Philosophy
In our initial paper on this topic, we introduced a toy model in physics that conceptualizes the space-time of classical physics as a section of a universal fiber space ˜E This fiber space is defined on the moduli space H, specifically Hilb(2)(E³), which pertains to physical systems involving both an observer and an observed entity situated in Euclidean 3-space, E³ The moduli space can be readily computed and is characterized by the structure H = ˜H/Z₂.
The space H = k[t1, , t6], where k = RandH := Spec(H), represents all ordered pairs of points in E³ The blow-up of the diagonal is denoted as ˜H, and Z2 signifies the inherent group action Collectively, H, along with its extensions H and ˜H, is referred to as the time-space of the model.
In this mathematical model, measurable time is represented as a metric ρ on the time-space, which quantifies all potential infinitesimal changes in the states of the studied objects Additionally, relative velocity is defined as an oriented line within the tangent space of a point in ˜H, indicating that the space of velocities is compact.
This leads to a physics framework that eliminates infinite velocities and inherently incorporates the principle of relativity The Galilean group operates on E3 and consequently on ˜H The Abelian Lie-algebra of translations establishes a three-dimensional distribution, ˜∆, in the tangent bundle of ˜H, which relates to zero velocities By defining a metric on ˜H, we can identify the distribution ˜c, representing light velocities, as the normal space of ˜∆ This framework allows us to conceptualize classical space-time as a universal space that is confined to a specific subspace.
S(l)of ˜˜ H, defined by a fixed line l ⊂ E 3 In chapter 4, under the section
In this article, we explore the relationship between time-space and space-times, highlighting how the generator τ ∈ Z2 connects to the classical physics operators C, P, and T, with the relation τ² = τ P T = id Additionally, we note that the three fundamental gauge groups of contemporary quantum theory, U(1), SU(2), and SU(3), are integral components of the fiber space structure.
For any point \( t = (o, x) \) in \( H \) that lies outside the diagonal \( \Delta \), we can examine the line \( l \) in \( E^3 \) defined by the pair of points \( (o, x) \) Additionally, we can analyze the action of \( U(1) \) on the normal plane \( B_o(l) \) of this line, which is oriented by the normal \( (o, x) \), as well as on the plane \( B_x(l) \), oriented by the normal \( (x, o) \) By utilizing parallel transport in \( E^3 \), we establish an isomorphism of bundles.
The partition isomorphism can be expressed as Po,x:Bo→Bx and P:Bo⊕Bx→Bo⊕Bx Utilizing P, we can denote (v, v) as (v, Po,x(v)) and P((v,0)) Additionally, it has been established that the line l uniquely defines a subscheme H(l) within H The tangent space corresponding to the point (o, x) is referred to as the tangent space at that location.
A(o,x) Together this define a decomposition of the tangent space ofH,
Ift = (o, o)∈∆, and if we consider a pointo 0 in the exceptional fiberEo of ˜H we find that the tangent bundle decomposes into,
The equation TH,o ˜ 0 = Co 0 ⊕ Ao 0 ⊕ ∆,˜ defines the tangent space Co 0 of Eo, with Ao 0 representing the light velocity at o 0 and ˜∆ denoting the 0-velocities The bundles Bo and Bx, along with the structure C(o,x) = {(ψ,−ψ) ∈ Bo ⊕ Bx}, are established as complex line bundles over H−∆ Additionally, C(o,x) extends throughout the entirety of ˜H, and its restriction to Eo corresponds with the tangent bundle.
Tensorising with C(o,x), we complexify all bundles In particular we find complex 2-bundlesCBo and CBx, on H−∆, and we obtain a canonical decomposition of the complexified tangent bundle Any real metric on
H will decompose the tangent space into the light-velocities ˜cand the 0- velocities, ˜∆, and obviously,
This decomposition can also be extended to the complexified tangent bundle of ˜H Clearly,U(1) acts on TH, and SU(2) and SU(3) acts naturally on
CBo⊕CBxandC∆ respectively Moreover˜ SU(2) acts onCCo 0 , in such a way that their actions should bephysicallyirrelevant U(1), SU(2), SU(3) are our elementarygauge groups.
