# Probability in many-worlds theories

Anthony J. Short<EMAIL_ADDRESS>H.H. Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, U.K.

###### Abstract

We consider how to define a natural probability distribution over worlds
within a simple class of deterministic many-worlds theories. This can help us
understand the typical properties of worlds within such states, and hence
explain the empirical success of quantum theory within a many-worlds
framework. We give three reasonable axioms which lead to the Born rule in the
case of quantum theory, and also yield natural results in other cases,
including a many-worlds variant of classical stochastic dynamics.

##  1 Introduction

Despite the amazing empirical success of quantum theory, its implications for
the nature of reality remain controversial. From a realist perspective, the
key issue is to replace the measurement postulates of textbook quantum theory
with a more objective and well-defined structure. Supplementing the theory
with hidden variables [1, 2], or including spontaneous collapse laws [3, 4, 5]
are two possible approaches. However, arguably the simplest approach,
initially proposed by Everett [6], is to drop the measurement postulates
altogether. The theory then simply describes a vector in Hilbert space
undergoing unitary evolution. Considering the kinds of unitary interaction
that constitute measurements, we find that they can be described by a
branching of the initial state into a superposition of ‘worlds’ each
containing a different measurement result. This is the many-worlds
interpretation of quantum theory [6, 7]. Observers in each world will see a
distinct result, hence on a qualitative level this is consistent with our
experiences. However, a key challenge is to recover the probabilistic
predictions of quantum theory within this deterministic many-worlds setting.
In particular, we want to explain why the observed relative frequencies of
quantum results in our world, for a huge variety of different scenarios, are
very close to those predicted by the Born rule [8] (i.e. $\textrm{relative
frequency}\approx|\textrm{quantum amplitude}|^{2}$).

The approach we pursue here is to argue for a ‘natural’ way to pick one world
at random from a many-worlds state. i.e. a natural probability distribution
over worlds. We can then consider which worlds are typical with respect to
this probability distribution. If worlds containing relative frequencies in
quantum experiments consistent with the Born rule are typical, this would
offer an explanation for our observations - that we are living in a typical
world, or are a typical instance of ourself. A probability distribution over
worlds can also be interpreted as a measure, allowing one to discuss
properties of ‘most’ or ‘almost all’ worlds. The theory predicts that atypical
worlds also exist, in which other instances of ourself have seen strange
results, and perhaps do not believe quantum theory. However, this seems
comparable to the situation in standard quantum theory, where we explain the
observed measurement results by noting that they are typical with respect to
the Born rule, and concede that it is possible we could have obtained very
strange results instead, and perhaps arrived at a different theory.

We identify three reasonable axioms, not specific to quantum theory, which we
would expect a natural probability distribution over worlds to obey. In the
case of quantum theory, we find that the unique probability distribution
obeying those axioms is that given by the Born rule. If the axioms we require
for naturalness are convincing, this could therefore explain the empirical
success of the Born rule.

As the probabilistic axioms are intended to be general in nature, we consider
a simple class of many-worlds theories which includes other possibilities in
addition to quantum theory. We consider four specific models within this
framework, and highlight any differences in the resulting probability
distributions, as well as one case in which a natural probability distribution
obeying our axioms does not exist.

In order to focus solely on the issue of probability, rather than on precisely
how to decompose the state into different worlds (often known as the preferred
basis problem), we consider a toy-theory in which the set of worlds are given.
Intuitively, we can think of these worlds as states which could be understood
classically on a macroscopic scale (e.g. tables and pointers have well-defined
locations, and observers have well-defined experiences). These states are
important in relating the content of the theory to our experiences, and are
related to the locality of physical interactions and decoherence [9].

Previous approaches to understanding probability in many-worlds include
Everett’s original work [6], which also sought to define a natural measure
over worlds in order to consider a typical observer, although Everett’s
assumptions (that the measure depends only on the corresponding amplitude, and
is additive when several orthogonal states are relabelled as a single state)
are not completely intuitive. Typicality is also considered in a very
different approach based on matter density [10]. Vaidman discusses many
different approaches to deriving the Born rule in [11]. He has proposed the
‘measure of existence of a world’ [12] and notes that deriving the Born rule
requires additional assumptions, that may be based on symmetries [13] or other
natural properties [11]. Zurek [14, 15] uses envariance to derive the Born
rule (which involves symmetries when the state is decomposed into a system and
environment). An alternative approach is based on relative frequencies for
infinitely repeated experiments [6, 16], but this is challenging to relate to
finite experiments. More recently, an approach based on decision theory was
proposed by Deutsch and developed by Wallace and others [17, 18, 19, 20, 21].
Several of the technical steps in our proofs are similar to those in this
decision-theoretic approach. However this approach has led to criticisms [22,
23], and because of its focus on the future rather than the past, it seems
that a decision-theoretic picture alone cannot explain why we find ourselves
in a world in which the historical record of quantum experiments agrees with
the Born rule. Very recently, Saunders has proposed an approach [24] based on
dividing the state into equal-amplitude branches based on the decoherence
structure and following similar counting arguments to those in statistical
mechanics.

##  2 A simple class of many-worlds theories

Consider a Hilbert space with a countably infinite orthonormal basis of
possible ‘worlds’ $|n\rangle$ labelled by non-negative integers,
$n\in\\{0,1,2,\ldots\\}$. The space may be over the real or complex numbers
depending on the particular theory under consideration. A many-worlds state is
a vector $|v\rangle$ in this space. We will refer to $v_{n}=\langle
n|v\rangle$ as the _amplitude_ of world $n$. For simplicity, we restrict
ourselves to _linear many-worlds theories_ for which the evolution over a
finite interval is given by a linear operator $T$, such that

$|v^{\prime}\rangle=T|v\rangle,$ (1)

We will denote the matrix elements of $T$ in the basis of worlds by
$T_{ij}=\langle i|T|j\rangle$. Our aim will be to derive a natural probability
distribution $p_{n}$ over worlds in the state $|v\rangle$. Similarly, we will
denote the probability distribution over worlds in $|v^{\prime}\rangle$ by
$p_{n}^{\prime}$. Different theories will be characterised by giving the set
of allowed state vectors and transformations in the theory (with the
requirement that the allowed transformations must map allowed states into
allowed states). The cases we consider here are:

  1. 1.

Quantum many-worlds theory. The allowed states are all complex vectors
satisfying $\langle v|v\rangle=1$, and the allowed transformations are all
unitary operators (for which $TT^{\dagger}=T^{\dagger}T=I$, where I is the
identity transformation). We will show that in this case $p_{n}=|v_{n}|^{2}$.

  2. 2.

Unnormalised quantum many-worlds theory. An alternative version of the quantum
many-worlds theory with unnormalised states. The allowed states are all
complex vectors satisfying $0<\langle v|v\rangle<\infty$, and the allowed
transformations are all unitary operators. We will show that in this case
$p_{n}=\frac{|v_{n}|^{2}}{\sum_{k}|v_{k}|^{2}}$. Note that in this case the
probability $p_{n}$ corresponding to a given world depends on all amplitudes,
and not only on $v_{n}$.

  3. 3.

Stochastic many-worlds theory This represents a many-worlds version of a
classical probabilistic world. The allowed states are those with real
amplitudes satisfying $v_{n}\geq 0\;\forall n$ and $\sum_{n}v_{n}=1$, and the
allowed transformations are those satisfying $T_{ij}\geq 0\;\forall i,j$ and
$\sum_{i}T_{ij}=1\;\forall j$. Note that although $v_{n}$ and $T_{ij}$ obey
the same mathematical properties as probability distributions and stochastic
maps respectively, we are not assuming that they represent probabilities.
However, we will show that in this case $p_{n}=v_{n}$.

  4. 4.

Discrete many-worlds theory A many-worlds theory in which there are an integer
number of copies of each world, with the dynamics transforming each world in
the original state into a finite number of new worlds. The allowed states are
all real vectors for which $v_{n}$ is a non-negative integer for each $n$ and
$\sum_{n}v_{n}<\infty$, and the allowed dynamics are those for which $T_{ij}$
is a non-negative integer for all $i,j$ and $\sum_{i}T_{ij}<\infty\;\forall
j$. We will show that in this theory there is no natural probability
distribution $p_{n}$ obeying our axioms.

##  3 Natural axioms for a probability measure

For each of the theories considered above, we will investigate whether one can
define a probability distribution over worlds $p_{n}$ obeying the ‘reasonable’
axioms given below.

  1. 1.

_Present state dependence -_ $p_{n}$ depends only on the present state
$|v\rangle$, and not on how that state was generated.

  2. 2.

_Weak connection with amplitudes -_ $v_{n}=0$ implies that $p_{n}=0$. Hence
only worlds with non-zero amplitudes are considered ‘real’ components of the
many-worlds state.

  3. 3.

_Weak connection with transformations -_ If the set of worlds can be
partitioned in such a way that a transformation $T$ acts separately on each
part, then $T$ will preserve the total probability of each part. This captures
the intuition that probability cannot ‘flow’ between worlds that are uncoupled
by the dynamics. Expressing this more formally, if there exists a partition of
the non-negative integers into subsets $\mathcal{S}_{k}$ such that $T_{ij}=0$
whenever $i$ and $j$ are in different subsets, then
$\sum_{n\in\mathcal{S}_{k}}p_{n}=\sum_{n\in\mathcal{S}_{k}}p^{\prime}_{n}$ for
all $k$.

Of these axioms, 1 and 2 both seem relatively uncontroversial, and are usually
assumed without comment in approaches to probability in many-worlds. Axiom 3
is more subtle and powerful, but also seems a natural property in the context
of transformations. We will discuss these axioms further at the end of the
paper. However, we will first show how they can be used to derive probability
measures for the theories given in section 2.

##  4 Deriving probability measures

For ease of understanding, in this section we describe informally how axioms 1
\- 3 allow us to derive probability rules for the many-worlds theories we
consider, highlighting the key steps in the proof. Detailed formal proofs of
all of the results can be found in appendix A.

We first prove two helpful lemmas which we use in deriving the probability
rules, that apply to _bounded_ states (those which contain only a finite
number of worlds with non-zero amplitude)

  1. L1

_Equal amplitudes give equal probabilities_. For bounded states in all of the
theories we consider, if two worlds have equal amplitude then they have the
same probability.

To prove this, we first consider a transformation which swaps two worlds, one
of which has zero amplitude and the other has non-zero amplitude. Due to axiom
3, the total probability of the two swapped worlds must stay the same, and due
to axiom 2 the worlds with zero amplitude must have zero probability. Hence
the probabilities are also swapped.

