By Abhay Ashtekar

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M. Gitman. Quantization of the relativistic particle. Class. Quant. , 17:L133, 2000. hep-th/0005249.

That is, the H function associated with Lagrangians of the form (129) vanishes identically. So what governs the evolution of such systems? If we differentiate (130) with respect to Q˙ B , we obtain Q˙ A ∂2L = 0. ∂ Q˙ A Q˙ B 29 (132) This means the mass matrix associated with our Lagrangian has a zero eigenvector and is hence non-invertible. 2 that such theories necessarily involve constraints. So, the total Hamiltonian for this theory must be a linear combination of constraints: HT = uI φI ∼ 0.

2 Dirac quantization Let us now pursue the quantization of our system. First we tackle the Dirac programme. We must first choose a representation of our Hilbert space. A standard selection is the space of functions of the coordinates x. Let a vector in the space be denoted by Ψ(x). Now, we need representations of the operators x ˆ and pˆ that satisfy the commutation relation [ˆ xα , pˆβ ] = i {xα , pβ }X=Xˆ = i δβα . (149) Keeping the notion of general covariance in mind, we choose x ˆα Ψ(x) = xα Ψ(x), pˆα Ψ(x) = −i ∇α Ψ(x).