Beyond Relativistic Quantum String Theory or
Discreteness and Determinism in Superstrings Gerard ’t Hooft Institute for Theoretical Physics Utrecht University and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail:
[email protected] internet: http://www.phys.uu.nl/~thooft
Summary The question “What lies beyond the Quantized String or Superstring Theory?” and the question “What lies beyond Quantum Mechanics itself?” might have one common answer: a discretized, classical version of string theory, which lives on a lattice in Minkowski space. The size a of the meshes on this lattice in Minkowski space is determined by the string slope parameter, α0 .
1.
Introduction
In spite of the beautiful mathematical underpinnings of string- and superstring theory[1][2], the physical foundations of these ideas are embarrassingly weak. Here, I am not referring to the lack of experimental evidence; the fact that this theory is almost impossible to subject to non-trivial tests or observations is an inevitable aspect of the domain of the physical phenomena it is designed to describe, and none of the competing approaches towards ‘physics at the Planck scale’ are faring any better than (super)string theory. Rather, the weakness of its foundations is the fact the we do not understand in a satisfactory way what the equations mean physically, as soon as one attempts to go beyond the level of perturbation theory. Term-by-term, the string perturbation theory appears to be finite and well-behaved, but what the theory means beyond this perturbation expansion is obscure. In particular, we have the fact that strings propagate in a space-time that itself gets curved because of the presence of strings, and the fact that the energy quanta of curved space-time, the gravitons, themselves must be strings again, to be precise, the lightest spin-two excitations of a closed string. If string theory is finite perturbatively, what does this mean for the non-perturbative system, whatever that is? From its formulation, one would have thought that, at least 1
in principle, strings should be able to form infinitely complex curves in arbitrarily small regions of space, and, when one includes such complex structures in a functional integral formalism, this should be expected to give rise to quite complex infinities. The most experienced practitioners of the theory usually respond to this that “the theory takes care of such infinities by itself”, and indeed, this appears to be exactly what happens in any explicit calculation. Now, in these lectures, we arrive at an even simpler answer: everything is finite because the string actually is a deterministic system that √ lives in a lattice space-time; target space is a lattice, whose meshes have a length 2π α0 . 1.1.
The Discrete Lorentz Group
Consider a theory with all its dynamical variables defined on a lattice: (t, x, y, z) ∈ Z4 . Equations of motion could for instance take the form: φ(x, y, z, t + 1) = F φ(x ± 1, y ± 1, z ± 1, t) . (1.1) Question: is there any such discrete theory for which an invariance, or covariance, property can be formulated under discrete Lorentz transformations (and indeed also under discrete Poincar´e transformations)? X µ → Lµν X ν ,
(1.2)
(where all matrix elements Lµν of the O(D − 1, 1) matrix L are integers: L ∈ O(D − 1, 1, Z) . 2 1 1 1 1 1 0 1 1 0 Examples of such matrices are: L1 = 1 1 0 1 or R1 = 0 1 1 1 0 0 The discrete Lorentz group is non compact.
2.
(1.3) 0 0 0 1
0 1 0 0
0 0 . 1 0
String Theory in a Nutshell 2.1.
Bosonic String Theory
Anticipating the notation that will be useful later, points in a continuous D dimensional space-time will be indicated as q µ , µ = 0, 1, · · · , D − 1 . The 2 dimensional world sheet is given two coordinates (x, t) . A string is given by a function q µ (x, t) . Its equations of motion are defined by means of the string action. Defining light cone coordinates on the world sheet, x± = √12 (t ± x) , we have Z p S = T dx+ dx− (∂+ q µ ∂− q µ )2 −(∂+ q µ )2 (∂− q ν )2 . 2
(2.4)
Here, T = 1/(2πα0 ) is the string tension constant. Since this action is invariant under all reparametrizations of the two coordinates x+ and x− , they can be chosen at will. Choosing them to follow light signals in the two directions on the string, implies that we demand (∂+ q µ )2 = (∂− q ν )2 = 0 ;
furthermore, t =
√1 q + 2
,
(2.5)
the latter implying that we also use light cone coordinates in target space. This simplifies the action: S = T ∂+ q µ ∂− q µ = T ( 21 (∂t q µ )2 − 21 (∂x q µ )2 ) .
