The finite-element approach to lattice field theory is both highly accurate
(relative errors $\sim1/N^2$, where $N$ is the number of lattice points) and
exactly unitary (in the sense that canonical commutation relations are exactly
preserved at the lattice sites). In this talk I construct matrix elements for
dynamical variables and for the time evolution operator for the anharmonic
oscillator, for which the continuum Hamiltonian is $H=p^2/2+\lambda q^4/4.$
Construction of such matrix elements does not require solving the implicit
equations of motion. Low order approximations turn out to be extremely
accurate. For example, the matrix element of the time evolution operator in the
harmonic oscillator ground state gives a result for the anharmonic oscillator
ground state energy accurate to better than 1\%, while a two-state
approximation reduces the error to less than 0.1\%.Comment: Contribution to Harmonic Oscillators II, Cocoyoc, Mexico, March
  23-25, 1994, 8 pages, OKHEP-94-01, LaTeX, one uuencoded figur

Milton, Kimball A.

English

arXiv

The finite-element approach to lattice field theory is both highly accurate (relative errors approximately 1/N(exp 2), where N is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is H = p(exp 2)/2 + lambda q(exp 4)/4. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a results for the anharmonic oscillator ground state energy accurate to better than 1 percent, while a two-state approximation reduces the error to less than 0.1 percent

