Adaptive Perturbation Theory: Quantum Mechanics and Field Theory

0.42 MB·English

by  Marvin Weinstein

SLAC-PUB-11532
Adaptive Perturbation Theory: Quantum Mechanics and Field Theory
6 Marvin Weinsteina∗
0
0 aStanford Linear Accelerator Center, Stanford University
2
Stanford, California 94309

n USA
a
J
Adaptiveperturbationisanewmethodforperturbativelycomputingtheeigenvaluesandeigenstatesofquantum

4 mechanical Hamiltonians that are widely believed not to be solvable by such methods[1]. The novel feature of
adaptive perturbation theory is that it decomposes a given Hamiltonian, H, into an unperturbed part and a
2
perturbation in a way which extracts the leading non-perturbativebehavior of the problem exactly. In this talk
v
I will introduce the method in the context of the pure anharmonic oscillator and then apply it to the case of
0
tunneling between symmetric minima. After that, I will show how this method can be applied to field theory.
6
1 In that discussion I will show how one can non-perturbatively extract the structure of mass, wavefunction and
0 coupling constant renormalization.
1
5
0
/
h 1. Introduction can avoid such complication by exhibiting a con-
t vergent perturbation expansion for each eigen-
-
p To avoid controversy I have titled this talk state of the Hamiltonian. An expansion that,
e Adaptive Perturbation Theory, but I could have
moveover, captures all non-perturbative features
h equally well called it Non-Perturbative Perturba-
of the problem in zeroth order. The case of the
:
v tion Theory (exceptthatIwastoldpeople would
anharmonic oscillator with a negative mass term
i dismiss it as crazy). Actually I hope you will
X will allow me to extend the simplest adaptive
agreebytheendofthetalkthatsuchatitlewould
techniquetohandlethecaseoftunnelingbetween
r
a have been eminently defensible. symmetric minima. In this talk I will not discuss
the case of tunneling between very asymmetric
2. Two Topics
minima,howeverthistopiciscoveredinthepaper
to appear on this work.
This talk is divided into two parts. First I in-
troducethegeneralideaswithintheframeworkof
3. The Harmonic Oscillator
ordinary quantum mechanics. Next, I show how
to extend these ideas to the case of scalar field
Let me begin by reminding you of a some sim-
theory in any number of space-time dimensions.
ple facts pertaining to the ordinary harmonic os-
Iwillbeginthequantummechanicsdiscussionby
cillator. The Hamiltonian of the ordinary har-
talkingaboutthe pureanharmonicoscillatorand
monic oscillator is:
then extend this discussion to cover the anhar-
monicoscillatorwitheitherapositiveornegative H = 1 p2+ mω2 x2, where [p,x]= i. (1)
mass term. 2m 2 −
The theory of an anharmonic oscillator with
Next,letmeintroduceγ-dependentoperatorsA†
a positive mass term has a rich literature. The γ
and A by
γ
various authors usually attempt to resum the
ordinary perturbation expansion to obtain the 1
x = A† +A ,
non-perturbative behavior. I will show that one √2γ γ γ
∗This work was supported by the U. S. DOE, Contract p = i γ(cid:0)A† A (cid:1) . (2)
No.DE-AC03-76SF00515. 2 γ − γ
r
(cid:0) (cid:1)
1
2
Substituting this into Eq. 1 yields (5)
H = γ A†2 A 2+2A†A +1 whereIhavedefinedtheγ-dependentnumberop-
4m − γ − γ γ γ
erator
+ mω2(cid:16) A†2+A 2+2A†A +1(cid:17). (3)
4γ γ γ γ γ N =A†A . (6)
γ γ γ
(cid:16) (cid:17)
It iscustomaryto chooseγ =mω soasto cancel
Given this expression I define the γ-dependent
the terms A†γ2 and Aγ2 and render the Hamil- unperturbed Hamiltonian, H0 and the perturba-
tonian diagonal in the number operator. This tion V by
immediately tells us that the eigenstates of the
Hamiltonian are the eigenstates of the number H =H0(γ)+V(γ),
operatorand that the energiesofthese states are γ λ λN (N 1)
γ γ
H (γ)= + (2N +1)+ − ,
given by En =(n+1/2)ω. 0 4 4γ2 γ 4γ2
(cid:18) (cid:19)
4. The Anharmonic Oscillator V(γ)= γ λ (A†2+A 2)
4 − 2γ2 γ γ
(cid:18) (cid:19)
Inthecaseofthegeneralanharmonicoscillator + λ 1 (A†4+A 4)+ 2(A†3A +A†A 3) .