(2.6)
Let us normalize our units such that T ≡1.
(2.7)
This action is a bilinear one (the sign ambiguity in the square root can easily be resolved in the classical theory), and hence the equation of motion is linear: ∂+ ∂− q µ (x+ , x− ) = 0 ;
∂t2 q µ − ∂x2 q µ = 0 ,
(2.8)
which is easily solved: q µ (x, t) = qLµ (x + t) + qRµ (x − t) ;
(2.9)
this however must still must obey the constraints (2.5), which imply: (∂x qLµ )2 = (∂x qRµ )2 = 0 ;
qL+ = x+ ,
qR+ = x− .
(2.10)
Apart from these constraints, the action (2.6) is that of a massless, non-interacting bosonic field theory in 1+1 dimensions. As such it is easy to quantize. Note that the constraints do not directly affect the quantum dynamics of this system; they merely ± a determine the functions qL,R when we know the D − 2 transverse functions qL,R . 2.2.
Superstrings
Fermionic excitations are obtained if, besides the coordinate fields q µ , we put fermionic fields on the string world sheet. Since the world sheet has only one space dimension, a fermionic field is a two component spinor, ψ = (ψ1 , ψ2 ) , and if we assume that they are massless, they form Majorana fermions, and the world sheet Dirac equation also gives rise to left movers and right movers: ψ1 (x, t) = ψL (x+ ) ,
ψ2 (x, t) = ψR (x− ) .
(2.11)
If these spinors can furthermore be labeled with a Lorentz index µ just like the ‘bosonic’ fields q µ , they together form N = 1 supermultiplets. We thus obtain the superstring. Supersymmetry then demands that the ‘bosonic’ constraints (2.10) are replaced by: (∂+ qLm )2 + 21 ψLµ κ ˆ ψLµ = 0 ,
(∂− qRµ )2 + 12 ψRµ κ ˆ ψRµ = 0 , 3
∂+ qL+ = ∂− qR+ = 1 , (2.12)
and that there are similar ‘fermionic’ constraints: (∂+ qLµ ) ψLµ = 0 ,
(∂− qRµ ) ψRµ = 0 ,
+ ψL,R =0.
(2.13)
Both bosonic strings and superstrings may be subject to different types of boundary conditions. In the absence of D -branes, one has only periodic boundary conditions, that is, strings form closed loops. Effectively, this reduces the independent degrees of freedom in the superstring to D − 2 bosonic variables, the transverse coordinate fields q a , and D − 2 transverse fermionic fields ψ a , moving to the left or to the right. Interactions can be introduced by allowing strings to ‘exchange their legs’ when they coincide at a space-time point. This interaction is then associated with a constant, the string-interaction constant gs .
3.
Strings on a lattice 3.1.
Bosonic String Theory on a Lattice
t
For this discussion, it is useful to replace the world sheet coordinates (x, t) by a lattice[3]: x ∈ Z , t ∈ Z . We may or may not decide later to replace this lattice by a continuum1 . The discretized equations of motion (2.8) turn into q µ (x, t + 1) = q µ (x − 1, t) + q µ (x + 1, t) − q µ (x, t − 1) .
x (3.1)
Again, we find left-movers and right-movers: q µ (x, t) = qLµ (x + t) + qRµ (x − t) .
(3.2)
To define the discretized constraint, we may define aµL (x) = qLµ (x + 1) − qLµ (x − 1) , aµR (x) = qRµ (x − 1) − qRµ (x + 1) ,
(aµL )2 = 0 ; (aµR )2 = 0 .
(3.3)
A genuine lattice string theory emerges if we also put target space on a lattice: q µ → Qµ ∈ ZD ,
aµ → Aµ ∈ ZD .