In contemporary physics, the simplest example illustrates our approach to understanding natural phenomena, where we perceive a distinction between the observer and the observed within a spatial context Today, the most intuitive framework for this observation is the three-dimensional Euclidean space.
To study a natural phenomenon, referred to as P, it is essential to describe it mathematically as an object, X, which depends on specific parameters The variations in P should align with changes in the parameter values of X Consequently, X serves as a model for P, meaning that any selection of parameter values for X should reflect a potentially observable aspect of P.
Mathematical objects X(1) and X(2) that represent the same aspect of P are considered equivalent The collection of equivalence classes of these objects is referred to as the moduli space of the models, X.
The investigation of the natural phenomenon P parallels the exploration of the structure of M Specifically, the concept of time serves as a metric within this framework, aligning with the perspectives of Aristotle and St Augustine.
With this philosophy, and thistoy-model in mind we embarked on the study of moduli spaces of representations (modules) of associative algebras in general, see Chapter 3.
The concept of dynamical structure within the space M is introduced through the development of Phase Spaces, as outlined in Chapter 2 This framework provides a comprehensive theoretical basis for analyzing the phenomenon P and its associated dynamics.
Phase Spaces, and the Dirac Derivation
For any associativek-algebraA we have, in [20], and Chapter 2, defined a phase spaceP h(A), i.e a universal pair of a morphismι:A→P h(A), and anι- derivation, d:A→P h(A), such that for any morphism of algebras,
A → R, any derivation of A into R decomposes into d followed by an
In the study of associative k-algebras, we encounter homomorphisms P h(A) → R, which can be either trivial or non-commutative, as referenced in sources [20] and [21] These algebras provide a foundational framework for quantization in physics By iterating this construction, we derive a limit morphism ι n : P h n (A) → P h ∞ (A), resulting in the image P h (n) (A) Additionally, we identify a universal derivation δ ∈ Derk(P h ∞ (A), P h ∞ (A)), known as the Dirac derivation.
This Dirac derivation will, as we shall see, create the dynamics in our different geometries, on which we shall build our theory For details, see
Chapter 2 explores the concept of superspace, which can be easily derived from the Ph-construction An affine superspace is associated with a quotient of the Ph(A) structure, where A represents the affine k-algebra of a particular scheme.
Non-commutative Algebraic Geometry, and Moduli of Sim-
This article discusses the foundations of affine non-commutative algebraic geometry in relation to associative k-algebras, where k is any field It establishes the existence of a non-commutative scheme structure on the set of isomorphism classes of simple finite-dimensional representations of a finitely generated algebra A It is demonstrated that any geometric k-algebra A can be derived from the non-commutative structure of its simple representations, and a quasi-affine scheme structure exists on each component of the simple representations Furthermore, a commutative algebra C(n) is identified, featuring an open subvariety and an étale covering that leads to a universal representation, which serves as a morphism of algebras that generates all isomorphism classes of simple n-dimensional A-modules.
In accordance with our philosophy of exploring the moduli space of mathematical models, we define \( A \) as the affine \( k \)-algebra representing this space, which encompasses all relevant parameters This framework establishes the geometric landscape that serves as the foundation for our quantum theory.
Obviously,EndC(n)( ˜V)'Mn(C(n)), and we shall use this isomorphism without further warning.
Dynamical Structures
In this article, we have introduced moduli spaces that represent both our mathematical objects and the dynamical variables relevant to our study The next step is to integrate these elements to develop a coherent dynamic framework within our geometric context.
Adynamical structure, see Definition (3.1), defined for aspace, or any associativek-algebra A, is now an ideal (σ) ⊂P h ∞ (A), stable under the
Dirac derivation, and the quotient algebraA(σ) :=P h ∞ (A)/(σ), will be called adynamical system.