the Hamiltonian is 4γ2 6 γ γ 3 γ γ γ γ
(cid:18) (cid:19)
H = 1p2+ mω2 x2+ 1λx4. (4) (7)
2 2 6
Now, since I have a different perturbation theory
The customary way to deal with this problem is
definedforeachchoiceofγ,I needaprinciple for
twointroduceannihilationandcreationoperators
fixing γ. This is where the adaptive comes into
chosentodiagonalizethefirsttwotermsandthen
adaptive perturbation theory. The key notion is
construct a perturbation theory based upon ex-
that I will use a simple variational calculation,
panding in the coupling λ. This approach has a
adapted to the eigenvalue to be calculated,in or-
well known problem; namely, it is known to di-
der to pick that value of γ that gives the most
verge, no matter how small λ, due to n! growth
convergent perturbation expansion.
oftheperturbationseries. Iwillnowshowhowto
To set up this variationalcalculationI firstde-
avoidthis problemandproducearapidlyconver-
fine a γ-dependent family of Fock-states, N .
gent adaptive perturbation theory expansion for | γi
The γ-dependent vacuum state, 0 , is defined
each eigenstate and eigenvalue of H. To empha- | γi
by the condition
sizethatthisisnotanexpansioninλIwillbegin
by specializing to the case m=0.
A 0 =0, (8)
γ γ
Setting m=0 in Eq. 4 and making the substi- | i
tution for x and p given in Eq. 2, leads to and the γ-dependent n-particle state, i.e., the
H = γ A†2 A 2+2A†A +1 state for which
4 − γ − γ γ γ
N n =nn , (9)
+ λ (cid:16)A†2A 2+2A†A +1+ 1((cid:17)A†4+A 4) γ| γi | γi
4γ2 γ γ γ γ 6 γ γ is just
(cid:18)
+23( A†γ3Aγ +A†γAγ3)+(A†γ2+Aγ2) |nγi= √1n!A†γn|0γi. (10)
γ λ λ (cid:17)
= + (2N +1)+ N (N 1)
4 4γ2 γ 4γ2 γ γ − The value ofγ usedto define the adaptiveper-
(cid:16) (cid:19) turbation theory for the nth level of the anhar-
+ γ λ (A†2+A 2) monic oscillator is determined by requiring that
4 −2γ2 γ γ
(cid:16) (cid:19) it minimize the expectation value
+ λ 1(A†4+A 4)+ 2(A†3A +A†A 3) ,
4γ2 6 γ γ 3 γ γ γ γ En(γ)= nγ H nγ . (11)
(cid:18) (cid:19) h | | i
3
Eqs.7-7showthatthisexpectationvalueisequal H =λ1/3 1p2+ 1x4 , (16)
to 2 6
(cid:18) (cid:19)
γ λ thusprovingtheclaim. Acomparisonofthevari-
E (γ)= n H (γ)n = + (2n+1)
n h γ| 0 | γi 4 4γ2 ational computation and the result of a second-
(cid:18) (cid:19)
order perturbation theory for λ = 1 and widely
λ
+ n(n 1). (12) differing values of n is given in Table 1. As ad-
4γ2 −
vertised, we see that the adaptive perturbation
Minimizing E (γ) with respect to γ gives theory for each level converges rapidly, indepen-
n
dent of λ and n.
2(n2+n+1) 1/3 Obviously, the same method can be used to
γ =λ1/3 2n+1 . (13) study the case where m2 > 0, except that now
(cid:18) (cid:19)
one has to solve a cubic equation to determine γ
At this point I substitute this value into Eq. 11 as a function of λ,m2 and n. The general result
to obtain thatforeachn,secondorderperturbationtheory
is accurate to better than one percent still holds
En(γ)min= 3λ1/3(2n+1)2/3 2n2+2(n+1) 1/3 true.
