CC BY
1MATRIX KERNEL REPRESENTATIONS AND APPLICATIONS
JOHN GOUGH
Department of Physics, Aberystwyth University, Wales, United Kingdom.
Abstract: We set up a notation which allows us to represent operators on a tensor product of Hilbert spaces in a convenient form. We apply this to system-bath models and show how it gives a transparent derivation of the Feynman-Vernon influence functional theory.
Key words: Representation of operator algebras, Feynman path integrals, Open quantum
systems, functional integration, Feynman-Vernon influence functionals.
1. Introduction
The aim of this note is to introduce some representations that simplify the description of a bipartite quantum system, that is, a system comprised of two separate subsystems A and B. The representation - which we refer to as the matrix kernel representation - is, itself, rather trivial, however, it does important work in helping to manipulate expressions. We demonstrate this by rederiving some well-known results where path integrals have been applied to open quantum systems. Feynman's path integral calculus [1] offers a very effective and versatile description of quantum mechanics which has found widespread applications [2]. However, it does sometimes get unwieldly, especially when dealing with bipartite models of a system coupled to a bath. The representation here allows us to separate out the path integral elements from the bipartite elements of the calculation. We will establish the Feynman-Vernon influence theory [3,4] as an application.
2. Matrix Kernel Represntation Let A and B be quantum systems with (separable) Hilbert spaces fyAand , respectively. The algebra of bounded operators on these spaces are denoted as and Ъ($в). Given a complete
orthonormal basis (|n)} for we define the matrix kernel representation of the algebra onto as follows: for each pair of indices (n,m) and each operator К E
ясьащв), let Mnm(x) be the unique operator in Ъ($в~) for which
(гр1мпт(х)\ф') = (пЗ^Щт®^'),
(1)
for all E
The following properties are immediate:
a) Mnm(A®B) = (пЩт) В;
b) Мпт(хУ = M mn(X*);
c) Mnm(X Y) = ЪкМпк{х)Мкт{?).
In the special case = L2(K,dx) where we have the resolution of identity Цх)(х1 dx = 1 we may define the kernel as the measurable mapping
M: Ж2Х Ъ($А®Ъв) ^ \(x,x',X) ^ MXXI(X).
Here c) will be replaced by c') Mxx,,(X Y) = f MxX'(x) Mxx„(?) dx'.
In principle, the representation is trivially expressing bipartite operators as matrices/kernels over the first factor A with entries in the B(i)B). However, we will see that the representation is valuable in describing and manipulating expressions for bipartite quantum systems.
3. Feynman- Vernon Influence Functionals We recall the path integral formulation for a quantum mechanical system A with canonical
observables q and p where the Hamiltonian splits as H = T + V with kinetic part T = — p2 and
potential V = V(q). By virtue of the Trotter formula, we have
(e-1 NTe-iNv) ^ e-itH
(2)
as N ^ ro. For a rigorous formulation, see [5]. Inserting a resolution of identity ¡lx)(xl dx = I between each of the N factors leads to
N-1
'mN\N/2 \it sr \m/xi+1 - x^2
(3)
which we write in the shorthand
(4f¿)
(qfle-itÑlqi) = ¡ Vq
(qi,0)
(4)
where SA[q] = [j q(T)2 — V(q(r))] dr is the classical action functional associated to a path q =
iq(r): 0 <t < t] and the integral is over all such paths satisfying the boundary conditions q(0) = qt and q(T) = qf.
Now suppose that our system, A, is coupled to a heat bath, B, and that the total interaction takes the form
HaB = (T + V)®!b + Ia®Hb + f(q)®C
(5)
where the system Hamiltonian T + V is as above.
Proposition In the matrix kernel representation, we have
е-иЙАВ) = ^$ЪЧ eiSAlq] U[q]
1 (6)
where Ut[q] is the family of unitary operators on Ъ(i)B), parameterised by time t and dependent on
path q, given by
Ut[q] = f e-if0>["B+f(q(T))'^] dT.
(7)
Demonstration The starting point is the Trotter formula
/ t~ - t ~ - - —- \N (e-i1jT®iBe-i1j[v®iB+iA®HB+f(q)®c]\ ^ e-itHAB
and this time we insert multiple resolutions of identity flx)(xl®fB dx = lA ®lB. This leads to
(<P®xPle-itRABl<p'®xp') = J dqfdqi <p(qr)* <p(qt)
The desired form is now readily obtained.