The same argument doesn’t apply if we swap two worlds with non-zero amplitudes
directly, as probabilities could move between the worlds. However, by
performing a sequence of three swaps, each of which involves one world with
zero amplitude, we can swap any two worlds and their corresponding amplitudes
and probabilities whilst leaving the remaining worlds unchanged. If the
amplitudes of the two swapped worlds were initially the same, then the final
state will be the same as the initial one, but with the probabilities swapped.
From axiom 1 it then follows that the probabilities for these two worlds must
be the same.

  2. L2

_Larger amplitudes cannot lead to smaller probabilities_. For bounded states
in all the theories we consider except discrete many-worlds theory, if one
world has a larger amplitude than another, its probability must be at least as
large.

To prove this, we perform a transformation which branches the higher amplitude
world into two final worlds, one of which has the same amplitude as the
smaller amplitude world. From axiom 3 the sum of the probabilities in the two
branches must be equal to the probability of the initial higher amplitude
world. However, one of the branches has the same probability as the smaller
amplitude world, hence the probability of the higher amplitude world must be
at least as large.

Next, we use these results to derive probability rules for our many-worlds
theories. We begin by considering a particularly simple class of quantum many-
worlds states in which the amplitudes are square roots of rational numbers.

$|v\rangle=\sum_{n=0}^{N-1}\sqrt{\frac{m_{n}}{M}}|n\rangle$ (2)

Following a branching strategy similar to Deutsch and Wallace [17, 20], we
perform a unitary transformation in which each world $|n\rangle$ evolves
independently of the others into $m_{n}$ new worlds with equal amplitude.

Each world in the final state has the same amplitude $\frac{1}{\sqrt{M}}$ and
thus equal probability $\frac{1}{M}$ from L1 above. Applying axiom 3 to the
transformation tells us that the probability in each set of branches is
conserved, hence we find that the probability of world $|n\rangle$ in the
initial state must be $\frac{m_{n}}{M}$, which is the standard quantum result.

To extend this result to arbitrary initial states in quantum many-worlds
theory, we must perform three additional steps. Firstly, if the initial state
contains an infinite number of worlds with non-zero amplitudes, we begin by
transforming it into a superposition of finitely many worlds in order to
create some ‘working space’ in which to apply the above results. For example,
when considering the probability $p_{k}$ we can perform a unitary
transformation that merges all of the worlds with labels greater than $k$ into
a single world, without affecting the other worlds (and hence not changing
$p_{k}$ due to axiom 3). For an arbitrary positive integer $M$, we can then
write the state as

$|v^{\prime}\rangle=\sum_{n=0}^{N-1}\sqrt{\frac{m_{n}+\epsilon_{n}}{M}}e^{i\phi_{n}}|n\rangle,$
(3)

where $m_{n}$ is the largest integer less than or equal to $M|v_{n}|^{2}$, and
$0\leq\epsilon_{n}<1$. Next, we eliminate the phase factors $e^{i\phi_{n}}$ by
applying a unitary which acts separately on each world, and therefore does not
affect the probabilities. Finally, we perform a branching unitary, in which
each initial world $|n\rangle$ is transformed into a superposition of $m_{n}$
new worlds with equal amplitude, and one new world with smaller amplitude. The
final state contains approximately $M$ worlds with equal amplitude
$\frac{1}{\sqrt{M}}$ and at most $N$ worlds with smaller amplitude. When $M\gg
N$, the smaller amplitude worlds are almost irrelevant (due to L2 above), and
the probability associated with world $n$ in the initial state is

$p_{n}\approx\frac{m_{n}}{M}\approx\frac{M|v_{n}|^{2}}{M}=|v_{n}|^{2}.$ (4)

By considering arbitrarily large $M$, this argument can be made exact, giving
the standard quantum probability rule $p_{k}=|v_{k}|^{2}$.

The derivation of the probability rules in unnormalised quantum theory and
stochastic many-worlds theory are very similar. In the former case, the main
difference is that the final state contains approximately $MX$ worlds with
equal amplitude, where $X=\sum_{m}|v_{m}|^{2}$. This gives
$p_{n}\approx\frac{m_{n}}{MX}\approx\frac{|v_{n}|^{2}}{X}$, and yields the
expected quantum probability rule for this case,
$p_{n}=\frac{|v_{n}|^{2}}{\sum_{m}|v_{m}|^{2}}$. In the case of stochastic
many-worlds theory, the steps are identical to those for quantum many-worlds
theory, except without phase factors, square roots, and absolute-values-
squared.

It is also important to note that the probability rules derived so far satisfy
the axioms for all allowed states and transformations, and not only the ones
considered when constructing the proof. For example, if a unitary
transformation is block-diagonal in quantum theory, it commutes with the
projectors onto each block, and hence conserves the total probability of each
block.

Finally, to see that no probability rule obeying our axioms is possible for
Discrete many-worlds theory, consider a transformation on the state
$|0\rangle+|1\rangle$ which takes $|1\rangle\rightarrow|1\rangle+|2\rangle$
whilst leaving all other worlds unchanged. Applying L1 to the initial state
and using axiom 3 gives $p_{0}^{\prime}=\frac{1}{2}$, while applying L1 to the
final state $|0\rangle+|1\rangle+|2\rangle$gives $p_{0}^{\prime}=\frac{1}{3}$.
As this leads to a contradiction, no probability rule obeying the axioms
exists111A similar argument can be used to rule out a ‘naive branch counting’
strategy in quantum many worlds theory, in which each world with non-zero
amplitude is assigned equal probability [25].

##  5 Discussion

In this section we present some additional discussion about the axioms,
theories beyond quantum theory, and decoherence in our approach.

###  5.1 Axioms

  1. 1.

_Present state dependence -_ This is a simplifying axiom, and incorporates the
fact that the state at a given time should be sufficient to make any
substantive claims about it, including the typical properties of worlds within
it. Arguably if historical information is important, it should form part of
the state, and our framework should be extended. Furthermore, this axiom
allows probabilities to be assigned to worlds in an arbitrary initial state,
without needing to know how that state was generated.

  2. 2.

_Weak connection with amplitudes -_ This seems the most compelling
requirement. Without this, one could simply assert that the state is
irrelevant, and $p_{0}=1$ in all cases.

  3. 3.

_Weak connection with transformations -_ This is the most complicated of the
three assumptions, but it is hard to see a weaker way of incorporating a
dependence on the dynamics of the theory. Without this, we could assign an
arbitrary probability distribution over the worlds appearing in every state
(for example we could always assign probability 1 to the world with non-zero
amplitude having the lowest numerical label). Within quantum theory, this also
fits nicely with the continuous time picture in which $T=e^{-iHt}$ for some
Hamiltonian $H$, as $T$ will act separately on each partition if $H$ does.
Note that in terms of the proofs, we only need this axiom to apply to a
specific set of unitaries involving branching, swapping, or merging of worlds.

An alternative to axiom 3, which is a stronger assumption but offers a nice
conceptual picture is:

  * 3’.

_Weak connection with transformations-_ For every state $|v\rangle$ and
transformation $T$ in the theory, there exists a conditional probability
distribution $P_{i|j}$ such that

$p_{n}^{\prime}=\sum_{m}P_{n|m}\,p_{m}$ (5)

satisfying $P_{i|j}=0$ whenever $T_{ij}=0$. This ensures that probability can
only ‘flow’ between states which are linked by the dynamics.

The existence of a conditional probability distribution $P_{i|j}$ for each
transformation of the state supports the idea that living within an evolving
many-worlds state could feel like undergoing a stochastic evolution. There is
also a nice symmetry between the claim that $T_{ij}=0\implies P_{i|j}=0$ and
$v_{n}=0\implies p_{n}=0$.

Axiom 3’ is strictly stronger than the original axiom 3 as it implies it but
is not implied by it. In particular when the worlds can be partitioned into
subsets on which $T$ acts separately, then the conditional probability
distribution $P_{i|j}$ can only redistribute probability within these subsets,
and hence the total probability of each subset is preserved. However, one can
imagine a trivial theory with only one allowed state
$|v\rangle=|1\rangle+|2\rangle$, and one allowed transformation
$T=|1\rangle\langle 2|+|2\rangle\langle 1|$. Under axiom 3 we could assign an
arbitrary probability $p_{1}$ for this state, but with axiom 3’ we must have
$p_{1}=p_{2}=\frac{1}{2}$.

Because axiom 3’ is stronger than axiom 3, we can derive all of the same
results as before (and Lemma 1 now follows directly by permuting the two
worlds in question). However, a more subtle point is whether the probability
rules derived previously actually satisfy axiom 3’ in all cases. For example,
for all states $|v\rangle$ and unitary transformations $T$ in quantum many-
worlds theory with $p_{n}=|v_{n}|^{2}$, can we always find a conditional
probability distribution $P_{i|j}$ satisfying the requirements of axiom 3’?
This is by no means trivial, but it is a consequence of results in [26, 27,
28] (in particular the Flow or Schrödinger Theory presented in [27]) that this
is indeed the case. In general, for a given $|v\rangle$ and $T$ within quantum
theory, there may be many different conditional probability distributions
$P_{i|j}$ satisfying axiom 3’. If conceptual importance is to be given to this
quantity, it may therefore be useful to consider adding additional axioms
which specify it uniquely, or properties which are independent of this choice.
Similar results will apply for unnormalised quantum theory, and in the case of
stochastic many-worlds theory we can simply take $P_{i|j}=T_{ij}$.

Overall, it is interesting that one can overlay a stochastic evolution onto a
many-worlds state with natural properties, and that doing so is helpful in
deriving the Born rule.

###  5.2 Non-quantum many-worlds theories

Recovering a natural probability rule in the case of stochastic many-worlds
theory shows that this approach is not specific to quantum theory.
Furthermore, this theory may also be interesting in its own right. In
particular, it is difficult to make sense of theories involving objective
probabilities at a fundamental level 222Frequentism only gives a definite
prediction for infinitely many trials, which is irrelevant for practical
situations, and the principal principle [29] seems somewhat unsatisfying.
Kent’s incompressible bit string provides another alternative approach [22].
This result suggests an interesting alternative possibility of treating
objective probabilities as amplitudes in a many-worlds state. The fact that we
find ourselves in a world which is typical with respect to the objective
probabilities, and that these are helpful in subjective decision making, would
then be a consequence of the natural probability distribution over worlds
matching the amplitudes.