These associative, but usually highly non-commutative,k-algebras are the models for the basicaffine algebras creating the geometric framework of our theory.
As an example, assume that A is generated by the space-coordinate functions, {ti} d i=1 of some configuration space, and consider a system of equations, δ n tp:=d n tp= Γ p (t i , dtj, d 2 tk, , d n−1 tl), p= 1,2, , d.
Let (σ) := (δ n tp−Γ p ) be the two-sidedδ-stable ideal ofP h ∞ (A), generated by the equations above, then (σ) will be called adynamical structure or a force law, of order n, and the k-algebra,
A(σ) :=P h ∞ (A)/(σ), will be referred to asa dynamical system of order n.
Producing dynamical systems of interest to physics, is now a major problem One way is to introduce the notion ofLagrangian, i.e any element
A δ-stable ideal (σ) ⊂ P h ∞ (A), satisfying δ(L) = 0 (mod(σ)), is defined as a solution to the Lagrange equation This concept represents a non-commutative approach to addressing the principles of parsimony established by Maupertuis and Fermat in the realm of physics.
In the commutative case, the Dirac derivation of dynamical systems of order 2 will have the form, δ=X i dti ∂
Whenever A is commutative and smooth, we may consider classical
Lagrangians, represented as L = 1/2P i,jgi,jdtidj ∈P h(A), utilize a non-degenerate metric within a regular coordinate system {ti} The resulting Lagrange equations yield a dynamical structure of order 2, expressed as d²ti = -X j,k Γ i j,k dtjdtk, where Γ denotes the Levi-Civita connection.
For a general Lagrangian \( L \in P h^2(A) \), one can set \( \delta \) as time and apply the Euler-Lagrange equations to derive force laws, as discussed in this introduction and further elaborated in section (4.5) General.
Quantum Fields, Lagrangians and Actions.
By definition,δinduces a derivationδσ∈Derk(A(σ),A(σ)), also called theDirac derivation, and usually just denotedδ.
Different Lagrangians can lead to various Dirac derivations on the same k-algebra A(σ), resulting in distinct dynamics for the universal families of the components of Simpn(A(σ)) for n ≥ 1, specifically concerning the particles within the system.
Quantum Fields and Dynamics
In the context of Simp(A(σ)), a family of components is referred to as a family of particles, represented by the versal family ˜V A section φ of the bundle ˜V functions on the moduli space Simp(A), extending beyond just the configuration space Simp1(A) or Simp1(A(σ)) The value φ(v) ∈ ˜V(v) at a point v ∈ Simpn(A) is identified as the state of the particle at the event v.
EndC(n)( ˜V) induces also a bundle, of operators, on the ´etale covering
In the context of the U(n) of Simpn(A(σ)), a section ψ of this bundle is referred to as a quantum field Specifically, each element a in A(σ) can be associated with a quantum field through the versal family map, denoted as ˜ρ The collection of these quantum fields constitutes a k-algebra.
Physicists will tend to be uncomfortable with this use of their language.
In classical quantum field theory, a traditional physicist typically defines a function ψ on a configuration space, distinct from Simpn(A(σ)), which takes values in a polynomial algebra created by specific creation and annihilation operators within a Fock space.
As we shall see, this interpretation may be viewed as a special case of our general set-up But first we have to introduce Planck’s constant(s) and
Fock-space Then in the section (4.6) Grand picture, Bosons, Fermions,
This article introduces the concepts of Introduction 7 and Supersymmetry, emphasizing the importance of locality of interaction A clear explanation of this intriguing non-quantum phenomenon within classical theory is provided by Cohen-Tannoudji on page 104 Additionally, the historical context surrounding these ideas is discussed, offering valuable insights into their development.
Notice also that in physics books, the Greek letterψis usually used for states, i.e sections of ˜V, or in singular cases, see below, for elements of the
Hilbert space serves as the foundational framework for observables in quantum mechanics and is also frequently applied to quantum fields In this context, we designate the quantum fields as ψ while reserving the symbol φ for states This terminology aligns with our discussion in section 4.6, which covers the grand picture of bosons, fermions, and supersymmetry.