8
(cid:0) (cid:1)(14) 4.1. What does this have to do with quasi-
particles?
which,forlargen, behavesas λ1/3n4/3 , whichis
An interesting corollary to the adaptive per-
the correct answer.
turbation theory technique is that it provides an
The fact that all energies scale as λ1/3 is an
explicit realization of the quasi-particle picture
easilyobtainedexactresultandso,thenon-trivial
underlying much of many-body theory. What we
part of variational computation is the derivation
have shown is that no matter how large the un-
ofthedependenceoftheenergyonn. Toseewhy
derlying coupling, the physics of the states near
all energies are proportional to λ1/3 it suffices to
a given n -particle state can be accurately de-
0
make the following canonical transformation
scribed in terms of perturbatively coupled eigen-
x states of an appropriately chosen harmonic oscil-
x ; p λ1/6p. (15)
→ λ1/6 → lator. Of course, as we have seen, the appropri-
ately chosen harmonic oscillator picture changes
In terms of these operators, the Hamiltonian of
as n changes. Furthermore, the perturbatively
0
the pure anharmonic oscillator becomes
coupledstates n , for n n , correspondto
| γ(n0)i ≈ 0
states containing an infinite number of particles,
if we choose as a basis those states that corre-
spond to a significantly different value of n .
0
Table 1
A comparison of the zeroth order and second or-
derperturbationresults,fortheenergyofthenth
levelofthepureanharmonicoscillator,tothe ex-
actanswerforλ=1andwidelyvaryingvaluesof
n.
λ n Variational 2nd Order Perturbation Exact Variational % Err Perturbative % Error
1.0 0 0.375 0.3712 0.3676 0.02 0.0098
1.0 1 1.334 1.3195 1.3173 0.01 0.0017
1.0 10 17.257 17.508 17.4228 -0.009 0.0049
1.0 40 104.317 105.888 105.360 -0.009 0.0050
4
5. The Double Well of γ that minimizes (γ,c); call this γ(c), and
E
thenplot (γ(c),c)forvariousvaluesofλandf2.
The most general negative mass version of the E
Three such plots are shown in Fig. 1. The first
anharmonic oscillator can, up to an irrelevant
plot is for a value of f which is large enough so
constant, be written as:
that the lowest energy is obtained for a gaussian
H = 1p2+ 1λ x2 f2 2. (17) shifted either to the right or left by the amount
2 6 − c = cmin. While there is a local minimum at
±
Clearly, for a no(cid:0)n-vanishi(cid:1)ng value of f2, this po- c = 0 it has a higher energy than the shifted
states. Things change as one lowers the value
tentialhastwominimalocatedatx= f. Thus,
± of f. Thus, the second plot shows that for lower
we would expect that the best gaussian approx-
f the shifted wavefunctions and the one centered
imation to the ground state of this system can’t
atzeroareessentiallydegenerateinenergy. Fora
be a gaussian centered at the origin; classical in-
slightly smaller value of f things reverse and the
tuitionwouldimply thatitisagaussiancentered
unshifted wavefunction has a lower energy than
about another point, x = c. In other words, if
the shifted ones.
0 is a state centered at x = 0, it is better to
γ
| i
adopt a trial state of the form l=1 l = 1
0.82 f = 2.4 0.804 f - 2.38
|c,Sγiin=cee−icp|0γi. (18) E(c)00..088.018551 E(c) 0000....877700999.24688
0.8 0.792
0.79
eicpxe−icp =x+c, (19) 0.795 0.788
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
c c
computing the expectation value of the Hamilto-
l =1
nian, Eq. 17, in the state specified in Eq. 18, is 0.796 f = 2.36
the same as computing the expectation value of E(c) 00..077.997249
the Hamiltonian obtained by replacing the oper- 0.788
0.786
atorxby x+c,inthe state 0γ . Inthe language 00..778824
| i 0.78
of the previous sections, this is equivalent to in- 0.778
-1 -0.5 0 0.5 1
c
troducingtheannihilationandcreationoperators
A† and A as follows:
γ γ Figure 1. Effective potential as a function of
1 c, showing two degenerate local minima at c =
x = A† +A +c,
√2γ γ γ cmin and one minimum at c = 0. The three
±
(cid:0) (cid:1) plots are for λ = 1 and different values of f and
γ
p = i A† A . (20) show how the minimum at c = 0 changes from
2 γ − γ
r lyinghigherthanthe minimaatc= cmin tobe-
(cid:0) (cid:1) ±
The expectation value of this Hamiltonian in the ing the globalminimum. If this were field theory
state 0 is this wouldbe characteristicof a first order phase
γ
| i
transition.