(3.4)
The discrete analogue of the constraints (2.10) would be A+ L = 1 ,
tr 2 1 A− L = 2 (AL ) ,
and L ↔ R ,
(3.5)
but there could be a problem with this: A+ L =
√1 (A0 L 2
+ AD−1 ), L
A− L =
1
√1 (A0 L 2
− AD−1 ), L
(3.6)
Physically, this lattice might be not so meaningful. The world sheet coordinates are in principle arbitrary, so that this ‘discretization’ of the string produces string ‘bits’ [3] of unspecified length. One will have to send these lengths to zero, eventually.
4
would modify the lattice√in Minkowski space (this could be remedied by modifying the constraints with factors 2 ), but a more fundamental problem would be that the world sheet lattice itself already implies a gauge choice, so adding the constraints (3.6) may over determine the gauge. A Lorentz transformation may multiply A+ with a factor and divide A− by the same factor, which would knock us out of the Minkowski lattice by generating fractions. It seems that the entire procedure of fixing the gauge on the world sheet may have to be reconsidered when employing this lattice. Later, it will be argued that the constraints should be used in an altogether different fashion. As yet, we observe that the constraints will produce some sort of lattice in Minkowski space, as described in Subsection 1.1. See the figure at the top of this Section, describing a string at a given world sheet time t . so if A0L + ALD−1 > 1 after a Lorentz transformation, imposing this gauge will require that x+ have a gap of length > 1 . This issue will not be pursued here. 3.2.
String interactions
Note that, our description of strings on the Minkowski space lattice is entirely classical, obeying the deterministic equations (3.1). It is tempting now to demand that also the string interactions are deterministic. This can be done in exactly one way, see the figure here. In a deterministic string theory living on a space lattice, we can demand that two strings exchange their legs when they meet at one lattice point. Originally, our theory only contained strings of infinite length. With this interaction, closed strings with finite lengths will emerge. Demanding that the interaction be deterministic has two other consequences. In quantum mechanics, superpositions are allowed. Here, we must choose how the legs are exchanged. This is unambiguous only if the strings have an orientation. So non-oriented strings are disallowed in this theory. Secondly, the strength of the interaction is no longer adjustable. When two strings meet at one point, the interaction must be certain, so, in units yet to be identified, the string coupling constant, gs , must be one. Quantization will now be realized in a very unconventional manner. Conventionally, the quantum theory is quite different from the classical one; one could say that the classical theory merely serves as an inspiration, an educated guess, for producing an altogether different system, which however does have the property that in the limit ~ → 0 the classical theory is reproduced. In our second lecture (see Sections 4—8), it will be explained that, under special conditions, lattice string theory is identical to a fully quantized string theory in the continuum, and it will be either a bosonic string or a superstring. The field equations do not change. A different way of saying this is that the states of the lattice system are promoted to being the elements of a basis in Hilbert space. While we are describing the fully quantized theory, these particular basis elements continue to evolve according to the classical equations of motion. 5
3.3.
Fermions
The classical counter parts of fermions are very simple. Besides the bosonic variables q µ , the superstring has fermions (2.11) living on the world sheet. There are D − 2 mutually independent fermion fields ψAa (x, t), A = 1, 2 , or equivalently, A = L, R . We saw in Subsection 2.2 that the solution on the world sheet behaves as µ ψL (x + t) µ ψA (x, t) = , (3.7) µ ψR (x − t) so, the spinors also have left-movers and right-movers. The deterministic lattice theory, instead, has Boolean degrees of freedom, σ a (x, t) = ±1 , a = 1, · · · , D − 2 , obeying equations similar to Eqs. (3.1): σ a (x, t + 1) = σ a (x − 1, t) σ a (x − 1, t) σ a (x, t − 1) .
(3.8)
This also splits up into left- and right-movers: σ a (x, t) = σLa (x + t) σRa (x − t) .
(3.9)
Our first impression would be that the superstring constraints (2.13) now have to be modified so as to determine the Boolean degrees of freedom σ D−1 and σ 0 , or equivalently, σ ± , on the world sheet, but we will do this differently later.
4.