For discrete many world theory there is no natural probability distribution
over worlds obeying our axioms. Two alternatives for understanding
probabilities within this theory are to violate axiom 1 or axiom 3. In the
former case, we could assume some initial probability distribution, and then
update this probability distribution during transformations according to

$p_{n}^{\prime}=\sum_{m}\left(\frac{T_{nm}}{\sum_{n^{\prime}}T_{n^{\prime}m}}\right)p_{m}.$
(6)

This would lead to the same natural probability distribution as a Stochastic
many-worlds theory for which the transition matrix elements $\tilde{T}_{nm}$
are given by the bracketed expression in (6). This would lead to a natural
flow of probability between worlds (obeying axiom 3), but the same state could
yield many different probability distributions according to how it was
generated. In the latter case, we could instead drop axiom 3, and take

$p_{n}=\frac{v_{n}}{\sum_{n}^{\prime}v_{n}}$ (7)

This is a function of only the current state (obeying axiom 1), but the
probability of a world can change even when the transformation acts on it like
the identity.

###  5.3 Decoherence

Note that this derivation of the Born rule does not rely on decoherence, and
indeed employs transformations such as permutations of worlds which would be
essentially impossible to achieve in practice. However, such transformations
are possible in principle, and it is helpful to use the full strength of the
theory to generate constraints upon possible probability rules. The two
particular instances where this is helpful are in proving Lemma 1, and when
compressing states with support over all worlds in order to generate ‘working
space’. An alternative for the former is to take Lemma 1 directly as an
additional axiom, perhaps motivated by symmetry, but the internal structure of
the two worlds may look very different, and it seems quite strong to assume
this is irrelevant in determining $p_{n}$. The latter could possibly be
eliminated by adding a continuity axiom, but then one would have to choose a
particular distance measure on states. Decoherence also plays a key role in
explaining ‘collapse’ in realistic situations, as it becomes practically
impossible to re-interfere macroscopically distinct states.

##  6 Conclusions

If reality has a deterministic many-worlds structure, as in the Everett
interpretation, then there are no objective probabilities in the theory. In
this case, how should we understand the fact that we are living in a world in
which the relative frequencies of outcomes in past quantum experiments are
very close to the probabilities predicted by the Born rule? Although it is
consistent to say that this is a mere coincidence (as a world with such
results must exist), it would be good to give a deeper explanation of this
fact.

The approach pursued here is to establish a natural way of picking a world at
random from the many-worlds state, and then observing that with very high
probability such randomly chosen worlds have the property that they agree with
the Born rule. i.e. that this is a typical property of the worlds. The key is
to motivate such a natural way of picking worlds at random. One apparently
natural way (at least for states containing a finite number of worlds) is to
simply assign equal weight to each world with non-zero amplitude, but this
ignores some of the mathematical structure in the state, and it violates one
of our reasonable axioms (axiom 3). Other problems with this strategy have
been discussed in [20].

It seems impossible to give a completely compelling set of requirements which
a probability distribution over worlds must obey, but we have defined three
natural axioms which are sufficient to recover the Born rule in the context of
quantum theory, and also give an appealing result for classical stochastic
theory. These axioms are that probabilities: only depend on the current state,
are zero for worlds which appear with zero amplitude, and cannot flow between
sets of worlds which are uncoupled by the dynamics. It would be interesting to
explore alternative possible requirements, and would also be good to
reconsider issues relating to the choice of ‘world’ basis in this context.

Are results such as these sufficient to explain the empirical success of the
Born rule? An interesting perspective is to consider a universe in which the
Everett interpretation is true, described by a single unitarily evolving
quantum state. Under reasonable conditions, such a state could be described by
a superposition of branching worlds, many of which contain structures which
look like people. What would it be like to live as one of those people in such
a universe? If it is like our own experiences then this supports the many-
worlds interpretation. If it is very different then this would rule it out.
The strangest situation would be if we cannot in principle say what it is
like, given that we know the correct mathematical theory describing the
universe.

Acknowledgments. This work was supported by an FQXi “physics of what happens”
grant. The author is grateful to A.Kent, J.Barrett, and Ll. Massanes for
helpful discussions.

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##  Appendix A Detailed proof of probabilities

We begin by proving two useful lemmas which apply to probability distributions
obeying our axioms for the many-worlds theories we consider. These lemmas
focus on _bounded_ states, which have a finite number of non-zero amplitudes
(such that $v_{n}=0$ for $n\geq N$), but the final theorems will apply to any
state.

We first show that for any theory in which all permutations are allowed
transformations (which includes all of the theories in this paper), any two
worlds with the same amplitude have the same probability.

######  Lemma 1 (equal amplitudes).

Consider a theory in which all permutations of worlds are allowed
transformations ($T$ is a permutation of worlds if $T_{ij}=\delta_{i,\pi[j]}$
where $\pi$ is a bijection of the non-negative integers), and a state
$|v\rangle=\sum_{n=0}^{N-1}v_{n}|n\rangle$ for which $v_{n}=v_{m}$. Then
$p_{n}=p_{m}$.

###### Proof.

Denote by $T^{n\leftrightarrow m}$ the transformation which swaps worlds $n$
and $m$ and leaves all other worlds unchanged ($T^{n\leftrightarrow
m}=|n\rangle\langle m|+|m\rangle\langle
n|+\sum_{k\notin\\{n,m\\}}|k\rangle\langle k|$). Hence in this case
$T^{n\leftrightarrow m}|v\rangle=|v\rangle$. Directly applying axiom 3 to this
transformation is not sufficient, as this doesn’t tell us how probabilities
change inside the $n,m$ subspace. Instead, we note that $v_{N}=0$, and that
$T^{n\leftrightarrow m}=T^{n\leftrightarrow N}T^{n\leftrightarrow
m}T^{N\leftrightarrow m}$. For the first transformation $T^{n\leftrightarrow
N}$, consider a partition of worlds in which $n$ and $N$ are in one part, and
each other world is in its own part. Due to axiom 3, we know that
$p_{n}+p_{N}=p^{\prime}_{n}+p^{\prime}_{N}$, and that $p_{k}=p^{\prime}_{k}$
for all $k\notin\\{n,m\\}$. Furthermore, from axiom 2 and the fact that
$v_{N}=v^{\prime}_{n}=0$, we know that $p_{N}=p^{\prime}_{n}=0$. Hence
$p^{\prime}_{N}=p_{n}$ and $T^{n\leftrightarrow N}$ permutes the corresponding
probabilities. Following a similar logic for the two subsequent
transformations $T^{n\leftrightarrow m}$ and $T^{N\leftrightarrow m}$ (each of
which permutes two amplitudes, one of which is zero), we find that the
probabilities $p^{\prime}_{n}$ after the sequence $T^{n\leftrightarrow
N}T^{n\leftrightarrow m}T^{N\leftrightarrow m}$are the permutation of the
original probabilities by $T^{n\leftrightarrow m}$, hence
$p_{n}^{\prime}=p_{m}$. However, due to axiom 1 and the fact that the initial
state is the same as the final state in this case, $p_{n}^{\prime}=p_{n}$.
Hence $p_{n}=p_{m}$. ∎

Our second lemma shows that, in all except the discrete many-worlds theory, if
one world has larger amplitudes than another it cannot have a smaller
probability. This is useful in deriving the probability rule for cases where
the amplitudes are not related to rational numbers in a convenient way.

######  Lemma 2 (larger amplitudes cannot lead to smaller probabilities).

Consider a state $|v\rangle=\sum_{n=0}^{N-1}v_{n}|n\rangle$ in quantum many-
worlds theory, unnormalised quantum many-worlds theory, or stochastic many-
worlds theory for which $|v_{l}|>|v_{k}|$. Then $p_{l}\geq p_{k}$.

###### Proof.

For quantum many-worlds theory, or unnormalised quantum many-worlds theory,
consider the unitary evolution which acts in the $\\{|l\rangle,|N\rangle\\}$
subspace as

$\displaystyle T|l\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{v_{k}}{v_{l}}\right)|l\rangle+\sqrt{1-\left|\frac{v_{k}}{v_{l}}\right|^{2}}|N\rangle$
$\displaystyle T|N\rangle$ $\displaystyle=$
$\displaystyle\sqrt{1-\left|\frac{v_{k}}{v_{l}}\right|^{2}}|l\rangle-\left(\frac{v_{k}}{v_{l}}\right)|N\rangle,$
(8)

and satisfies $T|m\rangle=|m\rangle$ if $m\notin\\{l,N\\}$. For stochastic
many-worlds theory consider a stochastic evolution which acts on $|l\rangle$
as

$T|l\rangle=\left(\frac{v_{k}}{v_{l}}\right)|l\rangle+\left(1-\frac{v_{k}}{v_{l}}\right)|N\rangle$
(9)

and satisfies $T|m\rangle=|m\rangle$ if $m\neq l$. In both cases this means
that for $|v^{\prime}\rangle=T|v\rangle$ , $v_{l}^{\prime}=v_{k}^{\prime}$ and
hence $p_{l}^{\prime}=p_{k}^{\prime}$ from Lemma 1. From axiom 3 we have that
$p_{N}^{\prime}+p_{l}^{\prime}=p_{N}+p_{l}$ and $p_{k}^{\prime}=p_{k}$, and
from axiom 2 we have $p_{N}=0$. Hence

$p_{l}=p_{l}^{\prime}+p_{N}^{\prime}=p_{k}^{\prime}+p_{N}^{\prime}=p_{k}+p_{N}^{\prime}\geq
p_{k}.$ (10)

∎

Using the above lemmas and the probability rule axioms, we now derive the
appropriate probability rules for our many-worlds theories (or in the case of
discrete many-worlds theory show that this is impossible). To illustrate the
key idea, we first derive the probability rule for quantum many-worlds theory
in the case in which all amplitudes are square roots of rational numbers,
using a similar branching strategy to Deutsch and Wallace [17, 20] (although
without any decision theoretic component).

######  Theorem 1 (probabilities when $v_{n}$ are square-roots of rational
numbers).

Consider a state of the form
$|v\rangle=\sum_{n=0}^{N-1}\sqrt{\frac{m_{n}}{M}}|n\rangle$ in quantum many-
worlds theory, where $m_{n}$ and $M$ are positive integers. Then
$p_{n}=\frac{m_{n}}{M}$.

###### Proof.

To prove this result, we branch each world in the original state into $m_{n}$
new worlds in the final state. Consider a partition of the non-negative
integers into the sets

$\mathcal{S}_{k}=\left\\{\begin{array}[]{cl}\\{k\\}\cup\mathcal{L}_{k}&\textrm{for
$0\leq k<N$}\\\ \\{k\\}&\textrm{for $k\geq N+MN$}\end{array}\right..$ (11)

where

$\mathcal{L}_{k}=\\{N+Mk,N+Mk+1,\ldots,N+Mk+(M-1)\\}$ (12)

Now consider a unitary $T$ whose matrix elements $T_{ij}$ are nonzero only
when $i$ and $j$ are in the same subset $\mathcal{S}_{k}$, and which has the
property that

$T|n\rangle=\frac{1}{\sqrt{m_{n}}}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle\qquad\textrm{for
$0\leq n<N$}.$ (13)

Then

$T|v\rangle=\frac{1}{\sqrt{M}}\sum_{n=0}^{N-1}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle.$
(14)

As this is a superposition of worlds with equal amplitude it follows from
Lemma 1 that the probability assigned to each world in this state must be
identical. Hence $p_{m}^{\prime}=\frac{1}{M}$ for each world in this state.
From axiom 3 and equation (13) it then follows that $p_{n}=\frac{m_{n}}{N}$. ∎

Next, we use Lemma 2 to extend this result to the general case of arbitrary
states.

