QUANTUM $\bm{\mathbb{F}_{\mathrm{un}}}$ :
THE $q=1$ LIMIT OF GALOIS FIELD QUANTUM MECHANICS, PROJECTIVE GEOMETRY & THE FIELD WITH ONE ELEMENT

Since the details of GQM can be found in Refs. \refciteChang:2012eh and \refciteChang:2012gg, here we only present the basic outline.

Consider the vector space $\mathbb{F}_{q}^{N}$. Vectors in $\mathbb{F}_{q}^{N}$ represent states of the model system, while dual vectors in the dual vector space $\mathbb{F}_{q}^{N*}$ represent possible outcomes of measurements. The probability of obtaining the outcome represented by the dual-vector $\langle x|\in\mathbb{F}_{q}^{N*}$ when a measurement is performed on the state represented by the vector $|\psi\rangle\in\mathbb{F}_{q}^{N}$ is given by

where the sum in the denominator runs over all the dual vectors $\langle y|$ in a basis of $\mathbb{F}_{q}^{N*}$ which includes $\langle x|$. The choice of basis of $\mathbb{F}_{q}^{N*}$ corresponds to an observable, each dual vector in the basis representing a different outcome.

The absolute value function in the above expression converts elements of $\mathbb{F}_{q}$ into either zero or one in $\mathbb{R}$:

Here, underlined numbers and symbols represent elements of $\mathbb{F}_{q}$, to distinguish them from elements of $\mathbb{R}$ or $\mathbb{C}$. Since $\mathbb{F}_{q}\backslash\{\underline{0}\}$ is a cyclic multiplicative group, this assignment of ‘absolute values’ is the only one consistent with the requirement that the map from $\mathbb{F}_{q}$ to non-negative $\mathbb{R}$ be product preserving, that is: $|\underline{k}\underline{l}|=|\underline{k}||\underline{l}|$.¹⁰¹⁰10The product preserving nature of the absolute value function guarantees that the probabilities of product observables on product states factorize in multi-particle systems. This property is crucial if we want to have isolated particle states. Since the same absolute value is assigned to all non-zero brackets, all outcomes $\langle x|$ for which the bracket with the state $|\psi\rangle$ is non-zero are given equal probabilistic weight.

Note also that the multiplication of $|\psi\rangle$ with a non-zero element of $\mathbb{F}_{q}$ will not affect the probability. Thus, vectors that differ by non-zero multiplicative constants are identified as representing the same physical state, and the state space is endowed with the finite projective geometry [89, 90, 91, 92]

where each ‘line’ going through the origin of $\mathbb{F}_{q}^{N}$ is identified as a ‘point,’ in close analogy to the complex projective geometry $\mathbb{C}P^{N-1}$ of canonical $N$-level QM defined on $\mathbb{C}^{N}$.

To give a concrete example of our proposal, let us construct a 2-level system, analogous to spin, on $\mathbb{F}_{q}^{2}$ for which the state space is $PG(1,q)$. This geometry consists of $q+1$ ‘points,’ which can be represented by the vectors

$r=2,3,\cdots,q$, where $\underline{a}$ is the generator of the multiplicative group $\mathbb{F}_{q}\backslash\{\underline{0}\}$ with $\underline{a}^{q-1}=1$. The number $q+1$ results from the fact that of the $q^{2}-1$ non-zero vectors in $\mathbb{F}_{q}^{2}$, every $q-1$ are equivalent, thus the number of inequivalent vectors is $(q^{2}-1)/(q-1)=(q+1)$. Similarly, the $q+1$ inequivalent dual-vectors can be represented as:

where the minus signs are dropped when the characteristic of $\mathbb{F}_{q}$ is two.¹¹¹¹11The ‘characteristic’ of a field is the smallest non-negative integer $m$ such that $\underbrace{\underline{1}+\underline{1}+\cdots+\underline{1}}_{m\;\mathrm{times}}=\underline{0}$, where $\underline{1}$ is the multiplicative unit and $\underline{0}$ is the additive unit. For example, the characteristic of $\mathbb{F}_{q}$ with $q=p^{n}$ is the prime $p$. The characteristics of $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are defined to be zero. From these definitions, we find:

and

Observables are associated with a choice of basis of $\mathbb{F}_{q}^{2*}$:

We assign the outcome $+1$ to the first dual-vector of the pair, and the outcome $-1$ to the second to make these observables spin-like. This assignment implies $A_{sr}=-A_{rs}$. The indices $rs$ can be considered as indicating the direction of the ‘spin,’ and the interchange of the indices as indicating a reversal of this direction.

