Let me begin this Sunday morning post with a thought-provoking quote from Britannica:
monad, (from Greek monas, “unit”), an elementary individual substance that reflects the order of the world and from which material properties are derived. The term was first used by the Pythagoreans as the name of the beginning number of a series, from which all following numbers derived. Giordano Bruno, in De monade, numero et figura liber (1591; “On the Monad, Number, and Figure”), described three fundamental types: God, souls, and atoms. The idea of monads was popularized by Gottfried Wilhelm Leibniz in Monadologia (1714). In Leibniz’s system of metaphysics, monads are basic substances that make up the universe but lack spatial extension and hence are immaterial. Each monad is a unique, indestructible, dynamic, soul-like entity whose properties are a function of its perceptions and appetites. Monads have no true causal relation with other monads, but all are perfectly synchronized with each other by God in a preestablished harmony. The objects of the material world are simply appearances of collections of monads.
God in a preestablished harmony
This rich and multilayered description of monads has fueled centuries of metaphysical speculation, philosophical debates, and even mathematical explorations. Today, we’ll delve into an intriguing reinterpretation of the monad concept—not in the metaphysical sense, but through the lens of mathematics, specifically the Clifford geometric algebra Cl(V), which we’ve symbolically denoted by a single letter: A.
Our focus is not merely academic. We aim to explore whether A can serve as a foundational model for the “objects of the material world.” In doing so, we confront a compelling question: How does one move from a singular monad to a collection of monads? After all, the material world seems to consist of interactions among many such entities.
Here, two possibilities come to mind. The first is divine intervention—if God, with infinite power, can create one monad, then it is presumably trivial to create an infinite supply of them, stored in some vast metaphysical “container.” The second possibility, however, is far more intriguing: the monad A might possess a self-replicating property. If so, understanding this self-replication mechanism would require us to study the monad in exquisite detail. This is precisely the journey we are embarking on: a careful and meticulous examination of A.
Nowadays, variations of have found widespread application in fields like quantum computing and quantum cryptography. Increasingly, papers and books are being published that explore A—or, as it’s often called in this context, the “Qubit.” However, for most researchers in these areas, A is not viewed as a monad in the philosophical or foundational sense. Instead, it is seen as a practical and versatile tool, a means to develop cutting-edge technologies, often driven by the demands of the military-industrial complex.
The questions these researchers ask about A
I place myself firmly in the second category of people—those who are drawn to the pursuit of understanding. That said, I wouldn’t hesitate to use the book for self-defense if the situation demanded it. But the essence of our endeavor here is not pragmatic or utilitarian; it’s a journey of curiosity and exploration, seeking to uncover the subtle and surprising truths that lie hidden within.
A contains geometry, algebra, and has a distinct
quantum-mechanical smell. Let us follow our noses and look closer into
the quantum-mechanical aspects of A.
A is an algebra, an algebra with involution "*". In fact, it is a C*-algebra and a von Neumann algebra - the baby version of them. There is a lot of publications about algebraic approach to quantum mechanics. It has started around 1936 with Birkhoff and von Neumann paper about the "The logic of quantum mechanics" (Ann. Math. 37, No. 4, pp. 823-843, 1936). The textbook by J.M. Jauch, "Foundations of Quantum Mechanics", Addison-Weley 1968, developed these ideas further, from a physicist's perspective. Algebras are used in quantum mechanics as "algebras of observables", which is somewhat confusing since ordinarily only selfadjoint elements of the algebra are being considered as "real observables". The product ab of two self-adjoint observable will not, in general, be self-adjoint, so real observables do not form an algebra under the algebra product (that is why Birkhoff and von Neuman were mainly focused on the Jordan product (ab+ba)/2). But there as simple observables, whose values are only 0 and 2. These form "quantum logic". Jauch calls them "propositions". They are presented by self-adjoint idempotents: p = p* = p2. Possible eigenvalues of self-adjoint idempotents are 1 and 0. They are treated as logical "yes" and "no". Let us concentrate on such elements of A and see what we can get this way?
We write a general element of A as p = (p0,p) ( or (p,p4) as in the previous post ), where p0 is a complex scalar, and p is a complex vector. Then p*= (p0*,p*), where * inside the parenthesis stands for complex conjugation. The p*=p means that p0*=p0 and p*=p. In other words p0 and p must be real.
Now we recall a general multiplication formula for A:
for p = (p0,p), q =(q0,q)
pq = (p0q0+p·q, p0q + q0p + i p⨯q) .
In particular for pp we get
pp = (p02 + p·p, 2p0p)
since p⨯p=0.
Thus pp = p implies
p02 + p·p = p0,
and
2p0p = p.
Then either p = 0 or not. If p =0, then p02 =p0, and we have two solutions p0=0 or p0=1. They correspond to trivial Hermitian idempotents p = 0 and p=1. On the other hand, if p is not a zero vector, then from the second equation we get p0 = 1/2. Substituting this value of p0 into the first equation we get
1/4 + p·p = 1/2, or
p·p = 1/4.
We deduce that p = 1/2n, where n is a unit vector in V.
Therefore a general form of a nontrivial Hermitian idempotent is:
p = (1+n)/2.
This is our general propositional question in A: it is something about a direction in V. How to interpret it? We will be returning to this question again and again.
Exercise 1. Let n be a unit vector in V. Let Tn denote the space tangent to the unit sphere at n. Thus Tn can be identified with the subspace of V consisting of all vectors of V perpendicular to n. Let J be defined as a linear operator on Tn defined by
Ju = u⨯n
Show that J is well defined and it defines a complex structure on Tn (i.e. that J2 = -1). Show that J is an isometry, that is that <Ju,Jv> = <u,v> for all u,v in Tn.
Exercise 2. Read Wikipedia article "Linear complex structure", section "Relation to complexifications". Take J from the previous exercise and extend it by linearity to TnC = Tn+iTn. Find a solution of up = u (see the discussion under the previous post) for p=(1+n)/2 within TnC . Express it as an eigenvector of the operator iJ - to which eigenvalue?
Exercise 3. Let n i p be as above. Show that up=u if and only if un=u.
Exercise 4. Find an error in the proof below. I can't, but it looks suspicious to me.
Statement. Let n and n' bet two different unit vectors. Then u satisfies both un=u and un'=u if and only if u=0.
Proof. Set n-n' = v. Suppose v is a nonzero vector. Then we have uv = 0. That means (u.v, u0v+iuxv)=0, Thus u is perpendicular to v. Now, in u0v+iuxv = 0 the first term is proportional to v, the second is perpendicular. Thus both must be zero. It follows that u0=0, and uxv=0. Thus if v is not zero, then u=0.
Exercise 5. Consider the following example:
Choose n = (0,0,1) - e3. Choose e1 = (1,0,0) for the real part of u1. Get the expression for u1. Define
E1=(1+n)
E2=u1
Show that E1,and E2 span the left ideal of A. For this calculate the action of e0,e1,e2,e3 on E1 and E2 and express the results as a linear combinations of E1,E2.
Exercise 6. Is the representation L of A on our left ideal from Exercise 5 irreducible or reducible?
To be continued.