Real numbers and integers
Imagine that, in contrast to appearances, the real world, at its most fundamental level, were not based on real numbers at all. We here consider systems where only integers describe what happens at a deeper level. Can one understand why our world appears to be based on real numbers? A mathematical tool is needed[4]. We shall first formulate it entirely in terms of quantum mechanical states. Consider two kinds of quantum systems. System (D) is a quantum theory whose basis elements are entirely specified by a set of N pairs of integers, to be called Qi and Pi , where i = 1, · · · , N . So the Hilbert space is spanned by the products of N states of the form |Qi , Pi i , with (Pi , Qi ) ∈ Z2N . System (R) is also a quantum system, but its Hilbert space is spanned by all wave functions ψ(qi ) , or equivalently, by all states |qi i , where again i = 1, · · · , N . Here qi ∈ RN . We also have operators pi in this Hilbert space. Here, we define pi = − 2πi ∂/∂qi .
(4.1)
Our mathematical tool is a mapping between the states of system (D) and system (R) . The mapping is a standard application of Fourier transforms. Mappings of this sort may become very important for instance when black holes are considered: it is generally believed that black holes come in a denumerable set of quantum states, whereas the particles we usually consider sit in spaces with continuous coordinates. 6
4.1.
Notation
Define a useful symbol: ≡ e2π ≈ 535.49165 · · · ;
ipx = e2πipx ;
iZ = 1 if Z ∈ Z .
(4.2)
Writing complex exponentials this way makes all periods 1 instead of 2π , which comes out handy when functions are normalized. In systems of the type (R) , we have (suppressing the index i ): hq|ipa |pi = hq + a | pi . hq1 |q2 i = δ(q1 − q2 ) , If N > 1 insert δij .
2π[q, p] = i , hq|pi = ipq , hp1 |p2 i = δ(p1 − p2 ) .
(4.3)
Note that this means that Planck’s constant h , not ~ , is normalized to one. Let now in general N, P, Q, X, · · · , p, q, x, · · · , α, η, ξ, λ, · · · ,
cap. Latin letters, l.c. Latin letters, l.c. Greek letters,
indicate integers, indicate real numbers, are numbers modulo 1;
the latter will also be called ‘fractional numbers’. Typically, − 12 < η ≤
1 2
(4.4)
.
In system (D) , we define the operators ηQ and ηP , with eigenvalues ηQ and ηP , such that [P, Q] = 0 ,
4.2.
hQ| ηQ i = iQ ηQ , hQ1 | Q2 i = δQ1 Q2 ,
hP | ηP i = iP ηP , 1 2 1 2 hηQ | ηQ i = δ(ηQ − ηQ ).
(4.5)
Q and P states
Until here, we have only considered quantum systems. These are systems whose states are described by Hilbert space, and where one may expect the evolution law te be described either by a hamiltonian, or more directly by some finite-time evolution operator U (t1 −t2 ) . However, particularly systems of the type (D) are also suitable to describe deterministic cellular automata. A (deterministic) cellular automaton is here defined to be a discrete system where the evolution operator U takes a very special form. At regularly chosen time intervals, the evolution operator is assumed to be a pure permutation operator acting on the (discrete) basis elements that we picked out. Thus, in this basis of Hilbert space, at these time intervals, the evolution operator is required to be a unitary matrix containing only ones and zeros. This condition implies that cellular automata are simply a subset of all quantum systems. However, being described by simple permutations, also means that these systems are fundamentally classical. They are as deterministic, and as classical, as an ordinary laptop. The only difference is that a laptop has a delete button. Unitarity forbids the erasure of information; this makes cellular automata even more restrictive than laptops. 7
The essential theme of the idea displayed in these lectures is, that we can switch between the quantum notation and the entirely classical notation where a cellular automaton simply permutes its states |Qi . We stress that, physically, we are then always talking about the same thing. If a classical automaton is described using a quantum state |ψi , this means nothing more than that the probabilities that different states Q are realized, are |hQ|ψi|2 . At the end of a calculation we will exclusively be interested in amplitudes out hQi |Qj iin . The classical and quantum probabilities will be the same. In spite of the obvious correctness of such a procedure, the reader must be warned that most of the practitioners of the foundations of quantum mechanics have not yet relinquished their skepticism against it. 4.3.
