⁄4 0), and point instant (x0 1⁄4 0), forms
The instant form dynamics (IFD) of quantum field theories is based on the usual equal time t 1⁄4 x0 quantization, which provides a traditional approach evolved from the nonrelativistic dynamics
The equal light-front time τ ≡ ðt þ z=cÞ= 2 1⁄4 xþ quantization yields the front form dynamics, nowadays more commonly called light-front dynamics (LFD), which provides an innovative approach to the study of relativistic dynamics
Summary
1949: i.e., the 1⁄4 0), and point instant (x0 1⁄4 0), (xμxμ 1⁄4 a2 > 0; x0 > 0) forms. To trace the forms of relativistic quantum field theory between IFD and LFD, we take the following convention of the space-time coordinates to define the interpolation angle [9,10,11,12,13]: xþ 1⁄4 cos δ x−ˆ sin δ sin δ x0 − cos δ x3 ; ð1Þ in which the interpolation angle is allowed to run from 0 through 45°, 0 ≤ δ ≤ π4. The annihilation of fermion and antifermion into two scalar particles, we show the characteristic behavior of the amplitudes as the form interpolates between IFD and LFD and examine the angular momentum conservation. In Appendix D, we discuss the noncollinear case of the annihilation of a fermion and antifermion into two scalar particles and provide the relation between the center of mass scattering angle and the apparent scattering angle in a boosted frame and correspond the angular distributions for. The angular distribution and the frame dependence of the eþe− → γγ helicity amplitudes are summarized in Appendices E and F, respectively
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