Applying Eq. (2) to this system, it is straightforward to show that

and thus,

So for each ‘spin’ $A_{rs}$ there exist two ‘eigenstates’: $|s\rangle$ for $+1$ (‘spin’ up) and $|r\rangle$ for $-1$ (‘spin’ down). For all other states the two outcomes $\pm 1$ are equally probable.

A two-‘spin’ system can be constructed on the tensor product space $\mathbb{F}_{q}^{2}\otimes\mathbb{F}_{q}^{2}=\mathbb{F}_{q}^{4}$. The number of non-zero vectors in this space is $q^{4}-1$, of which every $q-1$ are equivalent, so the number of inequivalent states is $(q^{4}-1)/(q-1)=q^{3}+q^{2}+q+1$. Of these, $(q+1)^{2}$ are product states, leaving $(q^{3}+q^{2}+q+1)-(q+1)^{2}=q(q^{2}-1)$ that are entangled. Of these, there exists an analog of the spin-singlet state given by

for any two states $|r\rangle$ and $|s\rangle$ up to a multiplicative constant. If the characteristic of $\mathbb{F}_{q}$ is two, the minus sign is replaced by a plus sign.

Products of the ‘spin’ observables are defined as

the four tensor products representing the outcomes $++$, $+-$, $-+$, and $--$, and the expectation value giving the correlation between the two ‘spins.’ The probabilities of the four outcomes are particularly easy to calculate for the state $|S\rangle$ since [42]

thus

and we obtain the probabilities and correlations listed in Table. 1. It is straightforward to show that these probabilities cannot be reproduced by any classical hidden variable theory.[40, 41] Thus, GQM is ‘quantum’ in this sense.

It should be noted, though, that GQM also has a common feature with CM when we look at the Clauser-Horne-Shimony-Holt (CHSH) version of Bell’s inequality.[93, 94] The CHSH bound[95] is the upper bound of the absolute value of the following combination of correlators:

where $A$ and $a$ are two observables of particle 1, and $B$ and $b$ are two observables of particle 2. All four observables are assumed to take on only the values $\pm 1$ upon measurement. For classical hidden variable theory the bound on $\left|\langle A,a;B,b\rangle\right|$ is 2,[95] while for canonical QM it is $2\sqrt{2}$.[96, 97] ¹²¹²12The largest possible value of the CHSH bound is 4.[98] See Ref. \refciteChang:2012we for a model which saturates this bound. Despite GQM not allowing a hidden variable minic, we can nevertheless show, again using Table 1, that its CHSH bound is the ‘classical’ value of 2. This bound is independent of the value of $q$ chosen for our Galois field $\mathbb{F}_{q}$ and is thus also independent of the size of the vector space $\mathbb{F}_{q}^{2}\otimes\mathbb{F}_{q}^{2}=\mathbb{F}_{q}^{4}$.[40, 41] So the limitation on the correlations is not due to the limited number of ‘spin’ direction available in the model. This GQM example shows that the absence of hidden variable mimics does not necessarily guarantee the violation of the classical CHSH bound.

The Galois fields $\mathbb{F}_{q}$ are only defined for $q=p^{n}$, that is, the order $q$ must be a power of a prime $p$. Thus, setting $q=1$ is illegitimate from the Galois field point of view. Indeed, $\mathbb{F}_{q}$ consists of $q$ elements so taking the naive $q=1$ limit, one expects $\mathbb{F}_{1}$ to consist of only one element (which could be denoted by either $\underline{0}$ or $\underline{1}$) with no distinction between addition and multiplication. Such an object is obviously not a “field.” In other words, a “field with one element,” in the usual sense of the term, does not and cannot exist. However, a different and more interesting picture emerges if instead of trying to define $\mathbb{F}_{1}$ first, we set $q=1$ directly in the $N=2$ ‘spin’ model we constructed above.

First, the number of states given in Eq. (5) will be reduced to $q+1=2$,

as are the possible outcomes listed in Eq. (8):

The sole observable of the system, Eq. (12), will be

for which $|\uparrow\;\rangle$ and $|\downarrow\;\rangle$ are ‘eigenstates’:

Thus, a measurement of $A$ on $|\uparrow\;\rangle$ will always yield $+1$, while that on $|\downarrow\;\rangle$ will always yield $-1$. No superpositions of these states exist, so the system reduces to a ‘classical’ one where each states has a definite outcome upon measurement.

The ‘two’-spin system will reduce to $q^{3}+q^{2}+q+1=4$ states, of which $q(q^{2}-1)=0$ are entangled. These four are the product states

The sole product observable, Eq. (20), is

where

The four states are all ‘eigenstates’ of $AA$ with definite outcomes when $AA$ is measured. Thus, the system is ‘classical’ and its CHSH bound is trivially 2.