γ λf4 λf2c2 λc4 λc2
(c,γ) = + + +
E 4 6 − 3 6 2γ While, there would seem to be nothing wrong
(cid:18)
with the situation shown in Fig. 1, it is prob-
λ λf2
+ , (21) lematic when one applies the same sort of anal-
8γ2 − 6γ
(cid:19) ysis to negative mass φ4 field theory in 1 + 1-
which should be minimized with respect to both dimensions. In this case, if one were to add a
γ and c in order to define the starting point of termlikeJφtotheHamiltonian,thisresultwould
the adaptive perturbation theory computation. imply the existence of a first order phase transi-
To see what these equations tell us, it is con- tion when J reached some finite value. At this
venient to hold c fixed and solve for the value point the expectation value of φ in the ground-
5
state would jump discontinuously from zero to
a non-zero value. It has been rigorously shown
that such a first order phase transition at a non-
vanishing value of J cannot occur[5,6].
6. Doing Better
Clearly,todobetter,itisnecessarytodosome-
thing different. The solution is remarkably sim-
ple. The trick is to take advantage of the fact
that H contains a term linear in A† and A .
γ γ
γ λf2 λf4 λ λf2c2
H = + +
4 − 6γ 6 8γ2 − 3
λc2 λc4 γ λ λ(3c2 f2)
+ + + + + − N
2γ 6 2 2γ2 3γ γ
(cid:20) (cid:21)
λ
+ N (N 1)
4γ2 γ γ −
√2λc(c2 f2)
+ − †+ Figure 2. Effective potential as a function o f c,
" 3√γ # A A
one global minimum.
(cid:0) (cid:1)
(22)
To do this introduce a trial state of the form.
ψ(γ,c,α)=eicp(cos(α) 0 +sin(α) 1 ). (23)
γ γ
| i | i e−icp(cos(α) n sin(α) n+1 ).
γ γ
Varyingoverαisthesameasminimizingthe2 2 − | i− | i
× (25)
Hamiltonianofthezeroandoneparticlestatesfor
fixed γ. Then, having done this, minimize over γ
The result of such a computation is to show that
andc. The typicalresult is seen inFig. 2, where,
as n grows the splitting between these states
as youcan see, the minimum atc=0 has all but
grows. Eventually, no matter what value is as-
disappeared.
signedtof (solongasitisfinite)thereisavalue
7. Large n of n for which the splitting between the states
ψ and ψ becomes of order unity. This
even odd
| i | i
For larger values of n, things change, even for is the point at which it makes no sense to talk
f 0. Generically, what happens is that as n abouttunneling betweenstates definedononeor
≫
grows cmin tends to zero, but doesn’t quite get the other side of the potential barrier.
there. Thus, as for the case n=0, there are still
two degenerate minima corresponding to equal
and opposite values of cmin and so it is possible 8. Applying It To Field Theory
to lower the energy by forming the states
For the purpose of this talk I will limit my
ψeven = eicp(cos(α) nγ +sin(α) n+1γ ) discussion to the case of φ4-field theory whose
| i | i | i
+ e−icp(cos(α) n sin(α) n+1 ) Hamiltonian is
γ γ
| i− | i
(24)
1 1 λ
and H= dnx Πφ(x)2+ ( φ(x))2+ φ(x)4 .