Other places, we may turn this around, to fit better with the comparable notation used in physics.
Let v ∈ Simpn(A(σ)) correspond to the right A(σ)-module V, with structure homomorphismρv:A(σ)→Endk(V), then the Dirac derivation δcomposed with ρv, gives us an element, δv∈Derk(A(σ), Endk(V)).
Recall now that for any k-algebraB, and rightB-modulesV,W, there is an exact sequence,
HomB(V, W)→Homk(V, W)→Derk(B, Homk(V, W)→Ext 1 B (V, W)→0, where the image of, η:Homk(V, W)→Derk(B, Homk(V, W)) is the sub-vectorspace of trivial (or inner) derivations.
Modulo trivial inner derivations, the derivation δv establishes a class ξ(v) within Ext 1 A (σ)(V, V), representing a tangent vector to Simpn(A(σ)) at the point v Consequently, the Dirac derivation δ uniquely defines a one-dimensional distribution in ΘSimp n (A(σ)) By fixing a versal family, this leads to the definition of a vector field ξ in ΘSimp n (A(σ)), and in favorable circumstances, it results in a rational derivation ξ in Derk(C(n)), which induces a derivation.
[δ]∈Derk(A(σ), End C(n) ( ˜V)), lifting ξ, and, in the sequel, identified withξ By definition of [δ], there is now aHamiltonian operator
Q∈Mn(C(n)), satisfying the following fundamental equation, see Theorem (4.2.1), δ= [δ] + [Q,ρ(˜−)].
This equation means that for an element (an observable) a ∈ A(σ) the element δ(a) acts on ˜V 'C(n) n as [δ](a) =ξ(˜ρV(a)) plus the Lie-bracket
Any right A(σ)-module V can also be viewed as a P h ∞ (A)-module, leading to a series of P h n (A)-module structures on V for n ≥ 1 This includes an A-module V0 = V and an element ξ0 ∈ Ext 1 A (V, V), representing a tangent of the deformation functor for V0 Additionally, there are elements ξ1 ∈ Ext 1 P h(A) (V, V) and ξ2 ∈ Ext 1 P h 2 (A)(V, V), which serve as tangents for the deformation functors of V1 and V2, respectively, as P h(A)-modules and P h 2 (A)-modules.
In the context of an A-module V, a sequence of tangents or momenta {ξn}, including a momentum ξ0 and an acceleration vector ξ1, defines a coherent framework for higher-order momenta ξn Specifying a point v in Simpn(A(σ)) necessitates the identification of a formal curvature through the base-point v0, situated within the universal deformation space of the A-module V.
Knowing the dynamical structure, (σ), and the state of ourobjectV at atimeτ0, i.e knowing the structure of ourrepresentationV of the algebra
A(σ), at that time (which is a problem that we shall return to), the above makes it reasonable to believe that we, from this, may deduce the state of
The fundamental assumption that physical laws remain consistent over time is often overlooked in modern physics textbooks This paper aims to provide a solid rationale for this essential principle, reinforcing its significance in the foundation of scientific understanding.
The mystery is, of course, why Nature seems to be parsimonious, in the sense of Fermat and Maupertuis, giving us a chance of guessing dynamical structures.
The system's dynamics are described using the Dirac vector field [δ], which generates the vector field ξ on Simpn(A(σ)) An integral curve γ of ξ represents a solution to the equations of motion, beginning at v0 ∈ Simpn(A(σ)) and concluding at v1 ∈ Simpn(A(σ)), with a length defined by τ1 − τ0.
This is only meaningful for ordered fieldsk, and when we have given a met- ric (time) on the moduli spaceSimpn(A(σ)) Assume this is the situation.