######  Theorem 2 (general quantum probability rule).

For an arbitrary state $|v\rangle=\sum_{n}v_{n}|n\rangle$ in quantum many-
worlds theory, $p_{n}=|v_{n}|^{2}$.

###### Proof.

We first transform the initial state (which may have infinitely many non-zero
components) into a bounded state to create some working space. Consider a
particular probability $p_{k}$. In order to determine this, we first perform a
unitary $T$ with the property that

$\displaystyle T|n\rangle$ $\displaystyle=|n\rangle$ if $n\leq k$
$\displaystyle
T\left(\frac{\sum_{n=k+1}^{\infty}v_{n}|n\rangle}{\sqrt{\sum_{m=k+1}^{\infty}|v_{m}|^{2}}}\right)$
$\displaystyle=|k+1\rangle$ , (15)

which acts separately on each of the worlds 0 to $k$, and collectively on the
rest (merging them into a single world). This gives a bounded state containing
$N=k+2$ worlds with the property that $p_{k}^{\prime}=p_{k}$ from axiom 3.
Next, we pick a large non-negative integer $M$, and write the transformed
state $|v^{\prime}\rangle$ as

$|v^{\prime}\rangle=\sum_{n=0}^{N-1}\sqrt{\frac{m_{n}+\epsilon_{n}}{M}}e^{i\phi_{n}}|n\rangle$
(16)

in which $m_{n}=\left\lfloor M|v_{n}|^{2}\right\rfloor$ and
$\epsilon_{n}=M|v_{n}|^{2}-\left\lfloor M|v_{n}|^{2}\right\rfloor$, where
$\left\lfloor x\right\rfloor$ is the floor function (the greatest integer less
than or equal to $x$). Hence each $m_{n}$ is an integer and
$0\leq\epsilon_{n}<1$.

We then apply the unitary transformation
$T=\sum_{n}e^{-i\phi_{n}}|n\rangle\langle n|$ to remove the phase factors from
the state, yielding

$|v^{\prime\prime}\rangle=\sum_{n=0}^{N-1}\sqrt{\frac{m_{n}+\epsilon_{n}}{M}}|n\rangle.$
(17)

As this unitary acts separately on each world, it follows from axiom 3 that
$p_{k}^{\prime\prime}=p_{k}$. Similarly to the previous case, we then perform
a transformation which branches each world in the original state into $m_{n}$
or $m_{n}+1$ worlds in the final state (depending on whether $\epsilon_{n}=0$)
via a unitary acting on the same partition of worlds $\mathcal{S}_{k}$ as in
theorem 1 with the property that

$T|n\rangle=\frac{1}{\sqrt{m_{n}+\epsilon_{n}}}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle\\\
\quad+\sqrt{\epsilon_{n}}|N+nM+m_{n}\rangle\end{array}\right)$ (18)

for $0\leq n<N$. The final state is then

$T|v^{\prime\prime\prime}\rangle=\frac{1}{\sqrt{M}}\sum_{n=0}^{N-1}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle\\\
\quad+\sqrt{\epsilon_{n}}|N+nM+m_{n}\rangle\end{array}\right).$ (19)

This is a superposition of at most $M$ worlds of equal amplitude
$\frac{1}{\sqrt{M}}$ and at most $N$ worlds with smaller amplitude. Given
lemma 1 and lemma 2, if we denote the probability of one of the amplitude
$\frac{1}{\sqrt{M}}$ worlds by $\delta$ and consider the total probability it
follows that $\delta\geq\frac{1}{M+N}$. From axiom 3 it therefore follows that

$\displaystyle p_{k}$ $\displaystyle\geq m_{k}\delta$
$\displaystyle\geq\frac{m_{k}}{M+N}$
$\displaystyle\geq\frac{M|v_{k}|^{2}-1}{M+N}$
$\displaystyle\geq|v_{k}|^{2}-\frac{N|v_{k}|^{2}+1}{M+N}.$ (20)

As this holds for arbitrarily large $M$ (for fixed $N$ and $v_{k}$) it must be
the case that $p_{k}\geq|v_{k}|^{2}$. A similar approach could be employed to
upper bound $p_{k}$, but it is simpler to consider the total normalisation of
the state $\sum_{n}|v_{n}|^{2}=1=\sum_{n}p_{n}$. As $p_{k}\geq|v_{k}|^{2}$ for
each $k$, the only possible solution is $p_{k}=|v_{k}|^{2}$ for all $k$. ∎

This proof can be extended very straightforwardly to the case of unnormalised
quantum many-worlds theory, and stochastic many-worlds theory.

######  Theorem 3 (unnormalised quantum probability rule).

For an arbitrary state $|v\rangle=\sum_{n}v_{n}|n\rangle$ in unnormalised
quantum many-worlds theory, $p_{n}=\frac{|v_{n}|^{2}}{\sum_{m}|v_{m}|^{2}}$.

###### Proof.

Note that Lemmas 1 \- 2 apply identically in unnormalised quantum many-worlds
theory. The proof of Theorem 3 is the same as that for Theorem 2 up until
(17). Defining $X=\sum_{m}|v_{m}|^{2}$ and $M^{\prime}=\left\lceil
MX\right\rceil$, where $\left\lceil x\right\rceil$ is the smallest integer
greater than or equal to $x$, we then define sets $\mathcal{S}^{\prime}_{k}$
via

$\mathcal{S}^{\prime}_{k}=\left\\{\begin{array}[]{cl}\\{k\\}\cup\mathcal{L}_{k}&\textrm{for
$0\leq k<N$}\\\ \\{k\\}&\textrm{for $k\geq N+M^{\prime}N$}\end{array}\right..$
(21)

where

$\mathcal{L}^{\prime}_{k}=\\{N+M^{\prime}k,N+M^{\prime}k+1,\ldots,N+M^{\prime}k+(M^{\prime}-1)\\}.$
(22)

Applying a unitary which acts on the partition $\mathcal{S}^{\prime}_{k}$ with
the property that

$T|n\rangle=\frac{1}{\sqrt{m_{n}+\epsilon_{n}}}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM^{\prime}+l\rangle\\\
\quad+\sqrt{\epsilon_{n}}|N+nM^{\prime}+m_{n}\rangle\end{array}\right)$ (23)

for $0\leq n<N$. The final state is then

$T|v^{\prime\prime\prime}\rangle=\frac{1}{\sqrt{M}}\sum_{n=0}^{N-1}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM^{\prime}+l\rangle\\\
\quad+\sqrt{\epsilon_{n}}|N+nM^{\prime}+m_{n}\rangle\end{array}\right).$ (24)

this state is a sum of at most $MX$ worlds with equal amplitude
$\frac{1}{\sqrt{M}}$ and at most $N$ worlds with smaller amplitude. If we
denote the probability of one of the amplitude $\frac{1}{\sqrt{M}}$ worlds by
$\delta$ as before, then it follows from lemma 1 and lemma 2 that
$\delta\geq\frac{1}{MX+N}$. Hence from axiom 3,

$\displaystyle p_{k}$ $\displaystyle\geq m_{k}\delta$
$\displaystyle\geq\frac{m_{k}}{XM+N}$
$\displaystyle\geq\frac{M|v_{k}|^{2}-1}{XM+N}$
$\displaystyle\geq\frac{|v_{k}|^{2}}{X}-\frac{\frac{N}{X}|v_{k}|^{2}+1}{MX+N}.$
(25)

As this holds for arbitrarily large $M$ (for fixed $N$ and $|v\rangle$) it
must be the case that $p_{k}\geq\frac{|v_{k}|^{2}}{X}$. However, as this
applies to each $k$, given the total normalisation of the probabilities, the
only possible solution is
$p_{k}=\frac{|v_{k}|^{2}}{X}=\frac{|v_{n}|^{2}}{\sum_{m}|v_{m}|^{2}}$ for all
$k$. ∎

######  Theorem 4 (stochastic probability rule).

For an arbitrary state $|v\rangle=\sum_{n}v_{n}|n\rangle$ in stochastic many-
worlds theory, $p_{n}=v_{n}$.

###### Proof.

We first transform the initial state into a bounded state to create some
working space. In order to determine a particular probability $p_{k}$, we
first perform a transformation $T$ given by

$T|n\rangle=\left\\{\begin{array}[]{cl}|n\rangle&\textrm{if $0\leq n\leq
k$}\\\ |k+1\rangle&\textrm{if $n>k$}\end{array}\right.$ (26)

which acts separately on each of the worlds 0 to $k$, and collectively on the
rest (merging them into a single world). This gives a bounded state containing
$N=k+2$ worlds with the property that $p_{k}^{\prime}=p_{k}$ from axiom 3.
Next, we pick a large non-negative integer $M$, and write the transformed
state $|v^{\prime}\rangle$ as

$|v^{\prime}\rangle=\sum_{n=0}^{N-1}\frac{m_{n}+\epsilon_{n}}{M}|n\rangle$
(27)

in which $m_{n}=\left\lfloor Mv_{n}\right\rfloor$ and
$\epsilon_{n}=Mv_{n}-\left\lfloor Mv_{n}\right\rfloor$, where $\left\lfloor
x\right\rfloor$ is the floor function. Hence each $m_{n}$ is an integer and
$0\leq\epsilon_{n}<1$. We then perform a transformation which branches each
world in the original state into $m_{n}$ or $m_{n}+1$ worlds in the final
state (depending on whether $\epsilon_{n}=0$) via a stochastic transformation
acting on the partition of worlds $\mathcal{S}_{k}$ as in theorem 1,

$\mathcal{S}_{k}=\left\\{\begin{array}[]{cl}\\{k\\}\cup\mathcal{L}_{k}&\textrm{for
$0\leq k<N$}\\\ \\{k\\}&\textrm{for $k\geq N+M$}\end{array}\right..$ (28)

where

$\mathcal{L}_{k}=\\{N+Mk,N+Mk+1,\ldots,N+Mk+(M-1)\\}$ (29)

with the property that

$T|n\rangle=\frac{1}{m_{n}+\epsilon_{n}}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle\\\
\quad+\epsilon_{n}|N+nM+m_{n}\rangle\end{array}\right)$ (30)

for $0\leq n<N$. The final state is then

$T|v^{\prime}\rangle=\frac{1}{M}\sum_{n=0}^{N-1}\left(\begin{array}[]{c}\sum_{l=0}^{m_{n}-1}|N+nM+l\rangle\\\
\quad+\epsilon_{n}|N+nM+m_{n}\rangle\end{array}\right).$ (31)