Operators
There is nothing wrong with treating discrete systems, including cellular automata, using the full machinery of quantum mechanics. Therefore, we simply introduce the operators ηQ and ηP . These will turn out to be handy for instance if we wish to identify some operator H such that U (t) = −iHt .
(4.6)
The definition (4.5) defines the operators ηQ,P uniquely if we also demand their eigenvalues to obey − 21 < η ≤
1 2
.
(4.7)
To find their matrix elements in the Q, P basis, one merely has to Fourier transform, Z 1 X X i(−1)N 2 iN η η dη −iN η = iN η , (4.8) η= 1 2πN − N N 6=0 2
and since iN ηQ |Qi = |Q − N i ,
(4.9)
we find the matrix elements: hQ1 |ηQ |Q2 i =
i (−1)Q1 −Q2 (1 − δQ1 Q2 ) , 2π Q1 − Q2
(4.10)
and similarly for ηP . Because the region (4.7) is symmetric, Eq. (4.8) vanishes ate N = 0 , and therefore the matrix elements (4.10) vanish when Q1 = Q2 . The commutator [ηQ , Q] has the matrix elements i i hQ1 |[ηQ , Q]|Q2 i = δQ1 Q2 − (−1)Q2 −Q1 = (I − |ψQ ihψQ |) , 2π 2π where the state |ψQ i is what we will call an edge state : hQ|ψQ i = (−1)Q ;
hη|ψi = δ(η − 12 ) .
(4.11)
(4.12)
So, one finds that the commutator [ηQ , Q] = i/(2π) only when it acts on a state that is orthogonal to the edge state. 8
4.4.
The mapping
Of course, the operators Q and ηP commute, so they can be diagonalized simultaneously. Now, take any real number q . We can split it into its closest integer Q and the remainder, which is a fractional number, call that ηP : q ≡ Q + ηP .
(4.13)
This split is unique and so it is a one-to-one mapping. This is the simplest way we can map (D) onto (R) : hQ, ηP |ψi = hq|ψi .
(4.14)
To find the operator p , all we have to do is calculate its eigen states2 , hq|pi = ipq .
(4.15)
All matrix elements can now be computed using Eqs. (4.3), (4.5) and (4.10).
5.
Orthogonality and Convergence 5.1.
Wave functions
The wave functions of the states |Q, P i in the q representation are easy to find. The state |0, 0i is hq|0, 0i = θ( 21 − |q| ) ,
(5.1)
see Fig. 1 a . Note, that it is properly normalized. The other states are hq|Q, P i = iP q θ( 12 − |q − Q| ) .
(5.2)
The wave functions in momentum space are obtained by acting with the matrix (4.15): Z sin π(p − P ) −ipQ . (5.3) hp|Q, P i = dq hp|qihq|Q, P i = π(p − P ) The wave function hp|0, 0i is displayed in Fig. 1 b . For our purposes, we are not quite content with the asymmetry between the states and their Fourier transforms, Eqs. (5.2) and (5.3), or, Figs. 1 a and b . We can do this better by first defining the operators q and p in the basis spanned by ηQ and ηP . If we take them to be periodic both in ηQ and in ηP , then a discontinuity arises at ηP = ± 21 , but not at ηQ = ± 12 . 2
It is in equations such as (4.14) and (4.15), that our notation using as in (4.2) is useful: the normalizations do not change.
9
q a)
p b)
-1
0
1
2
3
Figure 1: a ) The state |0, 0i in position space. b ) in momentum space.
To remedy this, we rotate over a phase angle φ(ηQ , ηP ) . It obeys φ(η, ξ + 1) = φ(η, ξ) + η;
φ(η + 1, ξ) = φ(η, ξ) .
(5.4)
It can also be forced to obey φ(η, ξ) = η ξ − φ(ξ, η) .
(5.5)
Using some mathematics, this phase angle can be obtained from an elliptic integral[4]. One finds: −i ∂ ∂ ∂ q = + φ(ηP , ηQ ) = Q + φ(ηP , ηQ ) ; 2π ∂ηQ ∂ηQ ∂ηQ −i ∂ ∂ ∂ p = − φ(ηQ , ηP ) = P − φ(ηQ , ηP ) . (5.6) 2π ∂ηP ∂ηP ∂ηP Eq. (5.5) implies that this obeys [q, p] = i/2π .