NASA Technical Reports Server

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FINITE-ELEMENT TIME EVOLUTION OPERATOR
FOR THE ANHARMONIC OSCILLATOR
Kimball A. Milton
Department of Physics and Astronomy
The University of Oklahoma, Norman OK 73019, USA
Abstract
The finite-element approach to lattice field theory is both highly accurate (relative errors
1/N 2, where N is the number of lattice points) and exactly unitary (in the sense that
canonical commutation relations are exactly preserved at the lattice sites). In this talk
I construct matrix elements for dynamical variables and for the time evolution operator
for the anharmonic oscillator, for which the continuum Hamiltonian is H = p2/2 + Aq4/4.
Construction of such matrix elements does not require solving the implicit equations of
motion. Low order approximations turn out to be extremely accurate. For exa_mple, the
matrix element of the time evolution operator in the harmonic oscillator ground state gives
a result for the anharmonic oscillator ground state energy accurate to better than 1%, while
a two-state approximation reduces the error to less than 0.1%.
1 Introduction
For over a decade now, the finite-element method has been developed for application to quantum
systems. (For a review of the program see [1].) The essence of the approach is to put the
Heisenberg equations of motion for the quantum system on a Minkowski space-time lattice in
such a way as to preserve exactly the canonical commutation relations at each lattice site. Doing
so corresponds precisely to the classical finite-element prescription of requiring continuity at the
lattice sites while imposing the equations of motion at the Gaussian knots, a prescription chosen
to minimize numerical error. We have applied this technique to examples in quantum mechanics
and to quantum field thhories in two and four space-time dimensions. In particular, recent work
has concentrated on Abelian and non-Abelian gauge theories [2, 3, 4].
Because it is the equations of motion that are discretized, a lattice Lagrangian does not exist
in Minkowski space. This is because the equations of motion are in general nonlocal, involving
fields at all previous (but not later) times. Similarly, a lattice Hamiltonian does not exist, in the
sense of an operator from which the equations of motion can be derived.
However, because the formulation is unitary, a unitary time-evolution operator must exist
which carries fields from one lattice time to the next. For linear finite elements this operator in
quantum mechanics has been explicitly constructed [5]. Construction of this operator requires
solving the equations of motion, which are implicit. Therefore, it is most useful, and perhaps
surprising, that when matrix elements of the time evolution operator are constructed in a harmonic
oscillator basis, they do not require the: solution of the equations of motion [6]. Although these
..... :.. <,'i
223
https://ntrs.nasa.gov/search.jsp?R=19950016566 2020-06-16T08:17:03+00:00Z
general formulas were derived some years ago, it seems they have not been exploited. My purpose
here is to study, in a simple context, the matrix elements of the evolution .operator, and see
how accurately spectral information may be extracted. My goal, of course, is to apply similar
techniques in gauge theories, for example, to study chiral symmetry breaking in QCD.
2 Review of the Finite-Element Method
Let us consider a quantum mechanical system with one degree of freedom governed by the con-
tinuum Hamiltonian
p2
H= -_ + V(q), (1)
from which follow the Heisenberg equations
[_ = -V'(q), gl = P. (2)
These equations are to be solved subject to the initial condition
[q(0),p(0)] = i. (3)
It immediately follows from (2) that the same relation holds at any later time
[q(t),p(t)] = i. (4)
Now suppose we introduce a time lattice by subdividing the interval (0, T) into N subintervals
each of length h. On each subinterval ("finite element") we express the dynamical variables as
rth degree polynomials
p(t) = _ ak(t/h) k, q(t) = _ bk(t/h) k, (5)
k=O k=O
where t is a local variable ranging from 0 to h. We determine the 2(r + 1) operator coeMcients
ak, bk, as follows:
1. On the first finite element let
a0=Po=p(0), bo=qo =q(0)- (6)
2. Impose the equations of motion (2) at r points within the finite element, at aih, i =
1,2, .... r, where 0 < cq < c_2 < "" < c_ < 1. This then gives
p(h) .._ pl = ak, q(k) _ ql = _ bk. (7)
k=0 k=0
3. Proceed to the next finite element: by requiring continuity (but not continuity of derivatives)
at the lattice sites, that is, on the second finite element, set
a0=pl, bo=ql, (8)
and again impose the equations of motion at c_ih, and so on.
224
How are the ai's determined? By requiring preservation of the canonical commutation relations
at each lattice site,
[qa,Pl] = [q0, Po] = i,