This discussion can be easily generalized to generic $N$-level GQM, in which the $q=1$ case reduces to a situation where all surviving states are simultaneous ‘eigenstates’ of all surviving observables.¹³¹³13Note that the vector space $\mathbb{F}_{q}^{N}$ has $q^{N}-1$ non-zero vectors, of which every $q-1$ of them are physically equivalent in GQM. Thus, the number of physically distinct states is $(q^{N}-1)/(q-1)=[N]_{q}$, which reduces to $N$ in the $q=1$ limit. What this exercise shows is that the $q=1$ limit of GQM, though admittedly fairly trivial, does make sense as a ‘classical’ theory. Note, however, that in letting $q=1$ we have retained the formalism of GQM as is from the ‘quantum’ $q\neq 1$ cases. The only thing that has changed is the value of $q$, which changes the number field over which the state space is constructed, formally, from $\mathbb{F}_{q}$ to $\mathbb{F}_{1}$. But what is $\mathbb{F}_{1}$?

The concept of $\mathbb{F}_{1}$ first appeared in a 1957 paper by Tits[58] as the “corps de caractéristique 1.” The main objective of the Tits paper was to use projective geometries to define the then recently discovered Chevalley groups geometrically.¹⁴¹⁴14Chevalley originally used purely algebraic methods in his definition.[99] To each Dynkin diagram and choice of finite field $\mathbb{F}_{q}$ a projective geometry was associated, and the Chevalley group identified with the group of projective transformations on the geometry. For instance, the projective geometry $PG(N-1,q)$ is associated with the Dynkin diagram for $A_{N-1}$, and the corresponding Chevalley group is the projective linear group $PGL(N,q)$. Towards the end of the paper, Tits argues that the $q=1$ limit of $PG(N-1,q)$ actually makes sense as a projective geometry,¹⁵¹⁵15With a minor modification to the axioms. See Ref. \refciteCohn:2004. and that the corresponding group of projective transformations is $S_{N}$.

To see this, let us first list a few properties of $PG(N-1,q)$ for $q\neq 1$:

• •

  The number of points in the geometry $PG(N-1,q)$ is

  $$\left[\begin{array}[]{c}N\\ 1\end{array}\right]_{q}\;=\;[N]_{q}\;=\;\dfrac{q^{N}-1}{q-1}\;=\;1+q+q^{2}+\cdots+q^{N-1}\;.$$

  (35)

  (See chapter 9 of Ref. \refciteCameron:1994.) Here, the notation is:

  $$\left[\begin{array}[]{c}N\\ M\end{array}\right]_{q}\;=\;\dfrac{[N]_{q}!}{[M]_{q}![N-M]_{q}!}\;,$$

  (36)

  with

  	$\displaystyle[N]_{q}$	$\displaystyle\equiv$	$\displaystyle 1+q+q^{2}+\cdots+q^{N-1}\;=\;\dfrac{q^{N}-1}{q-1}\;,$		(37)
  	$\displaystyle[N]_{q}!$	$\displaystyle\equiv$	$\displaystyle[N]_{q}[N-1]_{q}\cdots[2]_{q}[1]_{q}\;.$		(38)

  Eq. (36) is known as the Gaussian binomial coefficient, $[N]_{q}$ the $q$-analogue of the natural number $N$, and $[N]_{q}!$ the $q$-factorial.

  
  
• •

  The number of $k$-dimensional subspaces (points, lines, planes, etc.) in $PG(N-1,q)$ are

  	$\displaystyle\left[\begin{array}[]{c}N\\ k+1\end{array}\right]_{q}$	$\displaystyle=$	$\displaystyle\dfrac{[N]_{q}[N-1]_{q}\cdots[N-k]_{q}}{[k+1]_{q}[k]_{q}\cdots[1]_{q}}$		(41)
  		$\displaystyle=$	$\displaystyle\dfrac{(q^{N}-1)(q^{N-1}-1)\cdots(q^{N-k}-1)}{(q^{k+1}-1)(q^{k}-1)\cdots(q-1)}\;.$		(42)

  Each subspace contains

  $$[k+1]_{q}\;=\;\dfrac{q^{k+1}-1}{q-1}\;=\;1+q+q^{2}+\cdots+q^{k}$$

  (43)

  points.

For instance, say $N=3$: $PG(2,1)$ would consist of three ‘points.’ Let us call them $a$, $b$, and $c$. The ‘lines’ in this geometry would be $\{a,b\}$, $\{b,c\}$, and $\{c,a\}$, and the ‘plane’ would be the entire space $\{a,b,c\}$. There is a nice discussion in Ref. \refciteCohn:2004 on how this definition of ‘geometry’ satisfies all the required axioms. All possible ways to map the three points onto themselves form the group $S_{3}$.