(5.7)
The form of the state |0, 0i in this q space is pictured in Fig, 2. An explicit calculation of the matrix elements of q and p in Hilbert space spanned by the |Q, P i states, one finds: q = Q + aQ
,
hQ1 , P1 |aQ |Q2 , P2 i = hQ1 , P1 |aP |Q2 , P2 i =
10
p = P + aP , (−1)P +Q+1 iP 2π(P 2 + Q2 ) (−1)P +Q iQ . 2π(P 2 + Q2 )
(5.8)
a)
(X+.5).ψ 0.4
0.2
b) -1
0
(X+.5)(q−X−.5)
Figure 2: a) The modified wave function in q space, using Eqs. (5.6). It is equal to its own Fourier transform, so in p space, it is the same. b) The asymptotic form of the little peaks; they shrink rapidly.
From these, the commutator follows: hQ1 , P1 | [q, p] |Q2 , P2 i = =
i (I 2π
i 2π
− |ψedge ihψedge |),
(−1)Q1 −Q2 +P1 −P2 (δQ1 Q2 δP1 P2 − 1) . with
hQ, P |ψedge i = (−1)Q+P .
(5.9)
This contradicts Eq. (5.7), a contradiction that comes about from the fact that the phase function φ(ηQ , ηP ) is singular at the point ηQ = ±ηP = ± 21 . This edge state cannot be removed, because the phase has a vortex there. The vortex is a necessary consequence of the twisted boundary condition (5.4). This is why, at at least one point on the (ηQ , ηP ) torus, Eq. (5.7) fails to hold.
6.
A harmonic oscillator
As harmonic oscillator hamiltonian we take3 H = π(p2 + q 2 ) . What are the matrix elements hQ1 , P1 |H|Q2 , P2 i ? We have Z ∞ 2 hQ1 , P1 |q |Q2 , P2 i = dq hQ1 , P1 |qiq 2 hq|Q2 , P2 i .
(6.1)
(6.2)
−∞
Inspecting Fig. 2, one finds that this has a logarithmic divergence proportional to (−1)Q1 −Q2 +P1 −P2 3
∝
hQ1 , P1 |ψedge ihψedge |Q2 , P2 i .
The normalization is such that, with our commutation rules, Eqs. (6.5) will hold.
11
(6.3)
Only this edge state has a divergent hamiltonian. All other elements of the hamiltonian converge rapidly. Therefore, we introduce a constraint, hψedge |ψi = 0 .
(6.4)
P
Q
Figure 3: The trajectories of states of the harmonic oscillator in Q, P space.
We can write |ψ(t)i = −iHt |ψ(0)i ;
H = a† a = N ;
|ψ(T )i = |ψ(0)i ,
(6.5)
where N and T are integers (the period of this oscillator is 1). Now we know that, after a quarter phase rotation, the momentum and position operator of a harmonic oscillator interchange, Therefore, at steps tn = 41 n , where n = 1, 2, 3, 4 , we have q → p → −q → −p → q , and, because of our complete Q, P symmetry: (Q, P ) → (P, −Q) → (−Q, −P ) → (−P, Q) → (Q, P ) .
(6.6)
Therefore, at steps of one quarter of a full phase rotation, this system is a “deterministic” system. We call such a system an “automaton”. Needless to say, this automaton is a very trivial one, but it will illustrate the way we argue: This discrete, deterministic automaton is described by the hamiltonian (6.1) when transformed to Q, P space. Of course, the states only perform a perfect permutation at time steps that are multiples of 1/4 . The importance of this finding is, that it is often far from easy to find such a hamiltonian that is properly bounded from below.
12
7.
QFT in 1+1 dimensions, and the 1+1 dimensional CA 7.1.