This is a superposition of at most $M$ worlds of equal amplitude $\frac{1}{M}$
and at most $N$ worlds with smaller amplitude. Given lemma 1 and lemma 2, if
we denote the probability of one of the amplitude $\frac{1}{M}$ worlds by
$\delta$ and consider the total probability it follows that
$\delta\geq\frac{1}{M+N}$. From axiom 3 it therefore follows that

$\displaystyle p_{k}$ $\displaystyle\geq m_{k}\delta$
$\displaystyle\geq\frac{m_{k}}{M+N}$ $\displaystyle\geq\frac{Mv_{k}-1}{M+N}$
$\displaystyle\geq v_{k}-\frac{Nv_{k}+1}{M+N}.$ (32)

As this holds for arbitrarily large $M$ (for fixed $N$ and $v_{k}$) it must be
the case that $p_{k}\geq v_{k}$. However, as this apples to each $k$, given
the total normalisation of the state $\sum_{n}v_{n}=1=\sum_{n}p_{n}$, the only
possible solution is $p_{k}=v_{k}$ for all $k$. ∎

Finally, we show that discrete many-worlds theory admits no probability
measure consistent with the axioms.

######  Theorem 5 (discrete theory admits no quantum probability).

There is no probability rule consistent with the axioms for discrete
probability theory

###### Proof.

Consider the initial state $|v\rangle=|0\rangle+|1\rangle$ and the
transformation

$\displaystyle T|1\rangle$ $\displaystyle=|1\rangle+|2\rangle$ $\displaystyle
T|n\rangle$ $\displaystyle=|n\rangle\qquad\textrm{if $n\neq 1$},$ (33)

giving

$T|v\rangle=|0\rangle+|1\rangle+|2\rangle$ (34)

This transformation is consistent with a partitioning of the worlds such that
worlds $1$ and $2$ are in one part, and each other world is in its own part.
Hence from axiom 3, we have $p_{0}=p_{0}^{\prime}$, and
$p_{1}^{\prime}+p_{2}^{\prime}=p_{1}$. However, discrete many-worlds theory
admits all permutations as allowed transformations, so Lemma 1 also applies
and worlds with equal amplitude must have equal probability. Hence
$p_{0}=p_{1}=\frac{1}{2}$ and
$p_{0}^{\prime}=p_{1}^{\prime}=p_{2}^{\prime}=\frac{1}{3}$. This creates a
contradiction with $p_{0}=p_{0}^{\prime}$, hence no probability rule
satisfying the axioms exists. ∎


# Elementary equivalence of Kac-Moody groups

Jun Morita  Jun Morita, Division of Mathematics, Faculty of Pure and Applied
Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JAPAN
<EMAIL_ADDRESS>and  Eugene Plotkin  Eugene Plotkin, Department of
Mathematics, Bar-Ilan University, 5290002 Ramat Gan, ISRAEL
<EMAIL_ADDRESS>
###### Abstract.

The paper is devoted to model-theoretic properties of Kac-Moody groups with
the focus on elementary equivalence of Kac-Moody groups. We show that
elementary equivalence of (untwisted) affine Kac-Moody groups implies
coincidence of their generalized Cartan matrices and the elementary
equivalence of their ground fields. We also show that elementary equivalence
of arbitrary Kac-Moody groups over finite fields implies coincidence of these
fields and an isomorphism of their twin root data. The similar result is
established for Kac-Moody groups defined over infinite subfields of the
algebraic closures of finite fields.

Keywords: Kac–Moody groups, Chevalley groups, elementary equivalence of
groups, generalized Cartan matrix, twin root data.

MSC: 20G44, 03C20

##  1\. Introduction. Elementary equivalence

Importance of logical classifications of algebraic structures goes back to the
famous works of A.Tarski and A.Malcev. Here the term “algebraic structure”or
just “algebra”means a set with operations, see [19]. The main problem is to
figure out what are the algebras logically equivalent to a given one. We
discuss this problem from the perspectives of Kac-Moody groups.

Given an algebra $H$, its elementary theory $Th(H)$ is the set of all
sentences (closed formulas) valid on $H$.

######  Definition 1.

Two groups $H_{1}$ and $H_{2}$ are said to be elementarily equivalent if their
elementary theories coincide.

Very often we fix a class of algebras $\mathcal{C}$ and ask what are the
algebras elementarily equivalent to a given algebra inside the class
$\mathcal{C}$.

Last years a lot of attention was concentrated around the elementary
equivalence questions for linear groups. It is a kind of folklore that a group
elementarily equivalent to a linear group is linear. However, the question
what is the group elementarily equivalent to a given linear group, is far from
being trivial

Throughout the paper we will denote the elementary equivalent objects $A_{1}$
and $A_{2}$ by $A_{1}\equiv A_{2}$ and isomorphic objects are denoted by
$A_{1}\simeq A_{2}$.

Recall that the classical Keisler-Shelah’s isomorphism theorem states, in
particular, that two algebras $A$ and $B$ are elementary equivalent if and
only if there is an ultrafilter such that the corresponding ultrapowers of $A$
and $B$ are isomorphic [9], [4].

Let $G$ be a group from a class of groups $\mathcal{C}$. In many cases one can
fix the cardinality of groups from $\mathcal{C}$. We call $G$ elementarily
rigid, if for any group $H$ in $\mathcal{C}$ the equivalence $G\equiv H$
implies isomorphism $G\simeq H$.

So, the rigidity question with respect to elementary equivalence looks as
follows.

######  Problem 2.

Let a class of algebras $\mathcal{C}$ and an algebra $H\in\mathcal{C}$ be
given. Suppose that the elementary theories of algebras $H$ and
$A\in\mathcal{C}$ coincide. Are they elementarily rigid, that is, are $H$ and
$A$ isomorphic?

In general, elementary rigidity of groups is a rather rare phenomenon.
However, studying the elementary properties of linear groups resulted in
various examples of such kind. Historically, the first results were obtained
in [20], [31]. Recently, a series of papers on elementary equivalence of
linear groups with the emphasis on Chevalley groups and arithmetic lattices
has been published, see [5], [1], [2], [29], [27]. Most of the results state a
kind of elementary rigidity of these groups with respect to this or that class
of groups $\mathcal{C}$. The major example is the class of finitely generated
groups, cf. [27], [1].

In the paper we focus our attention on the similar problems of elementary
equivalence for Kac-Moody groups and obtain a sort of rigidity results.

##  2\. Affine Kac-Moody groups

Kac-Moody groups can be viewed as infinite dimensional analogs of Chevalley
groups (see [28]). One can say that they are in the same relation with respect
to infinite-dimensional Kac-Moody Lie algebras (see [13],[11], [22]) as
Chevalley groups are related to semi-simple finite-dimensional Lie algebras.
It is useful to emphasize that from the geometric point of view the passage
from Chevalley groups to Kac-Moody groups means the passage from linear
algebraic groups to ind-algebraic groups (see [26]). We refer to monographs
[18], [21] for the detailed exposition of the above said.

Let $\widetilde{G}(K)$ be an (untwisted) affine Kac-Moody group over the field
$K$. We want to prove that if two affine Kac-Moody groups
$\widetilde{G}_{1}(K_{1})$ and $\widetilde{G}_{2}(K_{2})$ are elementary
equivalent then their generalized Cartan matrices coincide and the fields
$K_{1}$ and $K_{2}$ are elementary equivalent.

Let $A$ be an $n\times n$ indecomposable generalized Cartan matrix of
(untwisted) affine type. By an (untwisted) affine Kac-Moody group we mean the
value of the simply connected Tits functor [30], cf.[25], [14],[15],
corresponding to $A$. So, one can write
$\widetilde{G}(K)=\widetilde{G}_{sc}(A,K)$. Let $R$ be the ring of Laurent
polynomials $R=K[t,t^{-1}]$, where $K$ be a field. Denote by $G_{sc}(R)\simeq
G_{\Phi}(R)$ the simply connected Chevalley group over $R$, where $\Phi$ is a
finite root system, associated with $A$. The adjoint Chevalley group is
denoted by $G_{ad}^{\Phi}(R)$. As usual, the elementary subgroups of Chevalley
groups are denoted by $E_{sc}^{\Phi}(R)$ and $E_{ad}^{\Phi}(R)$.

Denote by $\widetilde{G}_{ad}(A,K)$ the adjoint Kac-Moody group
([23],[10],[30]). Its elementary subgroup is $\widetilde{E}_{ad}(A,K)$. Denote
by $Z(\widetilde{G}_{sc}(K))$ and $Z(\widetilde{E}_{sc}(K))$ the centers of
$\widetilde{G}_{sc}(K)=\widetilde{G}_{sc}(A,K)$ and
$\widetilde{E}_{sc}(K)=\widetilde{E}_{sc}(A,K)$, respectively.

We have the following global picture:

$\begin{array}[]{ccccccccc}&&&&1&&1&&\\\ &&&&\uparrow&&\uparrow&&\\\
&&1&\rightarrow&K^{\times}&\simeq&K^{\times}&\rightarrow&1\\\
&&\uparrow&&\uparrow&&\uparrow&&\\\
1&\rightarrow&Z(\tilde{G}_{sc}(K))&\rightarrow&\tilde{G}_{sc}(K)&\rightarrow&H(R)&\rightarrow&1\\\
&&||&&\uparrow&&\uparrow&&\\\
1&\rightarrow&Z(\tilde{E}_{sc}(K))&\rightarrow&\tilde{E}_{sc}(K)&\rightarrow&G_{ad}^{\Phi}(R)&\rightarrow&1\\\
&&\uparrow&&\uparrow&&\uparrow&&\\\ &&1&&1&&1&&\end{array}$

Here, by definition $H(R)$ is $\widetilde{G}_{ad}(A,K)$ and:

$\tilde{G}_{sc}(K)/Z(\tilde{G}_{sc}(K))\simeq H(R):=\left\langle
G_{ad}^{\Phi}(R),\ \zeta(u)\mid u\in K^{\times}\right\rangle\subset{\rm
Aut}_{K}(\mathfrak{g}(\dot{A})\otimes R),$

where $G_{ad}^{\Phi}(R)=G_{sc}^{\Phi}(R)/Z(G_{sc}^{\Phi}(R))$ is the
corresponding adjoint Chevalley group, and $\zeta(u)\in{\rm Aut}(R)$ is
defined by $t^{\pm 1}\mapsto u^{\pm 1}t^{\pm 1}$. Also here,
$\mathfrak{g}(\dot{A})\otimes R$ is the corresponding loop algebra, and
$\dot{A}$ is a Cartan matrix associated with $A$.