2 2 ∇ 6
Z (cid:20) (cid:21)
ψ = eicp(cos(α) n +sin(α) n+1 ) (26)
odd γ γ
| i | i | i
6
It iscustomarytorewritethe operatorsφ(x) and Obviously,iftherangeofthemomentaappear-
Π (x) in terms of their Fourier transforms; i.e. ing in these sums is unrestricted, these expres-
φ
sions diverge. It is customary to deal with this
1
φ(~k) = dnxe−i~k·~xφ(x), problem, in the context of ordinary perturbation
√V
Z theory, by regulating the integrals and adding
Π (~k) = 1 dnxe−i~k·~xΠ (x), (27) counterterms to the Lagrangian to cancel diver-
φ φ
√V Z gences. Since I wish to discuss this theory non-
where V stands for the volume of the system. (I perturbatively, I will adopt a different strategy.
have assumed the system is in a finite volume so I will render the theory well defined by restrict-
that the momenta become discrete and the oper- ing the operators φ(~k) and Π (~k) to be finite in
φ
ators become well defined.) number. Thiscanbeaccomplishedinavarietyof
This Hamiltonian can be rewritten in terms of ways, but all amount to restricting the range of
these momentum space operators as ~k’s which appear in the Fourier transform.
Minimizing Eq. 31 with respect to each γ(~k)
Π ( ~k)Π (~k) (~k2+m2)
H = φ − φ + φ(~k)φ( ~k) yields
" 2 2 − #
X~k
1 1
λ 1 4 γ(k)2 =~k ~k+m2+2λ . (32)
+ φ(~k )φ(~k )φ(~k )φ(~k )δ( ~k ). · V γ(~k′)
6 V 1 2 3 4 i X~k′
X~ki Xi=1  
In particular, the equation for~k=0 is
(28)
To define the adaptive perturbation theory cal- 1 1
culation I introduce γ(k)-dependent annihilation γ(0)2 =m2+2λ , (33)
V γ(~k′)
and creations operators as follows, X~k′
 
which can be substituted into Eq. 32 to give
γ(~k)
φ(~k) = is 2 (A(−~k)†−A(~k)) γ(~k)2 =~k ~k+γ(0)2. (34)
·
( ( ~k)†+ (~k)) If we use this to rewrite the equation for γ(0), it
Π(~k) = A − A , (29)
becomes the non-perturbative equation
2γ(~k)
1 1
The vacuum staqte associated with this choice of γ(0)2 =m2+2λ . (35)
γ(~k)’s, V X~k′ (~k′)2+γ(0)2
q
vac = 0 , (30) TakingV to infinity andconvertingthe sumover
| i | γ(k)i ~k to an integral, we obtain an equation which is
k
Y
reminiscentofthe NambuJona-Lasinioequation.
is defined by the condition that it be annihilated
by all the (~k)’s. (2λ)2p Λ 1
A γ(0)2 m2 = kp−1dk .
γ(kA)s’sinbythmeinsiimmipzlienrgetxhaempvlaecuIumdeteexrmpeicnteattiohne − (2π)p Z0 ~k ~k+γ(0)2
·
valueoftheHamiltonianinthistrialstate. Itfol- q (36)
lowsdirectlyfromthesedefinitionsthatthefunc-
In fact Eq. 36 should not be thought of as an
tion to be minimized is
equation for γ(0), but rather as an equation for
γ(~k) (~k ~k+m2) m2; i.e., it should be rewritten as
vac H vac = + ·
h | | i k " 4 4γ(~k) # (2λ)2p Λ 1
X 2 m2 =γ(0)2 kp−1dk .
λ 1 − (2π)p Z0 ~k ~k+γ(0)2
+ . (31) ·
4V  γ(~k) q (37)
X~k
 
7
This shows that for any arbitrarily chosen value Eq. 38; i.e.,
of γ(0), this equation determines the value of
H =H +λV, (40)
m2 for which the chosen value of γ(0) will min- 0
imize the ground state energy density. This is, it iseasyto showthatthe resolventoperatorsat-
of course, nothing but a non-perturbative way isfies the integral equation
of determining the leading mass renormalization
1 1 λ 1
counter term.