Bosons
The model of the previous section was still quite trivial. To get a more interesting system, we need many oscillators. A beautiful example is a bosonic massless scalar quantum field theory in one space- and one time dimension. The quantum system would be defined by fields φ(x, t) and their canonical momentum fields p(x, t) , obeying commutation rules [φ(x, t), p(y, t)] =
i δ(x − y) ; 2π
[φ(x, t), φ(y, t)] = [p(x, t), p(y, t)] = 0 .
(7.7)
The hamiltonian is taken to be Z H=π
dx(p(x)2 + (∂x φ)2 ) .
(7.8)
Let us now, for simplicity, take space-time to be a lattice; in particular, x is discrete: x ∈ Z . According to the field equations, φ(x, t + a) + φ(x, t − a) = φ(x − a, t) + φ(x + a, t) .
(7.9)
We would like to map this model one-to-one on the cellular automaton: Q(x, t + a) + Q(x, t − a) = Q(x − a, t) + Q(x + a, t) ,
(7.10)
where Q are integers. a is the mesh size of the space-time lattice, which will be taken to be 1. Applying our earlier methods directly does not work: adding real numbers does not imply exactly that their integer parts add up, since there will be rounding errors: our procedure is non-linear! It does work if real numbers are merely interchanged. This is exactly what happens if one considers left movers and right movers : φ(x, t) = φL (x + t) + φR (x − t) ; p(x, t) = 21 aL (x + t) + 12 aR (x − t) .
(7.11)
aL = p + ∂x φ ;
(7.12)
aR = p − ∂ x φ .
The hamiltonian obeys 2 2 1 π(aL + aR ) , 2 aL (y)] = πi ∂x δ(x − y)
H = π(p2 + (∂x φ)2 ) = [aL , aR ] = 0 ;
[aL (x),
(7.13) ;
(7.14)
If we replace space-time by a lattice: (x, t) ∈ Z . the commutators are replaced by4 i δ 2π x,y
(7.15)
[aL (x), aL (y)] = ± 2πi if y = x ± 1 , 0 otherwise.
(7.16)
[φ(x), p(y)] =
4
The lattice artifacts could be chosen differently, but by redefinitions the commutators can always be recast in the form given here.
13
Now, we replace the hamiltonian of the continuum by a lattice hamiltonian in such a way that its action over one time unit is a pure replacement. Write H = H L + H R . In momentum space : Z 1/2 πκ L 1 H =2 dk aL (k)aL (−k) M (k) ; M (κ) = . (7.17) sin(2πκ) 0 This hamiltonian turns aL (x) into a pure left-mover, and aR (x) into a right-mover. M (κ) shows a κ dependence in such a way that the equations of motion for aL (κ) are no longer modified by lattice artifacts. This is important. Now, we can try to express aL (x) in terms of the left-movers of the cellular automaton: AL (x + t) = Q(x, t + 1) − Q(x − 1, t) Demanding [aL (x), aL (y)] = ± 2πi if y = x ± 1 , and temporarily disregarding the effects of an edge state, we find that one can write, as one can easily check, aL (x) = AL (x) + ηAL (x − 1) .
(7.18)
A better procedure, in η space, minimizes the effect of the edge state, as explained earlier: ∂ L φ(ηA (x + 1), ηAL (x)) − φ(ηAL (x − 1), ηAL (x)) (7.19) aL (x) = AL (x) + L ∂ηA (x) We can now substitute this identity in all operators of the discrete system to see how it maps to the states of the quantum field theory, and vice versa. The edge states in this case are limited to those corners where two consecutive ηAL (x) are ± 12 7.2.
Fermions
Suppose we have left- and right moving boolean variables in our discrete deterministic theory: σL (x + t) = ±1 and σR (x − t) = ±1 .
(7.20)
To map this system onto a system of quantized fermions, introduce operators 1 0 σ3 (x, t) = σ(x, t) = ; (7.21) 0 −1 then we can also define the operators 0 1 σ1 (x, t) = 1 0
;
σ2 (x, t) =
0 −i i 0
,
(7.22)
which allows us to define the fermionic operators by means of the Jordan-Wigner[7] transformation, Y ψ(x, t) = σ1 (x, t) σ3 (y, t) ; {ψ(x, t), ψ(y, t)} = δx,y (7.23) y