Then, there is a surjective homomorphism $\pi:\widetilde{G}_{sc}(A,K)\to
G_{\Phi}(R)$. Its kernel $Ker\pi$ is isomorphic to $K^{\times}$. We have,
$Z(\widetilde{G}_{sc}(A,K))=Z(\widetilde{E}_{sc}(A,K))$ and

$1\to Z(\widetilde{E}_{sc}(A,K))\to\widetilde{E}_{sc}(A,K)\to
G_{ad}^{\Phi}(R)\to 1.$

Then,

$\widetilde{E}_{sc}(A,K)/Z(\widetilde{E}_{sc}(A,K))\simeq
G_{ad}^{\Phi}(R)\simeq E_{ad}^{\Phi}(R).$

The group $G_{ad}^{\Phi}(R)\simeq E_{ad}^{\Phi}(R)=E_{ad}^{\Phi}(K[t,t^{-1}])$
is used to call loop group [10]. The following commutator formulas are useful.
Assume $|K|\geq 4$. Then

$[\widetilde{G}_{sc}(A,K),\widetilde{G}_{sc}(A,K)]=\widetilde{E}_{sc}(A,K),\quad[\widetilde{E}_{sc}(A,K),\widetilde{E}_{sc}(A,K)]=\widetilde{E}_{sc}(A,K).$

In the similar manner

$[\widetilde{G}_{ad}(A,K),\widetilde{G}_{ad}(A,K)]=\widetilde{E}_{ad}(A,K),\quad[\widetilde{E}_{ad}(A,K),\widetilde{E}_{ad}(A,K)]=\widetilde{E}_{ad}(A,K),$

and $\widetilde{E}_{ad}(A,K)$ is isomorphic to the loop group.

Given a group $G$ a subgroup $H$ is called definable if there is a first order
formula $\phi$ such that $a\in H$, where $a\in G$ if and only if $a$ satisfies
a formula $\phi$.

######  Theorem 3.

Let $\widetilde{E}_{1_{sc}}(A_{1},K_{1})$ and
$\widetilde{E}_{2_{sc}}(A_{2},K_{2})$ be two elementary (simply connected)
affine Kac-Moody groups whose ranks are bigger than 2. Suppose that
$\widetilde{E}_{1_{sc}}(A_{1},K_{1})\equiv\widetilde{E}_{2_{sc}}(A_{2},K_{2})$.
Then $A_{1}=A_{2}$ and $K_{1}\equiv K_{2}$.

###### Proof.

Suppose that
$\widetilde{E}_{1_{sc}}(A_{1},K_{1})\equiv\widetilde{E}_{2_{sc}}(A_{2},K_{2})$.
Since center is a definable subgroup the group
$\widetilde{E}_{sc}(A,K)/Z(\widetilde{E}_{sc}(A,K))$ is interpretable in
$\widetilde{E}_{sc}(A,K)$ (see, for example, [17], page 131). Hence we have

$\widetilde{E}_{1_{sc}}(A_{1},K_{1})/Z(\widetilde{E}_{sc}(A_{1},K_{1}))\equiv\widetilde{E}_{2_{sc}}(A_{2},K_{2})/Z(\widetilde{E}_{sc}(A_{2},K_{2}))$

since both groups are interpretable in the elementary equivalent groups by the
same first order formulas. That is
$\widetilde{E}_{1_{ad}}(A_{1},K_{1})\equiv\widetilde{E}_{2_{ad}}(A_{2},K_{2})$
and according to the isomorphism with Chevalley groups
$E_{1_{ad}}^{\Phi}(R_{1})\equiv E_{2_{ad}}^{\Psi}(R_{2})$ .

Hence,

$E^{\Phi}_{ad}(K_{1}[t,t^{-1}])\equiv E^{\Psi}_{ad}(K_{2}[t,t^{-1}]).$

So, by Bunina (see [5], Theorem 11) we have $\Phi=\Psi$ and

$K_{1}[t,t^{-1}]\equiv K_{2}[t,t^{-1}].$

The Laurent polynomial ring $K[t,t^{-1}]$ is isomorphic to the group ring of
the group $\mathbb{Z}$ over the field $K$. Then by Theorem 2 of [16],

$K_{1}\equiv K_{2}.$

Hence, $\Phi=\Psi$, $K_{1}\equiv K_{2}$ as required.

∎

The converse statement to Theorem 3 is not true.

######  Proposition 4.

There exist fields $K_{1}$ and $K_{2}$ such that $K_{1}\equiv K_{2}$, but the
affine Kac-Moody groups $\widetilde{E}_{1_{sc}}(A,K_{1})$ and
$\widetilde{E}_{2_{sc}}(A,K_{2})$ are not elementary equivalent.

###### Proof.

Let $K_{1}=\mathbb{Q}$ be the field of rational numbers and let $K_{2}$ be
some its ultrapower. So these fields are elementary equivalent, i.e.,
$K_{1}\equiv K_{2}$. However, the polynomial rings $K_{1}[t]$ and $K_{2}[t]$
are not elementary equivalent (see Corollary 3.8 in [12]). Then the rings of
Laurent polynomials $K_{1}[t,t^{-1}]$ and $K_{2}[t,t^{-1}]$ are not elementary
equivalent as well, (see Lemma 2, [16]). Then, $E_{\Phi}(K_{1}[t,t^{-1}])$ is
not elementarily equivalent to $E_{\Phi}(K_{2}[t,t^{-1}])$ by [5]. Since
$\widetilde{E}_{ad}(A,K_{i})\simeq E^{\Phi}_{ad}(K_{i}[t,t^{-1}])$ the Kac-
Moody groups $\widetilde{E}_{1_{sc}}(A,K_{1})$ and
$\widetilde{E}_{2_{sc}}(A,K_{2})$ are not elementarily equivalent.

∎

######  Proposition 5.

Let $K_{1}$ and $K_{2}$ be algebraically closed fields of characteristic zero.
Then the loop groups $\widetilde{E}_{1_{ad}}(A_{1},K_{1})$ and
$\widetilde{E}_{2_{ad}}(A_{2},K_{2})$ of rank bigger than one are elementarily
equivalent if and only if $A_{1}=A_{2}$, $tr.deg_{\mathbb{Q}}(K_{1})$ and
$tr.deg_{\mathbb{Q}}(K_{2})$ either both infinite or both finite and equal.
Here $tr.deg_{\mathbb{Q}}(K)$ stands for the transcendence degree of the field
$K$ over $\mathbb{Q}$.

Since algebraically closed fields are elementarily equivalent if and only if
they have the same characteristic this means that for elementarily equivalent
zero characteristic fields
$\widetilde{E}_{1}(A_{1},K_{1})\equiv\widetilde{E}_{2}(A_{2},K_{2})$ if and
only if $A_{1}=A_{2}$, $tr.deg_{\mathbb{Q}}(K_{1})$ and
$tr.deg_{\mathbb{Q}}(K_{2})$ either both infinite or both finite and
$tr.deg_{\mathbb{Q}}(K_{1})=tr.deg_{\mathbb{Q}}(K_{2})$.

The proof of Proposition 5 follows from the proof of Theorem 3 and the
following

######  Lemma 6.

Let $K_{1}$ and $K_{2}$ be algebraically closed fields of characteristic zero.
Then $K_{1}[t,t^{-1}]\equiv K_{2}[t,t^{-1}]$ if and only if
$tr.deg_{\mathbb{Q}}(K_{1})$ and $tr.deg_{\mathbb{Q}}(K_{2})$ either both
infinite or both finite and
$tr.deg_{\mathbb{Q}}(K_{1})=tr.deg_{\mathbb{Q}}(K_{2})$.

###### Proof.

Let $K_{1}[t,t^{-1}]\equiv K_{2}[t,t^{-1}]$. Then $K_{1}[t]\equiv K_{2}[t]$
(see Lemma 2, [16]). Then $tr.deg_{\mathbb{Q}}(K_{1})$ and
$tr.deg_{\mathbb{Q}}(K_{2})$ either both infinite or both finite and
$tr.deg_{\mathbb{Q}}(K_{1})=tr.deg_{\mathbb{Q}}(K_{2})$, (see Corollary 3.11,
[12]). Note, that this direction is valid for arbitrary infinite fields.

Let $tr.deg_{\mathbb{Q}}(K_{1})$ and $tr.deg_{\mathbb{Q}}(K_{2})$ either both
infinite or both finite and

$tr.deg_{\mathbb{Q}}(K_{1})=tr.deg_{\mathbb{Q}}(K_{2}).$

Then Proposition 3.19 of [12] remains true for Laurent polynomials and implies
$K_{1}[t,t^{-1}]\equiv K_{2}[t,t^{-1}]$.

∎

######  Question 7.

Does Proposition 5 still valid if we replace
$\widetilde{E}_{i_{ad}}(A_{i},K_{i})$ by
$\widetilde{E}_{i_{sc}}(A_{i},K_{i})$, in it.

##  3\.  Elementary Equivalence for Kac-Moody groups, I

Let $A$ be an $n\times n$ indecomposable generalized Cartan matrix of non-
spherical type, and $\tilde{E}_{sc}(A,K)$ the simply connected elementary Kac-
Moody group over a finite field $K$. Let $\Omega=\\{\,x_{\pm\alpha_{i}}(r)\mid
1\leq i\leq n,\ r\in K\\}$ where $x_{\pm\alpha_{i}}(r)$ are root elements of
root subgroups corresponding to simple roots. Then
$\tilde{E}_{sc}(A,K)=\langle\,\Omega\,\rangle$. For $g\in\tilde{E}_{sc}(A,K)$,
we set $\ell(g)=\mbox{min}\\{\,l\\!\mid g=g_{1}g_{2}\cdots g_{l},\
g_{k}\in\Omega\\}$. Note that $\Omega$ is a finite set, since $|\Omega|\leq
2n\\!\cdot\\!|K|$. Let $\mathcal{F}$ be a non-principal ultra-filter over the
set $J$ and ${\prod}_{\mathcal{F}}\tilde{E}_{sc}(A,K)$ the ultrapower of
$\tilde{E}_{sc}(A,K)$. Elements of ${\prod}_{\mathcal{F}}\tilde{E}_{sc}(A,K)$
will be denoted by $[x_{j}]_{\mathcal{F}}$, where $x_{j}$ lies in
$\tilde{E}_{sc}(A,K)$, or just by columns indexed by $\mathcal{F}$. Define the
subgroup $X$ of ${\prod}_{\mathcal{F}}\tilde{E}_{sc}(A,K)$ by the following
rule:

$X={\prod}_{\mathcal{F}}^{\flat}\tilde{E}_{sc}(A,K):=\left\\{\left.\left[\begin{array}[]{c}\vdots\\\
x_{j}\\\ \vdots\end{array}\right]_{\mathcal{F}}\ \right|\
x_{j}\in\tilde{E}_{sc}(A,K),\ \exists L\geq 0\ \mbox{s.t.}\ \forall j\in J,\
\ell(x_{j})\leq L\right\\}$

Here, ${\prod}_{\mathcal{F}}^{\flat}$ is the notation for the restricted
ultraproduct, defined by the formula above, where

$\left.\left[\begin{array}[]{c}\vdots\\\ x_{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\ \right|\ x_{j}\in\tilde{E}_{sc}(A,K),$

is an element of ${\prod}_{\mathcal{F}}\tilde{E}_{sc}(A,K)$.