Now, recalling that $PG(N-1,q)$ was obtained by identifying the lines through the origin of $\mathbb{F}_{q}^{N}$ as points, Tits argues that the ‘projective geometry’ $PG(N-1,1)$ should be interpretable as

where $\mathbb{F}_{1}$ is the “corps de caractéristique 1,” aka the field with one element.

Taken literally, “corps de caractéristique 1” implies¹⁶¹⁶16See footnote 11.

that is, the multiplicative unit and the additive unit are the same number, so it may seem that $\mathbb{F}_{1}=\{\underline{1}\}=\{\underline{0}\}$, where

Such an object would not have any structure to speak of and it is difficult to envision how one could construct any geometry, projective or otherwise, on it. Thus, a more sophisticated definition of $\mathbb{F}_{1}$ is called for.

Various definitions of $\mathbb{F}_{1}$ and $\mathbb{F}_{1}$-geometries exist in the literature, which endeavor to make sense of these concepts, a survey of which can be found in Ref. \refciteLopezPena:2009 by Lorscheid and Lopez-Pena. Here, we adopt the definition of $\mathbb{F}_{1}$ by Kurokawa and Koyama in Ref. \refciteKurokawa:2010 who argue that $\mathbb{F}_{1}$ should be interpreted as the set $\mathbb{F}_{1}=\{\underline{1}\}+\{\underline{0}\}$ on which only multiplication is defined. Note that $\underline{1}$ is the multiplicative unit, and $\underline{0}$ is the multiplicative zero, that is:

This is different from $\mathbb{F}_{2}=\{\underline{0},\underline{1}\}$ in that addition is not allowed.¹⁷¹⁷17It may be more precise to say that only the addition of $\underline{0}$ is allowed. $\underline{1}+\underline{1}$ is prohibited. Another definition referred to in the literature[62, 82, 83, 84] defines $\underline{1}+\underline{1}=\underline{1}$ with all other additions and multiplications among $\underline{0}$ and $\underline{1}$ defined in the usual way, cf. Eq. (46). This is the reason for the unfamiliar notation $\{\underline{1}\}+\{\underline{0}\}$: $\{\underline{1}\}$ is the multiplicative group by itself, and $\{\underline{0}\}$ is the multiplicative zero (but not the additive unit since there is no addition) which is tacked on. Thus, $\mathbb{F}_{1}$ is not really a field but what is known as a ‘monoid.’

Since addition is not allowed, ‘vector’ spaces over $\mathbb{F}_{1}^{N}$ are not truly ‘vector’ spaces in the usual sense of the term. However, lacking a better alternative, let us use vector space terminology anyway. Since $\mathbb{F}_{1}^{N}$ is an $N$-dimensional space, it will have $N$ basis vectors, which we denote

The linear combinations of these basis vectors are of the form

but since we should have no addition, only one of the $c_{i}$’s is allowed to be non-zero at a time. So the $N$ basis vectors are the only (non-zero) vectors in this space. They will each be represented by column vectors with only one $\underline{1}$, the remaining elements being all $\underline{0}$. Note that these are precisely the states we obtained in taking the limit $q=1$ in GQM.

Linear transformations on $\mathbb{F}_{1}^{N}$ must not lead to addition of the basis vectors. So the only allowed transformations are those for which the $N\times N$ matrix representation has at most one $\underline{1}$ in each row. The non-singular ones, i.e. the automorphisms, are the ones in which there is one $\underline{1}$ in each row and each column. These are simply the matrices that permute the $N$ basis vectors. Since $\mathbb{F}_{1}\backslash\{\underline{0}\}=\{\underline{1}\}$, the projective space $PG(N-1,1)$ will simply consist of the non-zero vectors in $\mathbb{F}_{1}^{N}$. There are $N$ of these, as required for $PG(N-1,1)$. The projective linear group $PGL(N,1)$ would be $S_{N}$.

Linear maps from $\mathbb{F}_{1}^{N}$ to $\mathbb{F}_{1}$, i.e. the dual vectors, must also satisfy the no-addition requirement, so they would be $N\times 1$ row-vectors, each with only one $\underline{1}$, the remaining elements all $\underline{0}$. These are precisely the dual vectors representing outcomes in the $q=1$ limit of GQM.

Thus, the Kurokawa-Koyama approach matches well with the GQM construction, and the $q=1$ ‘classical’ limit can be constructed directly on $\mathbb{F}_{1}^{N}$ defined in this fashion. The resulting state space has the projective geometry $PG(N-1,1)$. Note, particularly, that the prohibition of addition in Kurokawa-Koyama leads to the lack of superposition of states in the ‘classical’ limit of GQM. Therefore, the $q=1$ limit of GQM can be considered to be ‘classical’ due to this very special feature of $\mathbb{F}_{1}$.