Then, given astate, φ(v0)∈V˜(v0)'V0, of the particleV˜, we prove that
Introduction 9 there is acanonical evolution map,U(τ0, τ1) transporting φ(v0) from time τ0, i.e from the point representing V0, to time τ1, i.e corresponding to some point representingV1, alongγ It is given as,
Qdτ)(φ(v0)), where exp(R γ) is the non-commutative version of the classical action in- tegral, related to the Dyson series, to be defined later, see the proof of
Theorem 4.2.3 and Section 4.6 provide a comprehensive overview of the relationships between bosons, fermions, and supersymmetry By utilizing unitary representations, we can draw parallels to the S-matrix, perturbation theory, and further extend these concepts to Feynman integrals and diagrams.
Classical Quantum Theory
Many classical physics models are either fundamentally commutative or singular, with singular models characterized by conditions such as Q=0 or [δ]=0 General relativity exemplifies the commutative category, while classical Yang-Mills theory represents the singular type Notably, theories that incorporate connections tend to be singular and infinite-dimensional However, by imposing singularity on a theory, it is sometimes possible to recover the classical infinite-dimensional models based on Hilbert space as a limit of finite-dimensional simple representations, which correspond to dynamic systems, as demonstrated in Examples 4.2-4.4, specifically in the context of the Harmonic oscillator.
Planck’s Constants, and Fock Space
This general model allows us also to define a general notion of aPlanck’s constant(s),~ l , as the generator(s) of thegeneralized monoid, Λ(σ) :={λ∈C(n)|∃fλ∈A(σ), fλ6= 0,
[Q,ρ(δ(f˜ λ))] = ˜ρ(δ(fλ))−[δ](˜ρ(fλ)) =λ˜ρ(fλ)} which has the property thatλ, λ 0 ∈Λ(σ), fλfλ 0 6= 0 impliesλ+λ 0 ∈Λ(σ).
The Fock algebra, or Fock space, can be defined as a sub-k-algebra of EndC(n)( ˜V), generated by the operators {a l + := f ~ l, a l − := f− ~ l} This concept is explored in detail through examples, including the one-dimensional harmonic oscillator and the quartic anharmonic oscillator in ranks 2 and 3 This framework serves as a natural extension of traditional classifications of representations of semi-simple Lie algebras, as discussed in the context of fundamental particles.
When A represents the coordinate k-algebra of a moduli space, it is essential to examine the family of Lie algebras associated with the essential automorphisms of the objects classified by Simp(A(σ)) By applying invariant theory, as discussed in references [18], we can derive a comprehensive framework for Yang-Mills theory, particularly in the context of plane curve singularities, as highlighted in sources [33] and [22] This approach provides a foundational model for understanding gauge particles and gauge fields, illustrating their interactions with ordinary particles through representations on corresponding simple modules.
General Quantum Fields, Lagrangians and Actions
Perfectly parallel with this theory of simple finite dimensional representa- tions, we might have considered, for given algebrasA, and B, the space of algebra homomorphisms, φ:A→B.
In the commutative, classical case, when A is generated by t1, , tr, and
B is the affine algebra of aconfiguration spacegenerated byx1, , xs,φis determined by the images φi := ˜φ(ti), and φ or {φi}is called a classical field Any suchfield,φinduces a unique commutative diagram of algebras,
Given dynamical structures, (say of order two), σ and à, defined on A, respectivelyB, we construct a vector field [δ] on thespace,F(A(σ), B(à)), of fields,φ:A(σ)→B(à) The singularities of [δ] defines a subset,
There are natural equations of motion, analogous to those we have seen above, see (3.2) Notice that a field φ∈M is said to beon shell, those of
F−M are off shell We shall explore the structure of M in some simple cases.
Choosing the appropriate dynamical structures (σ) and (à) for a specific physical setup is not straightforward While these structures can be defined by force laws, such laws do not typically arise naturally in most scenarios.
