Solution of Coulomb Path Integral in Momentum Space

Home , Hagen Kleinert

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(4) r −2r2 − 2 − s ( ) Y With this, the total euclidean action in the path integral (3) is

S 1 p2 2 1 1 [p, x,h]= ds ip˙ x + x2 E + f 2 fα . (5) Ae · 2 2 − 2r2 − r Z0 " # The path integrals over f and x in (15) are Gaussian and can be done, in this order, yielding a new euclidean action

S 2 1 4p˙ 2 e[p]= ds α , (6) 2 2 A 2 0 p2 − Z "( + pE) # where we have introduced pE = √ 2E, assuming E to be negative. The positive regime can always be obtained by analytic continuation. Now, a stereographic− projection

p p2 2 π 2pE pE 2 2 , π4 2 − 2 (7) ≡ p + pE ≡ p + pE transforms (8) to the form

S 1 1 2 [~π]= ds ~π˙ α2 , (8) Ae 2 p2 − Z0 E π 2 where ~π denotes the four-dimensional unit vectors ( , π4). This describes a point particle of pseudomass µ = 1/pE moving on a four-dimensional unit sphere. The pseudotime evolution amplitude of this system is

4 − 2 π −A (~π S ~π 0) = e SpE D e e[~π]. (9) b | a (2π)3/2p3 Z E There is an exponential prefactor arising from the transformation of the functional measure in (15) to the unit sphere. Let us see how this comes about. When integrating out the spatial fluctuations in going from (5) to (8), the canonical 3 3 3 3 3/2 2 2 measure in each time slice d p d x/(2π) becomes d p 8/(2π) (p + pE). From the stereographic projection (7) we see 4 3/2 3 4 that this is equal to d ~π/(2π) pE, where d ~π denotes the product of integrals over the solid angle on the surface of the unit sphere in four dimensions, with the integral d4~π yielding the total surface 2π2. From Chapter 10 in the textbook [1] we know that in a curved space, the time sliced measure of path integration is given by the product of invariant R integrals dq g(q) in each time slice, multiplied by an effective action contribution exp( eff ) = exp( dsR/¯ 6µ), where R¯ is the scalar curvature. For a sphere of radius r in D dimensions, R¯ = (D 1)(−AD 2)/r2, implying here R p 2 − − R exp( eff ) = exp( ds 1/µ) = exp( ds pE). Thus, when transforming the time-sliced measure in the original path integral−A (3) to the time-sliced measure on the sphere in (9) which contains the effective action, the exponent is modified accordingly. R R A complete set of orthonormal hyperspherical functions on this sphere may be denoted by Ynlm(~π), where n,l,m are the quantum numbers of the hydrogen atom with the well-known ranges (n = 1, 2, 3,..., l = 0,...,n 1, m = l,...,l). They can be expressed in terms of the three-dimensional representation Dj (u) of the SU(2)− matrices − m1m2 u = ~π~σ with the Pauli matrices ~σ (1, σ1, σ2, σ3) as ≡ 2j +1 Y (~π)= (j, m ; j, m l,m) Dj (u). (10) 2j+1,l,m 2π2 1 2| m1m2 r m1,m2=−j,...,j X The orthonormality and completeness relations are

∗ ′ (4) ′ d~πY ′ ′ ′ (~π)Y (~π)= δ ′ δ ′ δ ′ , Y (~π )Y (~π)= δ (~π ~π). (11) n l m nlm nn ll mm nlm nlm − Z n,l,mX where the δ-function satisﬁes d~πδ(4)(~π′ ~π) = 1. When restricting the complete sum to l and m only we obtain the four-dimensional analog of the Legendre− polynomial: R

2 n2 sin nϑ Y (~π′)Y (~π)= P (cos ϑ), P (cos ϑ)= , (12) nlm nlm 2π2 n n n sin ϑ Xl,m where ϑ is the angle between the four-vectors ~πb and ~πa:

2 2 2 2 2 (pb pE)(pa pE)+4pEpbpa cos ϑ = ~πb~πa = − 2 2− 2 2 (13) (pb + pE)(pa + pE) The path integral for a particle on the surface of a sphere was solved in [1]. The solution of (9) reads

∞ n2 S (~π S ~π 0) = (2π)3/2p3 P (cos ϑ) exp p2 n2 + α2 . (14) b | a E 2π2 n − E 2 n=1 X For the path integral itself in (9), the exponential contains the eigenvalue of the squared angular-momentum operator Lˆ2/2µ which in D dimensions is l(l + D 2)/2µ, l = 0, 1, 2,... . In our system with D = 4, l = n 1, these − − 2 2− − eigenvalues are n2 1, leading to an exponential e pE (n 1)S. Together with the exponential prefactor in (9), this leads to the exponential− in (14). The integral over S in (15) with (15) can now be done yielding the amplitude at zero ﬁxed pseudoenergy