= V . (41)
H z H z − H z H z
0 0
− − − −
9. Wavefunction Renormalization
This equation is customarily solved by iteration
to give the following series;
To this point I have shown how a simple vari-
ationalcalculationcaptures the generalnotionof 1 1 λ 1
= V
massrenormalizationinanon-perturbativeman- H z H z−H z H z
0 0 0
ner. I now wish to briefly describe what one has −λ2 1− 1− −
to do to capture wavefunction and coupling con- + V V
H z H z H z
0 0 0
stant renormalization. The trick is to proceed − − −
λ3 1 1 1
as in the double well problem and generalize the V V V
trialstatetoincludetheeffectsofthe †4and 4 −H0−z H0−z H0−z H0−z
A A λ4 1 1 1 1
termswhichappearinthenormalorderedHamil-
+ V V V V
tonian; i.e. terms of the form H0 z H0 z H0 z H0 z H0 z
− − − − −
1 +...... (42)
4 γ(k1)γ(k )γ(k )γ(k ) ×
k1X,k2,k3 2 3 4 SinceV onlylinksthevacuumtothefourparticle
p † † † † + states and then links four particle states back to
Ak1Ak2Ak3Ak4 Ak1Ak2Ak3Ak4
the vacuum, this series simplifies to
h (38i)
1 1
To allow these terms contribute to the ground 0 0 =
h |H z| i E z
0
state energy I have to add to the vacuum state − −
a general four particle state; i.e., I need a trial + 1 λ2 h0|V|nihn|V|0i
state of the general form E0 z" (En z)(E0 z)#
− n − −
X
2
vac + α(k ,k ,k )4 particle , (39) 1 0V n nV 0
| i 1 2 3 | − i + λ2 h | | ih | | i +...
E z (E z)(E z)
where thXe sum is assumed to go over all possible (cid:18) 0− " Xn n− 0− #!
j
four-particle states. The question is ”How do we ∞
1 0V n nV 0
minimize over the α(k ,k ,k )’s?”. = λ2 h | | ih | | i
Actuallythisquestio1nc2anb3efinessedsinceitis E0−zj=0" n (En−z)(E0−z)#
X X
equivalenttofindingthegroundstateenergyden-  1 
sity of a system where the Hamiltonian is trun- = .
E z λ2 h0|V|nihn|V|0i
cated to the vacuum state and all possible four 0− − n En−z
particlestates. While it is hardto do this for the P (43)
fullHamiltonian,itcanbedoneforthecasewhere
Clearly, since the poles of the resolvent operator
the full Hamiltonian is limited to the part indi-
correspondto the eigenvaluesofthe Hamiltonian
catedin Eq.38. Inthis casethe generalsolution,
it is only necessary to find that zero of the func-
whichwillresumthe importantnon-perturbative
tion
effects in λ, can be obtained by considering the
resolvent operator. Dividing H into the part di- E z λ2 h0|V|nihn|V|0i =0 (44)
0
agonal in the number operator and the terms in − − E z
n
n −
X
8
which lies to the value E in the limit λ 0. answer. Figure 3 shows how a single iteration
0
→
While this looks like a difficult problem it can manages to reproduce the behavior of the exact
be solvedtoarbitraryaccuracybyconvertingthe answer to pretty good accuracy over the whole
problem into an equation which can be solvedit- range of x.
eratively. ToshowthisIwillbeginbydefiningδz
as
z =E λ V 2 δz, (45)
0− | |0n
s n 1.0
X
and then use this definition to rewrite Eq. 44 as
an integral equation for δz; i.e.,
0.75
V¯ 2
δz = | |0n , (46)
En−E0 +δz
n λ
X
0.5
where I have defined V¯ to be
0n
0V n
V¯ = |h | | i| . (47)
0n
V 2 0.25
j| |0j
q
It is clearPthat with these definitions, so long as
the range of E E is bounded, in the limit
n − 0 0.0 2.5 5.0 7.5 10.0
λ the solution to the integral equation is
→ ∞ x
δz =1. Inotherwords,inthe limitoflargeλthe
energy shift is proportional to λ instead of the
perturbative behaviour which goes like λ2.