In the realm of theoretical physics, 11 physicists emphasize the significance of the Lagrangian, denoted as L ∈ P h(A), in their discussions The Lagrangian density, L, is regarded as a crucial component within the versal family of iso-classes of F(A, B), highlighting its fundamental role in the analysis of physical systems.
Assuming a local affine algebraic geometric structure in this space, parametrized by a ring C, we can define the versal family through a homomorphism of k-algebras, φ˜: P h(A) → C ⊗ k P h(B), and set L := ˜φ(L) Traditionally, a natural representation linked to a derivation of B is chosen, represented by ρ: P h(B) → B, leading to L := ρ(L) The Lagrangian density is treated as a function of φi and φi,j := ∂φ/∂xij, effectively functioning within the configuration space Simp1(B) with coefficients from C It is postulated that there exists a functional or action that assigns a real or complex value to every field φ.
S :=S(L(φi, φi,j)), usually given in terms of a trace, or as an integral of L on part of the configuration space, see below S should be considered as a function on
F:=F(A, B), i.e as an element ofC The parsimony principles of Fermat and Maupertuis is then applied to this function, and one wants to compute the vector field,
∇S∈Θ F , which mimic our [δ], derived from the Dirac derivations The equation of motion, i.e the equations picking out the subspaceM⊂F, is therefore,
Here is where classical calculus of variation enters, and where we obtain differential equations forφi, theEuler-Lagrange equations of motion.
Notice now that in an infinite dimensional representation, theT race is an integral on the spectrum The equation of motion definingM⊂F, now corresponds to, δS :=δ
The calculus of variations leads to the formulation of Euler-Lagrange equations, identifying the singularities of the gradient of the action, ∇S, while avoiding reliance on a specific dynamical framework or universal families For further illustration, refer to Examples (3.7).
In this section, we explore the harmonic oscillator, demonstrating that the classical infinite-dimensional representation emerges as a limit of finite-dimensional simple representations Additionally, we reveal that the Lagrangian of the harmonic oscillator generates a vector field ∇SonSimp2(A(σ)), which differs from the vector field produced by the Dirac derivation of the corresponding dynamical system derived from the Euler-Lagrange equations Despite these differences, we find that the sets of singularities for both vector fields coincide.
This should never the less be cause for worries, since the world we can test is finite The infinite dimensional mathematical machinery is obviously just a computational trick.
The reliance on the Parsimony Principle through Lagrangians and the Euler-Lagrange equations presents a challenge; without proving that ∇S = [δ] for a specific dynamical structure σ, the philosophical assurance that a preparation in A(σ) guarantees a deterministic future for our objects is lost.
Otherwise, it is clear that the theory becomes more flexible It is easy to cook up Lagrangians.
In Quantum Field Theory (QFT), physicists often exhibit vagueness when quantizing fields by treating functions on configuration space, such as {φi, φi,j}, as elements of a k-algebra They introduce commutation relations and operate under the assumption that these functions behave like operators This confusion may stem from a failure to differentiate between the role of B in classical physics and the role of P h(B) in quantum theory.
Grand Picture Bosons, Fermions, and Supersymmetry
This article outlines the overarching concepts of Quantum Field Theory (QFT) derived from previous discussions These foundational ideas serve as a philosophical framework for exploring various topics, including the harmonic oscillator, general relativity, electromagnetism, spin, and quarks, which are detailed in Examples (4.2) to (4.14).
In particular, we sketch, here and in Chapter 5, how we may treat the problems ofBosons, Fermions,Anyons, andSuper-symmetry.
Connections and the Generic Dynamical Structure
Moreover we shall see that, on a space with a non-degenerate metric, there is a uniquegeneric dynamical structure, (σ), which produces the most in-
This article explores 13 intriguing physical models, highlighting how any connection on a bundle induces a representation of A(σ) We will apply this method to quantize both the Electromagnetic Field and the Gravitational Field, leading to generalized outcomes.
Maxwell, Dirac and Einstein-type equations, with interesting properties, see
Examples (4.1), (4.13) and (4.14) The Levi-Civita connection turns out to be a very particular singular representation for which the Hamiltonian is identified with theLaplace-Beltrami operator.