∞ n2 2 (~π ~π ) = (2π)3/2p3 P (cos ϑ) . (15) b| a 0 − E 2π2 n 2En2 + α2 n=1 X This has poles displaying the hydrogen spectrum at energies: 1 E = , n =1, 2, 3,.... (16) n −2n2

3. Consider the following generalization of the ﬁnal action (8):

S 2 1 1 4p˙ 2 e[p]= ds α h , (17) A 2 h p2 2 2 − Z0 " ( + pE) # This action is invariant under reparametrizations s s′ if simultaneously h hds/ds′. The path integral with the action (8) in the exponent may thus be rewritten→ as a path integral with the→ gauge-invariant action (17) and an additional path integral dh Φ[h] with an arbitrary gauge-ﬁxing functional Φ[h]. Going back to a real-pseudotime parameter s = iτ, the action corresponding to (17) which describes the dynamics of the point particle in the Coulomb potential reads R

1 τb 1 4p˙ 2 [p]= dτ + α2h , (18) 2 2 2 A 2 τa h p Z " ( + pE) # At the extremum in h, this action reduces to

τb p˙ 2 [p]=2α dτ . (19) 2 2 2 A a p Zτ s ( + pE) This is the manifestly reparametrization invariant form of an action in a curved space with a metric gµν = µν 2 2 2 δ / p + pE . In fact, this action coincides with the classical eikonal in momentum space:

pb S(p , p ; E)= dτ px˙ . (20) b a − Zpa 2 2 Observing that the central attractive force makes p˙ point in the direction x, and inserting r = α(p + pE)/2, we ﬁnd precisely the action (19). In fact, the canonical quantization of a system− with the action (19) a la Dirac leads directly to a path integral with action (18) [7].

3 The eikonal (20), and thus the action (19), determines the classical orbits via the ﬁrst extremal principle of theo- retical mechanics found in 1744 by Maupertius.

4. Since the Coulomb path integral in momentum space is equivalent to that of a point particle on a sphere, we can use it to pass an experimental judgement on the possible presence of an extra R-term in the Hamiltonian operator of the Schrödinger equation in curved space which could be caused by various historic choices of the measure of path integration in the literatue [2–4]. In the exponent of (14), an extra term c µ¯h2R/2 in the Hamilton operator in addition to the Laplace-Beltrami term µ¯h2∆/2 would appear as an extra constant× 3c added to n2. The hydrogen − 2 spectrum would then have the energies En = 1/2(n +3c). The only theoretically proposed candidates for c are 1/24, 1/12, and 1/8 [2–4]. The resulting strong− distortions of the hydrogen spectrum would certainly have been noticed experimentally a long time ago, apart from the fact that they would contradict Schrödinger theory in x-space whose spectrum (16) as the first triumph of quantum theory in atomic physics. On fundamental level, the present discussion confirms the validity of the nonholonomic mapping principle [1,8] which predicted the extra factor exp( eff ) = exp( dsR/¯ 6µ) in the measure of the path integral in curved space, without which the correct spectrum in−A curved momentum space would not have been obtained–the energy would have had the unphysical form α/2(n2 1) with a singularityR at n = 1! − −

ACKNOWLEDGMENT This work was supported by Deutsche Forschungsgemeinschaft under contract Kl-256.

4 [1] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientiﬁc, Singapore 1995, Second extended edition, pp. 1–850. (http://www.physik.fu-berlin.de/˜kleinert/kleiner re.html#b5). [2] B.S. DeWitt, Rev. Mod. Phys. 29, 337 (1957). [3] K.S. Cheng, J. Math. Phys. 13, 1723 (1972). [4] For older treatments see the review M.S. Marinov, Phys. Rep. 60, 1 (1980). For a more recent and detailed discussion of the problem see Chapters 10 and 11 in the textbook [1]. [5] See Chapter 12 in [1]. [6] K. Fujikawa, Prog.Theor.Phys. 96 (1996) 863-868 (hep-th/9609029); (hep-th/9608052); (hep-th/9608052). [7] See the discussion in [6] and Chapter 19 of the textbook [1]. [8] H. Kleinert, Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion, FU-Berlin preprint 1997 (http://www.physik.fu-berlin.de/˜kleinert/kleiner re3#258), (APS eprint aps1997sep03 002), and Act. Phys. Pol. B 29, 1033 (1998) (gr-qc/9801003)