Given a number $L\geq 0$ let

$X_{L}:=\left\\{\left.\left[\begin{array}[]{c}\vdots\\\ x_{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\in X\ \right|\ \forall j\in J,\
\ell(x_{j})\leq L\right\\},\ \mbox{then}\ X=\bigcup_{L\geq 0}X_{L}.$

We also introduce
$A^{\prime},K^{\prime},\Omega^{\prime},\ell^{\prime},X^{\prime}$ and
$X_{L^{\prime}}^{\prime}$ etc. for $\widetilde{E}_{sc}(A^{\prime},K^{\prime})$
similarly. Then, we obtain the following.

######  Theorem 8.

Let $A,A^{\prime}$ be indecomposable generalized Cartan matrices of non-
spherical type, and let $K,K^{\prime}$ be finite fields. Suppose
$\widetilde{E}_{sc}(A,K)\equiv\widetilde{E}_{sc}(A^{\prime},K^{\prime})$.
Then, the twin root data of $A$ and $A^{\prime}$ coincide, that is
$\mbox{TRD}(A)\simeq\mbox{TRD}(A^{\prime})$, and $K\simeq
K^{\prime}\simeq\mathbb{F}_{q}$. Here, TRD means the twin root datum.

###### Proof.

. Since
$\widetilde{E}_{sc}(A,K)\equiv\widetilde{E}_{sc}(A^{\prime},K^{\prime})$, then
by Keisler-Shelah’s isomorphism theorem there is an isomorphism $\phi$ such
that

$\phi:{\prod}_{\mathcal{F}}\widetilde{E}_{sc}(A,K)\overset{\sim}{\rightarrow}{\prod}_{\mathcal{F}}\widetilde{E}_{sc}(A^{\prime},K^{\prime}),$

where $\mathcal{F}$ is a non-principal ultra-filter on $J$, see [9]. We note
here that since $\Omega$ and $\Omega^{\prime}$ are finite $\phi(X_{L})\subset
X_{LM}^{\prime}\ \mbox{and}\ \phi^{-1}(X_{L^{\prime}}^{\prime})\subset
X_{L^{\prime}M^{\prime}}$ if
$M=\mbox{max}\\{\ell^{\prime}\big{(}\phi(\omega)\big{)}\mid\omega\in\Omega\\}\
\mbox{and}\
M^{\prime}=\mbox{max}\\{\ell\big{(}\phi^{-1}(\omega^{\prime})\big{)}\mid\omega^{\prime}\in\Omega^{\prime}\\}$.
Therefore we obtain

$\phi\big{(}{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\big{)}\subset{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime}),\quad{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\supset\phi^{-1}\big{(}{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})\big{)}.$

That is, we have
$\phi\big{(}\prod_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\big{)}={\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})$.
This means that there is an isomorphism

$\phi^{\prime}=\phi|_{\prod_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)}$

such that

$\phi^{\prime}:{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\overset{\sim}{\rightarrow}{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime}).$

Let
$\mu:\tilde{E}_{sc}(A,{\prod}_{\mathcal{F}}K)\longrightarrow{\prod}_{\mathcal{F}}\tilde{E}_{sc}(A,K)$
be a natural map defined by

$g=x_{\beta_{1}}(\left[\begin{array}[]{c}\vdots\\\ r_{1}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}})\cdots
x_{\beta_{m}}(\left[\begin{array}[]{c}\vdots\\\ r_{m}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}})\overset{\mu}{\mapsto}\mu(g)=\left[\begin{array}[]{c}\vdots\\\
x_{\beta_{1}}(r_{1}^{j})\cdots x_{\beta_{m}}(r_{m}^{j})\\\
\vdots\end{array}\right]_{\mathcal{F}}\ $

for $\displaystyle{\left[\begin{array}[]{c}\vdots\\\ r_{i}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\in}$ ${\prod}_{\mathcal{F}}K$,
$\beta_{k}=\pm\alpha_{i}$ for some $i$, and $r_{i}^{j}\in K$. Here, we note
that $\mu$ is well-defined

(cf. [5], [24]). Let us check that $\mu$ is a monomorphism. Suppose
$\displaystyle{\mu(g)=1=\left[\begin{array}[]{c}\vdots\\\ 1\\\
\vdots\end{array}\right]_{\mathcal{F}}}$, and show that $g=1$.

If we put $\displaystyle{U=\left\\{j\in J\mid x_{\beta_{1}}(r_{1}^{j})\cdots
x_{\beta_{m}}(r_{m}^{j})=1\right\\}}$, then $U\in\mathcal{F}$. We define

$\displaystyle{s_{i}^{j}=\left\\{\begin{array}[]{lll}r_{i}^{j}&\mbox{if}&j\in
U;\\\\[5.69046pt] 0&\mbox{if}&j\not\in U.\end{array}\right.}$ Then, we see
$\displaystyle{U\subset U_{i}:=\left\\{j\in J\mid
s_{i}^{j}=r_{i}^{j}\right\\}}$ and $U_{i}\in\mathcal{F}$ for each $i$. Hence,
we obtain $\displaystyle{\left[\begin{array}[]{c}\vdots\\\ r_{i}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}=\left[\begin{array}[]{c}\vdots\\\
s_{i}^{j}\\\ \vdots\end{array}\right]_{\mathcal{F}}}$ in
${\prod}_{\mathcal{F}}K$ and

$g=x_{\beta_{1}}(\left[\begin{array}[]{c}\vdots\\\ r_{1}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}})\cdots
x_{\beta_{m}}(\left[\begin{array}[]{c}\vdots\\\ r_{m}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}})=x_{\beta_{1}}(\left[\begin{array}[]{c}\vdots\\\
s_{1}^{j}\\\ \vdots\end{array}\right]_{\mathcal{F}})\cdots
x_{\beta_{m}}(\left[\begin{array}[]{c}\vdots\\\ s_{m}^{j}\\\
\vdots\end{array}\right]_{\mathcal{F}})=1,$

On the other hand, we put $N=|\Omega|$ and set
$\Omega=\\{\omega_{1},\ldots,\omega_{N}\\}$. We need to check that $X$
coincides with the image of $\mu$. Let $g\in X_{L}$. Then, write
$g=g_{1}\cdots g_{L}$. We need to subdivide $g_{k}$ as follows:

$g_{k}=\left[\begin{array}[]{c}\vdots\\\ x_{jk}\\\
\vdots\end{array}\right]_{\mathcal{F}}=h_{k}^{(1)}\cdots h_{k}^{(i)}\cdots
h_{k}^{(N)},\quad h_{k}^{(i)}=\left[\begin{array}[]{c}\vdots\\\
h_{jk}^{(i)}\\\ \vdots\end{array}\right]_{\mathcal{F}},\quad
h_{jk}^{(i)}=\omega_{i}^{\delta(\omega_{i},x_{jk})},$

where $\delta(\omega_{i},x_{jk})=\delta_{\omega_{i},x_{jk}}$ (Kronecker delta)
and $x_{jk}\in\Omega$. Since both $\omega_{i}$ and $x_{jk}$ belong to
$\Omega$, we define $\delta(\omega_{i},x_{jk})=1$ if $\omega_{i}=x_{jk}$ and 0
otherwise.

By our definition, we note that if we write $w_{i}=x_{\beta_{i}}(r_{i})$, then

$h_{k}^{(i)}=\left[\begin{array}[]{c}\vdots\\\ \vdots\\\ x_{jk}\\\ \vdots\\\
\vdots\end{array}\right]_{\mathcal{F}}=\left[\begin{array}[]{c}\vdots\\\
w_{i}\\\ \vdots\\\ 1\\\
\vdots\end{array}\right]_{\mathcal{F}}=\left[\begin{array}[]{c}\vdots\\\
x_{\beta_{i}}(r_{i})\\\ \vdots\\\ 1\\\
\vdots\end{array}\right]_{\mathcal{F}}=x_{\beta_{i}}(\left[\begin{array}[]{c}\vdots\\\
r_{i}\\\ \vdots\\\ 0\\\ \vdots\end{array}\right]_{\mathcal{F}})$

Hence, we have $g_{k}\in\mu(\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K))$ and
$g\in\mu(\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K))$, which shows
$X_{L}\subset\mu(\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K))$. Another
inclusion $\mu(\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K))\subset X$ can be
obtained easily. In fact, let
$\dot{g}\in\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K)$ and we write
$\dot{g}=\dot{g}_{1}\cdots\dot{g}_{L}$ such that $L=\ell(\dot{g})$ and

$\dot{g}_{k}\in\dot{\Omega}:=\left\\{x_{\pm\alpha_{i}}(r)\left|1\leq i\leq n,\
r=\left[\begin{array}[]{c}\vdots\\\ r_{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\in{\prod}_{\mathcal{F}}K^{\prime}\right.\right\\}\quad(1\leq
k\leq L).$

Then, we see $\mu(\dot{g})\in X_{L}\subset X$ and
$\mu(\widetilde{E}_{sc}(A,\prod_{\mathcal{F}}K))\subset X$. Hence, we have

$X_{L}\subset\mu(\widetilde{E}_{sc}\left(A,{\prod}_{\mathcal{F}}K\right))\subset
X\quad\mbox{and}\quad
X=\mu(\widetilde{E}_{sc}\left(A,{\prod}_{\mathcal{F}}K\right)).$

Similarly we can show
$X_{L^{\prime}}^{\prime}\subset\mu(\widetilde{E}_{sc}(A^{\prime},\prod_{\mathcal{F}}K^{\prime}))\subset
X^{\prime}$ and
$X^{\prime}=\mu(\widetilde{E}_{sc}(A^{\prime},\prod_{\mathcal{F}}K^{\prime}))$.