one finds
(9)
1
= - (10)
r = 1 (linear finite elements) a 2
1 1
r = '2 (quadratic finite elements) a+ = _ -l- 2--_ (11)
1 v/3 1
r = 3 (cubic finite elements) al,3 = _ T zva'"/;;' a_ = _ (12)
These points are exactly the Gaussian knots, that is, the roots of the rth Legendre polynomial,
Pr(2a- 1)=0. (13)
Amazingly, these are precisely the points at which the numerical error is minimized. It is known
for classical equations that if one uses N rth degree finite elements the relative error goes like
N -2T, while imposing the equations a.t any other points would give errors like N -r.
Let us consider a simple example. The quartic anharmonic oscillator has continuu:n Hamilto-
ni an
l 1A4H
= _p2 + 4 q , (14)
for which the equations of motion are
0=P /b=-Aq 3. (15)
If we use the linear (r = 1) finite-element prescription given above, the corresponding discrete
lattice equations are
ql -- q0 Pl + PO Pl -- Po /_
_ (q, + q0) z. (16)
h 2 ' h 8
(Notice the easily remembered mnemonic for linear finite elements: Derivatives are replacea _5'
forward differences, while undifferentiated operators are replaced by forward averages.) By com-
muting the first of these equations with pl +p0 and the second with ql -4-qo the unitarity condition
(9) follows immediately. These equations are implicit, in the sense that we must solve a nonlinear
equation to find ql and pl in terms of q0 and p0- Although such a solution can be given, let use
make a simple approximation, by expanding the dynamical operators at time 1 in powers of h,
with operator coefficients at time 0. Those coefficients are determined by (16), and a very simple
calculation yields
_ 2 3
qa = qo + hpo - -_h qo + " " ,
pl = po - )_hqZo - ;)_h2qopoqo + .... (17)
We can define Fock space creation and annihilation operators in terms of the initial-time operators
(a+a t) (a-a t )
q0= ¢5 ' ;0 - ' (18)
225
which satisfy
[a,a t] = 1. (19)
Here we have introduced an arbitrary variational parameter 7. The Fock-space states (harmonic
oscillator states) are created and destroyed by these operators:
I_) - (_10), (20)
which states are not energy eigenstates of the anharmonic oscillator. We can now take matrix
elements in these states of the dynamical operators at lattice site 1, using (17):
(llPll0) _ (llpolO)(1 + i3hA7 4- 3h2172 + ...)i _ 4
1 2 2
(lJp0]0)(1 + iwh- -_ h +...), (21)
and
+i h
.2
(llq,10) (llq010)(1 4
-
(llq010)(1 + iwh- laj2h2 + ...), (22)
2
where we have assumed approximately exponential dependence on the energy difference w. Equat-
ing the coefficients of the terms through order h 2 constitutes four equations in two unknowns.
These equations are consistent and yield
3 4 1 (23)
= = 7-7,
so the energy difference between the ground state and the first excited state is approximately
w = A _ 1.145t '/3 (24)
t
which is only 5% higher than the exact result Em= 1.08845)0/3. A similar calculation using
quadratic finite elements (r = 2) reduces the error to 0.5%.
3 The Time-Evolution Operator
Because the canonical commutation relations are preserved at each lattice site, we know that there
is a unitary time evolution operator that carries dynamical variable forward in time:
q,_+, = Uq,_U t, Pn+l = UpnUf, (25)
For the system described by the continuum Hamiltonian (1) in the linear finite-element scheme,
we have found [5] the following formula for U:
U : eihp_/4eihA(q'_)eihp_n/4, (26)
226
where
2 [x- g-l(4x/h2)] 2 + V(g-l(4x/h2)), (27)A(x) = -£7
4
g(x) = Ux + V'(x). (28)
The implicit nature of t;he finite-element prescription is evident in the appearance of the inverse
of the function g.
Given the time evolution operator, a lattice Hamiltonian may be defined by U = exp(ih_).
For linear finite elements 7/ differs from the continuum Hamiltonian by terms of order h 2. For
example,
V= _rn q • 7-l- rr_htan-' _p +_m q ], (29)
(30)
(31)
If one uses quadratic finite elements 7-/differs from the continuum Hamiltonian by terms of order
h 4, etc.
4 Matrix Elements of Dynamical Variables
Remarkably, it is not necessary to solve the equations of motion to compute matrix elements of
the dynamical variable. Introduce creation and annihilation operators as in (18). Then, in terms
of harmonic oscillator states (20) the f011owing formula is easily derived [6] for a general matrix
element of ql"
(m]qlln> - x_2-(v/_£_,m_, + v/-_£_,n_l)
c-iO(m-n) /_+Rv/_r2_+mn!m ! _ dz ze-g2(z)/4m g'(z)Hn(g(z)/2R)Hm(g(z)/2R), (32)
where g is given by (28), H,_(x) is the nth Hermite polynomial, and we have introduced the
abbreviations
_ __ 1 e_iO 2"1 iR 2_ 472 +-- _ +-- (33)
h 4 h272' Rh 2 Rh_"
For the example of the harmonic oscillator, this formula gives for the ground state-first excited
state energy difference w = (2/h) tan-l(h/2), consistent with (29), while for.the anharmonic
oscillator if we expand in h we obtain precisely the expansion (22).
227