Given the large λ value of δz it is possible to
give an iteration procedure for solving for δz for Figure 3. Comparison of exact energy as a func-
arbitrary values of λ. This is done by defining a tionofx(redcurve)comparedtoδz (bluecurve)
1
sequence of values z by the recursion relation
j After three iterations it is much harder to dis-
δz = 1 tinguish the curves, see Fig. 4. Doing more iter-
0
V¯ 2 ations just produces better and better accuracy
δz = | |0n over the entire range. By twelve iterations one is
1 En−E0 +δz
n λ 0 accurate to very high accuracy except in a very
X
V¯ 2 small band around x 0.8, where the fractional
δz = | |0n , (48) ≈
n En−E0 +δz error is 0.015. The same computation can be
n λ n−1 ≈
X doneforthemorerealisticcasewhereV connects
The desired value of δz is obtained by taking the us to a large number of momenta, with similar
limit n . As an example, consider the case results.
→ ∞
in which H is a 2 2 matrix with two arbitrary Having established that it is in principle possi-
×
entries on the diagonal E0 and E1; i.e., ble to obtain the shift in the groundstate energy
due to the part of the Hamiltonian which mixes
E x
0
the vacuumwithfourparticlestates,itistime to
H = . (49)
  go back to our discussion of the adaptive pertur-
x E
1
bation theory scheme. Given the preceding dis-
 
Since it is trivial to diagonalize this matrix it is cussion it is easy to show that differentiating the
easytocompare,forthecaseE =1andE =2, expectation value of the Hamiltonian in our trial
0 1
the iterative solution for arbitraryx to the exact statewillyieldasetofequationsfortheγ(k)’sof
9
and then observe that this means that
1.0
1 1
γ(0)2 =m2 + 2λ
V γ(k)
" #
k
X
0.75 + λ2Σ(0,λ,γ(ki)). (52)
Substituting this into the equation for a generic
value of k yields
0.5
γ(k)2 = k2+γ(0)2
+ λ2 Σ(k2,λ,γ(k )) Σ(0,λ,γ(k )) .
i i
−
0.25 (53)
(cid:0) (cid:1)
Atthispointtheroleofwavefunctionrenormal-
ization becomes obvious. If, as is conventional,
0.0 2.5 5.0 7.5 10.0 we Taylor series expand Σ(k2,λ,γ(k )) so as to
i
x rewrite Eq. 53 as
∂
γ(k)2 = (1+ Σ(k2),λ,γ(k ))k2+γ(0)2
∂k2 i
+ λ2 Σ(k2,λ,γ(k )) Σ(0,λ,γ(k ))
i i
Figure 4. Comparison of exact energy as a func- −
∂
tionofx(redcurve)comparedtoδz3(bluecurve) − k2(cid:0)∂k2Σ(0,λ,γ(ki)) . (54)
(cid:19)
This can be put into a conventional form
γ(k)2 =k2+γ(0)2+λ2Σ¯(k2,λ,γ(k )) (55)
the general form i
by introducing a rescaling of the fields and the
1 (k2+m2) λ 1 1 overall Hamiltonian by
4 − 4γ(k)2 −2γ(k)2 "V Xk′ γ(k′)# φ Z1/2φ ; Π Πφ
λ2 Σ(k2,λ,γ(k ))=0, (50) → φ φ → Zφ1/2
−γ(k)2 i
H Z H. (56)
φ
→
where the function Σ(k2,λ,γ(k )) is obtained by Thus, we see that wavefunction renormalization
i
differentiating the solution for δz obtained from is just that, anoverallchangein theγ(k)’s. Note
our integral equation. In writing things in this thattherescalingoftheHamiltonianisnecessary
form I used the fact, which can be obtained becauseweareworkinginaHamiltonianandnot
fromtheintegralequation,thatdifferentiatingδz a manifestly covariant formalism.