We have just proved

$\widetilde{E}_{sc}(A,{\prod}_{\mathcal{F}}K)\simeq{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K),\qquad{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})\simeq\widetilde{E}_{sc}(A^{\prime},{\prod}_{\mathcal{F}}K^{\prime})$

and

$\widetilde{E}_{sc}(A,{\prod}_{\mathcal{F}}K)\simeq{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\simeq{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})\simeq\widetilde{E}_{sc}(A^{\prime},{\prod}_{\mathcal{F}}K^{\prime}).$

Then, by [6], Theorem 4.1, p.27, (cf. [7], [8]), we reach

${\rm TRD}(A)\simeq{\rm TRD}(A^{\prime})\ \mbox{and}\
{\prod}_{\mathcal{F}}K\simeq{\prod}_{\mathcal{F}}K^{\prime}.$

We obtain $K\equiv K^{\prime}$ and $K\simeq K^{\prime}\simeq\mathbb{F}_{q}$.∎

######  Corollary 9.

If $q\not=q^{\prime}$, then
$\widetilde{E}_{sc}(A,\mathbb{F}_{q})\not\equiv\widetilde{E}_{sc}(A^{\prime},\mathbb{F}_{q^{\prime}})$.

######  Remark 10.

It looks plausible that the ground field can be interpreted in the Kac-Moody
group. At least the finitness of the ground field and its characteristic can
be first order detected by means of [6], Proposition 4.16. We are thankful to
anonymous referee for this observation.

##  4\.  Elementary Equivalence for Kac-Moody groups, II

Let $A$ be an $n\times n$ indecomposable generalized Cartan matrix of non-
spherical type, and $\widetilde{E}_{sc}(A,K)$ the simply connected elementary
Kac-Moody group over an infinite subfield $K$ of the algebraic closure
$\overline{\mathbb{F}}_{p}$ of a finite prime field $\mathbb{F}_{p}$.

We will basically follow the streamline of Section 3. But, for the sake of
completeness we present the full proof for these fields as well.

It is well known [3] that every $K$ can be presented as the union of nested
finite fields $K=\bigcup_{\lambda\in\Lambda}K_{\lambda}$ for some sequence of
numbers $\Lambda$, where $K_{\lambda}=\mathbb{F}_{p^{\lambda}}$. Let
$\Omega_{\lambda}=\\{\,x_{\pm\alpha_{i}}(r)\mid 1\leq i\leq n,\ r\in
K_{\lambda}\\}$ and $\Omega=\bigcup_{\lambda\in\Lambda}\Omega_{\lambda}$, then
$\widetilde{E}_{sc}(A,K_{\lambda})=\langle\,\Omega_{\lambda}\,\rangle$ and
$\widetilde{E}_{sc}(A,K)=\langle\,\Omega\,\rangle=\bigcup_{\lambda\in\Lambda}\widetilde{E}_{sc}(A,K_{\lambda})$.

For $g\in\widetilde{E}_{sc}(A,K)$, we set $\ell(g)=\mbox{min}\\{\,l\\!\mid
g=g_{1}g_{2}\cdots g_{l},\ g_{k}\in\Omega\\}$. And also, for
$g\in\widetilde{E}_{sc}(A,K_{\lambda})$, we set
$\ell_{\lambda}(g)=\mbox{min}\\{\,l\\!\mid g=g_{1}g_{2}\cdots g_{l},\
g_{k}\in\Omega_{\lambda}\\}$. Note that $\Omega_{\lambda}$ is a finite set,
since $|\Omega_{\lambda}|\leq 2n\\!\cdot\\!|K_{\lambda}|$.

Similar to previous Section, put

$Y={\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)$
$:=\left\\{\left.\left[\begin{array}[]{c}\vdots\\\ y_{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\ \right|\exists\lambda\in\Lambda,\
\exists L\geq 0,\ \mbox{s.t.}\ \forall j\in J,\
y_{j}\in\widetilde{E}_{sc}(A,K_{\lambda}),\ \ell_{\lambda}(y_{j})\leq
L\right\\}$

and define

$Y_{L,\lambda}:=\left\\{\left.\left[\begin{array}[]{c}\vdots\\\ y_{j}\\\
\vdots\end{array}\right]_{\mathcal{F}}\in Y\ \right|\ \forall j\in J,\
y_{j}\in\widetilde{E}_{sc}(A,K_{\lambda}),\ \ell_{\lambda}(y_{j})\leq
L\right\\},\ \mbox{then}\ Y=\bigcup_{L,\lambda}Y_{L,\lambda}.$

We also introduce
$A^{\prime},K^{\prime},\Omega^{\prime},\Omega_{\lambda^{\prime}}^{\prime},\ell^{\prime},\ell_{\lambda^{\prime}},Y^{\prime}$
and $Y_{L^{\prime},\lambda^{\prime}}^{\prime}$ etc. for
$\widetilde{E}_{sc}(A^{\prime},K^{\prime})$ similarly. Then, we obtain the
following.

######  Theorem 11.

Let $A,A^{\prime}$ be indecomposable generalized Cartan matrices of non-
spherical type, and let $K,K^{\prime}$ be infinite subfields of the algebraic
closures of finite fields. Suppose
$\widetilde{E}_{sc}(A,K)\equiv\widetilde{E}_{sc}(A^{\prime},K^{\prime})$.
Then, $\mbox{TRD}(A)\simeq\mbox{TRD}(A^{\prime})$,
$\mbox{char}(K)=\mbox{char}(K^{\prime})$ and $K\equiv K^{\prime}$.

###### Proof.

Since
$\widetilde{E}_{sc}(A,K)\equiv\widetilde{E}_{sc}(A^{\prime},K^{\prime})$,
there is an isomorphism $\phi$ such that

$\phi:{\prod}_{\mathcal{F}}\widetilde{E}_{sc}(A,K)\overset{\sim}{\rightarrow}{\prod}_{\mathcal{F}}\widetilde{E}_{sc}(A^{\prime},K^{\prime}),$

where $\mathcal{F}$ is a non-principal ultra-filter on $J$.

We note here

$\phi(g)\in\phi(Y_{L,\,\lambda})\subset Y_{LM,\,\lambda^{\prime}}^{\prime}$

if $g\in Y_{L,\,\lambda}$ and

$M=\mbox{max}\\{\ell_{\lambda^{\prime}}^{\prime}\big{(}\phi(\omega)\big{)}\mid\omega\in\Omega_{\lambda}\\}$

for some $\lambda^{\prime}$ satisfying
$\phi(g)\in\widetilde{E}_{sc}(A^{\prime},K_{\lambda^{\prime}}^{\prime})$ and
$\phi(\Omega_{\lambda})\subset\widetilde{E}_{sc}(A^{\prime},K_{\lambda^{\prime}}^{\prime})$.
We also note

$\phi^{-1}(g^{\prime})\in\phi^{-1}(Y_{L^{\prime},\,\mu^{\prime}}^{\prime})\subset
Y_{L^{\prime}M^{\prime},\,\mu}$

if $g^{\prime}\in Y_{L^{\prime},\,\mu^{\prime}}^{\prime}$ and

$M^{\prime}=\mbox{max}\\{\ell_{\mu}\big{(}\phi^{-1}(\omega^{\prime})\big{)}\mid\omega^{\prime}\in\Omega_{\mu^{\prime}}^{\prime}\\}$

for some $\mu$ satisfying
$\phi^{-1}(g^{\prime})\in\widetilde{E}_{sc}(A,K_{\mu})$ and
$\phi^{-1}(\Omega_{\mu^{\prime}}^{\prime})\subset\widetilde{E}_{sc}(A,K_{\mu})$.
Therefore we obtain

$\phi\left({\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\right)\subset{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime}),\quad{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\supset\phi^{-1}\left({\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})\right).$

That is, we have
$\phi\big{(}\prod_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\big{)}={\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})$.
This means that there is an isomorphism

$\phi^{\prime}=\phi|_{\prod_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)}$

such that

$\phi^{\prime}:{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\overset{\sim}{\rightarrow}{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime}).$

Hence, we obtain

$\widetilde{E}_{sc}\left(A,\bigcup_{\lambda\in\Lambda}\left({\prod}_{\mathcal{F}}K_{\lambda}\right)\right)={\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A,K)\qquad\
$ $\
\qquad\simeq{\prod}_{\mathcal{F}}^{\flat}\widetilde{E}_{sc}(A^{\prime},K^{\prime})=\widetilde{E}_{sc}\left(A^{\prime},\bigcup_{\lambda^{\prime}\in\Lambda^{\prime}}\left({\prod}_{\mathcal{F}}K_{\lambda^{\prime}}^{\prime}\right)\right).$

Since $K_{\lambda}$ is finite, ${\prod}_{\mathcal{F}}K_{\lambda}\simeq
K_{\lambda}$. Hence
$\bigcup_{\lambda\in\Lambda}\left({\prod}_{\mathcal{F}}K_{\lambda}\right)\simeq\bigcup_{\lambda\in\Lambda}K_{\lambda}\simeq
K.$ Therefore,
$\widetilde{E}_{sc}(A,K)\simeq\widetilde{E}_{sc}(A^{\prime},K^{\prime})$.

Then, by [6], Theorem 4.1, p.27, we reach

${\rm TRD}(A)\simeq{\rm TRD}(A^{\prime})\quad{\rm and}\quad K\simeq
K^{\prime}.$

$\Box$

Corollary. Let $A,A^{\prime}$ be indecomposable generalized Cartan matrices of
non-spherical type. Suppose

$\widetilde{E}_{sc}(A,\overline{\mathbb{F}_{p}}\,)\equiv\widetilde{E}_{sc}(A^{\prime},\overline{\mathbb{F}_{p^{\prime}}}\,).$

Then, $\mbox{TRD}(A)\simeq\mbox{TRD}(A^{\prime})$ and $p=p^{\prime}$. ∎

In particular, we obtain

$\widetilde{E}_{sc}(A,\overline{\mathbb{F}_{p}}\,)\not\equiv\widetilde{E}_{sc}(A^{\prime},\overline{\mathbb{F}_{p^{\prime}}}\,)$

if $p\not=p^{\prime}$.

Acknowledgements. J. Morita was partially supported by the Grants-in-Aid for
Scientific Research of Japan (Grant No. 26400005). E.Plotkin was supported by
ISF grants 1994/20, 1623/16 and the Emmy Noether Research Institute for
Mathematics. We are very grateful to B.Kunyavskii for careful reading of the
final version of the manuscript and important remarks. Our special thanks go
to anonymous referee for the numerous valuable corrections.

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