5 Matrix Elements of the Time Evolution Operator
A similar formula can be derived for the harmonic oscillator matrix elements of the time evolution
operator. (There is an error in the formula printed in [6].)
1 1 e_i(_+,_+x)0
ro " _ ' 3 TI 2 2 2 --i0
× dzg'(z)H,_(g(z)/2R)H,_(g(z)lUR)c bhv(')+'_ _ (_) /8-h g(z)_ /s?R], (34)
oo
which again is expressed in terms of g not g-1.
Ebr the harmonic oscillator, where V = q2/2, (34) gives for the ground-state energy
1 I It
(olulo) -- d = tan- (35)
which follows from (29). For the anharmonic oscillator, V = .\qt/4, again, for a first look, we
expand in powers of h, with the result, for the harmonic oscillator ground state,
i 3i . 4 _ h2 ((OIU]O) = l+]t _72 +T6 A'7 ) + ---
/
.1_ 2 2
,_ l + iwoh - -_woh +...,
3 9 105 . 2 s _
32.),4 + _'_-,\_t'z ,_]-_ A 3/ )
(36)
which is also derivable from (31). Equating powers of h gives _2s two equations, which are to be
solved first for the dimensionless number A@ = a. Once tile number a is determined, the value
of wo is expressed as
w0 = A'/a/'(a), f(a)- 4c_,/3 1 + _c_ . (37)
For a first estimate, we use tim "principle of minimum sensilivity", that is, use the stationary
value of a,
2
f'(_) = 0 _ a,= 3 _ f((Q = 0.4293, (3S_
which is about 2% higher than the exact value of 0.42081 [7]. In fact, when we solve (36) for a
we find a complex value
1 i
a = _ + 2---_ =_ f(a) = 0.4178 T 0.0077i. (39)
The imaginary part is small, and the real part is only 0.7% low. The failure of (39) to be reM does
not indicate any breakdown of unitarity, but only that the one state approximation is not exact.
We do much better by making a two-state approximation, where we must diagonalize the 2 x 2
matrix ( 0o /40tU2o "
We then find the following relation between w0.2 and a = A_,6:
)0/3
_o0,2 = 1--6-a-1/3[12 + 21a _ 2_/_(8 + 16a + 33c_2)_/2], (41)
which, for the - sign, is plotted in Fig. 1:
228
0.70
0.60
0.50
0.40 ..... L ..... J I .... L........
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 1 Ground-state energy for the anharmonic oscillator as a function of _ = A76,
in the second approxi}nation. Here w0 = _/3/(c_), f(c_) given by (41).
This graph shows that the ground-state energy is very insensitive to the value of c_.
principle of minimum sensitivity give spectacular agreement with the exact result,
w0 = 0.421235X 1/3,
The
(42)
being only 0.1°£ high, while it gives a good value for the third state, w2 = 2.992_ 1/3. Solving for
c_ from the eigenvalues of (40) gives even better results:
wo = X'/a(0.42054 + 2 x 10-6i), w2 = _'/3(2.94328 -.022029i), (43)
where the ground state energy is now low by 0.06%, the imaginary part being negligible.
229
6 Conclusions
The simple calculations given here for the quantum-mechanical anharmonic oscillator are the be-
ginning of a program to develop use of lattice Hamiltonian techniques to explore gauge theories
in the finite-element context. The astute reader will note that the numerical results presented in
Sec. 5 also hold in the continuum, by virtue of (31). It is in two or more space-time dimensions
that the essential nature of the lattice in such calculations comes into play [3, 4, 8]. The high
accuracy contrasted with the simplicity of the approach leads us to expect that we can extract spec-
tral information, anomalies, and symmetry breaking from an examination of the time-evolution
operator.
Acknowledgments
I thank the US Department of Energy f()r partial financial support of this research. I am grateful
to Carl Bender, Pankaj Jain, and Dean Miller for useful conversations.
References
[1] C. M. Bender, L. R. Mead, and K. A. Milton, "Discrete Time Quantum Mechanics," preprint
OKHEP-93-07, hep-ph/9305246, to be published in the Journal of Computational and Ap-
plied Mathematics.
[2] K. A. Milton, in Proceedings of the XXVth International Conference on High-Energy Physics,
Singapore, 1990, edited by K. K. Phua and Y. Yamaguchi (World Scientific, Singapore, 1991),
p. 432.
[3] D. Miller, K. A. Milton, and S. Siegemund-Broka, Phys. Rev. D 46,806 (1993).
[4] D. Miller, K. A. Milton, and S. Siegemund-Broka, "Finite-Element Quantum Electrodynam-
ics. II. Lattice Propagators, Current Commutators. and Axial-Vector Anomalies," preprint
OKHEP-93-11, hep-pti/9401205, submitted to Phys] Rev. D. [
[5] C. M. Bender, K. A. Milton, D. H. Sharp, L. M. Simmons, Jr., and R. Stong, Phys. Rev. D
32, 1476 (1985).
[6] C. M. Bender, L. M. Simmons, Jr., and R. Stong, Phys. Rev. D 33, 2362 (1986).
[7] J. F. Barnes, H. J. Brascamp, and E. H. Lieb, in Studies in Mathematical Physics, edited
by E. H. Lieb, B. Simon, and A. S. Wightman (Princeton University Press, Princeton, N J,
1976).
[8] C. M. Bender and K. A. Milton, Phys. Rev. D 34, 3149 (1986).
230


Finite-Element Time Evolution Operator for the Anharmonic Oscillator

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