with respect to a given γ(k) produces a factor of Finally, I will just say a few words about cou-
γ(k)−2 multiplied by a function of k2,λ and all pling constant renormalization and why φ4 the-
the γ(k ). As in the earlier discussion of mass ory doesn’t exist in four dimenstions, at least if
i
renormalization, it is convenient to rewrite these one attempts to remove the cutoff. Clearly, once
equations as the γ(k)’s have been chosen, the only remaining
issue is determining those values of λ for which
some physical quantity come out finite. For the
1 1
γ(k)2 =k2+m2 + 2λ purposes of this discussion I can choose this to
V γ(k′)
" k′ # be the energy of a trial state containing only two
X
+ λ2Σ(k2,λ,γ(k )) (51) particles of momentum zero.
i
10
AsinearlierdiscussionsIwillrestrictthisanal- equation, requires much more thought.
ysistotheeffectoftermsinH whichtaketwopar-
ticles to two particles. In this case I can employ
11. Summary
asimilarargument,baseduponstudyingthema-
trixelementoftheresolventoperatorinthiszero- InthistalkIbeganbyshowingyouhowtocon-
momentumstate,toshowthatinfourdimensions vert problems,which in the pastwere thoughtto
this energy is given by a series in λ(λln(Λ))n. be impossible to deal with perturbatively, can be
From this it follows that the only way to have easily done using the method I have called adap-
this come out finite is to take λ 1/ln(Λ). But tive perturbation theory. I then went on to show
→
this, of course, implies that there is no interac- howthesemethodscanbeextendedtogiveavery
tion. pretty picture ofthe structureofrenormalization
in a non-perturbative context. These ideas can
10. The Spectrum Isn’t Boost Invariant be extended to a theory which includes fermions
by adding adding a general Bogoliubov transfor-
The final point which should be touched upon
mation for the fermion fields defined in momen-
before closing my talk is the observation that at
tum space. A bit more work has to be done to
first glance an equation of the form
extend the tricks to cover the case of compact
γ(k)2 =k2+γ(0)2+λ2Σ¯(k2,λ,γ(k )) (57) Abelian gauge-fields. However,a paper by David
i
Horn and myself[7], extended in the obvious way
would seem to be problematic, since γ(k) for to fit it into the frameworkof adaptive perturba-
k = 0 is not just related by a boost to the value tion theory, showed how to do this for the case
6
fork =0. Infact,thisisnotabug,itisafeature. of compact QED. This paper showed that the
What it says is that while it is possible to choose methodisquitecapableofextractingtheinterest-
the γ(k)’s to give a good perturbation perturba- ing non-perturbative structure of confinement in
tion theory, with this choice of γ(k)’s the opera- this theory in any number of space-time dimen-
tors †(k) donotcreatea trueasymptotic states sions. While the path towards extending these
A
which should be used to compute scattering am- ideas to the case of a non-abelian gauge theory
plitudes. Even if I assume that the state created is not so obvious, I nevertheless believe that it is
by †(0) is a good approximation to the zero possible.
A
momentum state, the state for k = 0 should be
6
obtained by applying the boost operator to this
REFERENCES
state. Foraninteractingtheorythisiscertainlya
multi-particle state. Thus, it follows that adopt- 1. Subsequent to completing this work I was
ingtheformalismofadaptiveperturbationtheory made aware of previous papers by I. D. Fer-
forces implies that the scattering problem must anchuk and collaborators, who, in a series of
be handled as in the parton picture. In other very interesting papers, developed a method
words, the computation of a scattering process whose basic idea is the same as the approach
intrinsicallyhastwoparts;first,itisnecessaryto described in this paper. These authors then
find the parton wavefunction for the asymptotic appliedthismethodtoawidevarietyofprob-
scattering states and then, this wavefunction, to- lems in quantum mechanics, demonstrating
gether with the explicit form of the Hamiltonian the rapid convergence of the approximation
written in terms of the annihilation and creation scheme. Although our way of implementing
operators associated with the specific choice of the basic idea differs in significant ways, the
γ(k)’s, shouldbe usedto compute scatteringam- fundamental ideas are the same and so, the
plitudes. This point, together with the observa- applicationstheseauthorsdiscusscanalsobe
tion that applying a boost (written in terms of treatedby the methods I describe.The intro-
the relevant annhilation and creation operators) ductionoftheirapproachandsomeoftheap-
to a given state implies a kind of Alterelli-Parisi plicationstheytreatinbeautifuldetailcanbe