The Feynman-Vernon Theory deals with the evolution of the reduced density matrix os the system. We take the initial state to be the factor state pAB(0) = pA(0)® pB(0) and the reduced state of the system at time t is the partial trace
pA(t) = trB[e-itR™ pAB(0) e+itR™}.
(8)
Its kernel can be represented as
(qflPA(t)lq/) = trB [m %fl(e-it»AB pAB(0) e+itB">)} = f dqidqi' trB [M\(e-it»™ ) Mqiqi,(pA(0)® pB(0)) M(e+^*)}
= f dqidqi' trB [M\(e-™AB ) pB(0) m%fl (e+^*)} (qilpAmqi').
Here we used properties b) and c') of the matrix kernel representation. This implies that we may propagate the kernel of the system density matrix forward in time according to
MPa^M = f dqidqi' 0(qf,q'f,tlqi,q'i,0) (qilpA(0)lqi')
where
3(qr, q'f, tk, q[, 0) = trB [m\(e-it»AB ) pB(0) Mq^ (e+^B)}
= f^<Dq f^Vq' eW]-sM']) T[q,q']
and the Feynman-Vernon influence functional is given by
(9)
(10)
T[q,q'] = trB{ Ut[q] pB(0) Ut[q'V }.
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(11)
One immediately sees the normalization property T[q, q] = 1, the sesquilinearity property T[q> q']* = ?[q', q], and the positivity condition
f^Vq f^V*q' c[q]* T[q,q'] c[q'] > 0.
4. Quantum Stochastic Processes It is convenient to transfer to the Heisenberg picture. For X a system operator, one sets
Xt = e+itnAB (X®IB) e-it»AB.
(12)
Similar to the Kolmogorov construction through finite-dimensional distributions, we say that we have specified the quantum stochastic once we have specified all moments [6]
trAB^ABm^ ... X^}
(13)
For arbitrary n > 0, times 11, ...,tn and (bounded) system operators X(1\ ...,X(n). As an illustrative example, let us take t1 > t2. Here we will compute
trAB{pAB(0)X(1 X™} = f dqidq'' Wa^M') x trB{ pB(0) M™. (e+^B (X(1)®IB) e-i(ti-t2)nM(X(2)®IB) e-^»™)}.
Here the Mqi'.(...) term must be broken up as the product of five separate terms
Mqi'(...) = | dqadqbdqcdqa X M%a ( e +i ^ab) (qa\X(1)\qb)Mqqc ( e-i(ti-t2)iJAB)(qc\x(2)\qd)Mqdq. (e-it2R™)
(qa.ti) (qb.ti-t2) (qdte) '
= I dqadqbdqcdqd | V*q | Vq' | Vq'
(qi.O) (qc0) (qi,0)
X e-istl [q]+iStl-t2 [q']+iSt2 [q'']
X trB{pB(0) Uti[qY Uti-t2[q'] Ut2[q"] }. Note that this involves a bath average over three unitaries, each with its own different path dependence. The model will, of course, be generally non-Markovian in nature.
5. Conclusion
We have introduced a shorthand notation to tackle the bipartite nature of an open quantum system (system + bath). In [7] we have already shown how Feynman diagrams can dealt with more systematically using a notational shorthand. The matrix kernel used here has the merit of simplifying the two separate components in path integral calculations. Once one becomes accustomed to this representation, then further redundancy conventions - similar to Einstein's - can be made to make expressions more precise.
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LIST OF USED SOURCES:
1. Feynman R.P., Hibbs, A.R. (1965), Quantum Mechanics and path integrals, McGraw-Hill, New York.
2. Kleinert, H. (2009) Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th Edition, World Scientific, New Jersey.
3. Feynman, R.P., Vernon, F.L. (1963) Ann. Phys. 24, 118-173pp.
4. Caldeira A.O., Leggett, A.J. (1983), Physica A 121, 587-616 pp.
5. Smolyanov, O.G, Tokarev, A.G., Truman, A. (2002) Hamiltonian Feynman path integrals via the Chernoff formula, Journ. Math. Phys., 43 (10), 5161-5171 pp.
6. Accardi, L., Frigerio, A., Lewis, J.T., Quantum Stochastic Processes (1982) PRIMS, 18, No. 1, 97-133 pp.
7. Gough, J., Kupsch, J. (2018) Quantum Fields and Processes: A Combinatorial Approach, Cambridge studies in advanced mathematics, 171, Cambridge University Press, Cambridge.
