New PDF release: A tutorial on Principal Component Analysis

By Shlens J.

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In general, Hamiltonian H(q(t), p(t)) is given by the quadratic form in the canonically conjugate momentum pr (t). 40) r=1 where the kernel {Dr,s (q(t))}fr,s=1 of the quadratic part is assumed to be real, symmetric and positive definite. 40), we have 36 1. Path Integral Representation of Quantum Mechanics f pr (t) r=1 d qr (t) − H(q(t), p(t)) dt f =− 1 pr (t)Dr,s (q(t))ps (t) 2 r,s=1 f pr (t) + r=1 d qr (t) − Cr (q(t)) − V (q(t)) . 39) as a quasi-Gaussian integral. We have D[p(t)] exp i f tb dt ta pr (t) r=1 d qr (t) − H(q(t), p(t)) dt = [DetD(q; tb , ta )]−1/2    stationary value of         i tb  f d  pr (t) qr (t) − H(q(t), p(t))  × exp  dt .

22) n=1 which is t-dependent. We note that the path integral with respect to ψn (t, x) D[ψ] 62 2. 21) is no longer constrained. Z[J] (W [J]) is the generating functional of (the connected parts of) the vacuum expectation values of the time-ordered ˆ (xk ). 23W) defines the connected parts. 12) is given by a quadratic form in πn (x), namely H[ψ(t), ∇ψ(t), π(t)] ≡ 1 d3 x d3 yπn (t, x)Dn,m (x, y; ψ)πm (t, y) 2 + a term linear in π + a term independent of π . 21) as a quasi-Gaussian integral, obtaining the determinant factor (DetDn,m (x, y; ψ))−1/2 in the integrand of D[ψ].

3 Weyl Correspondence In the discussions so far, we have utterly evaded the problem of operator ordering with the notion of a “well-ordered” operator in order to provide the essence of path integral quantization. In this section, we shall discuss the problem of operator ordering squarely with the notion of the Weyl correspondence. In Sect. 1, with the notion of the Weyl correspondence, we discuss the correspondence of analytical mechanics and quantum mechanics. In Sect. 2, using the result of the previous section, we reconsider the path integral formula in the Cartesian coordinate system and derive the mid-point rule as a natural consequence of the Weyl correspondence.

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A tutorial on Principal Component Analysis by Shlens J.


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