Clocks and Classical Dynamics
To effectively measure time, we must explore the concept of clocks, which leads us to two distinct models The first model, known as The Western clock, is based on the behavior of a free particle in one dimension, characterized by d²τ = 0 The second model presents an alternative perspective, highlighting the diversity in our understanding of time measurement.
Eastern Clock, modeled on the harmonic oscillator in dimension 1, i.e one withd 2 τ =τ
Time-Space and Space-Times
In Example (3.5), we explore the application of point-like particles within the ˜H-model, serving as an introduction to the Levi-Civita connection The intricate geometry of ˜H suggests the potential to define concepts such as mass and charge, represented by different colors, in relation to the diagonal structure ˜∆ This leads to the intriguing notion that every point in our reality can be viewed as a black hole, possessing a density of mass and charge Additionally, the five-dimensional nature of ˜∆ evokes theories reminiscent of Kaluza and Klein.
The concept of mass, viewed as a geometric property of ˜H, offers a promising foundation for deriving Newton’s law of gravitation, particularly through the analysis of blow-ups along the diagonal This approach can be illustrated with a straightforward example that results in a Schwarzschild-type geometry, where the associated equations of motion simplify to Kepler’s laws.
In this section, we revisit our toy model, highlighting the emergence of standard gauge groups U(1), SU(2), and SU(3) We demonstrate how the previously discussed results can be utilized to develop a comprehensive geometric theory that closely aligns with both general relativity and quantum theory, effectively generalizing both frameworks Further illustrations can be found in Examples (4.13) and (4.14), which explore the action of the natural gauge group on the canonically decomposed tangent bundle.
H presents a compelling theory for elementary particles, encompassing concepts such as spin, isospin, hypercharge, and quarks Central to this theory is the idea of non-commutative invariant space, which is crucial for understanding particle interactions Additionally, the article suggests potential models for both light and dark matter or energy, as illustrated in the provided examples.
In this toy model, light cannot be treated as point-particles, as there are no radars available for such particles, unlike in current general relativity However, the quantized electromagnetic (E-M) theory effectively explains communication through light Additionally, developing a coherent model of the universe's creation enhances our understanding of the phenomena we are studying, which will be explored in the next section.
Cosmology, Big Bang and All That
Our toy-model, i.e the moduli space, H, of two points in the Euclidean
3-space, or its ´etale covering, ˜H, turns out to be createdby the versal de- formation of the obvious (non-commutative) singularity in 3-dimensions,
The deformation functor of the k-algebra U exhibits a versal space that includes a flat component isomorphic to ˜H, which is part of the modular suite Notably, the modular stratum, or the inner room, is simplified to the base point This framework provides an effective model for The Universe, establishing straightforward connections to traditional cosmological models such as those proposed by Friedman-Robertson-Walker and Einstein-de Sitter.
Interaction and Non-commutative Algebraic Geometry
In section 1.4, we will explore the interactions, lifetimes, decay, and creation of particles, drawing inspiration from fundamental physics concepts such as cross-sections, resonance, and the cluster decomposition principle, as discussed by Weinberg.
The model discussed in this paper offers a compelling approach to treating interactions between fields by utilizing a non-commutative metric, replacing traditional metrics and connections This innovative perspective is perhaps the most intriguing feature of the proposed framework.
The essential point is that, in non-commutative algebraic geometry, say in the space of representations of an algebra B, there is atangent space,
The functor T(V, W) is defined as Ext¹_B(V, W) for any two points V and W Specifically, when B is the polynomial ring P h(k[x₁, , xₙ), a 1-dimensional representation of B can be expressed as a pair (q, ξ), where q is a closed point in Spec(k[x]) and ξ is a tangent at that point An analysis of two such points, (q₁, ξ₁) and (q₂, ξ₂), reveals that T((q₁, ξ₁), (q₂, ξ₂)) has a dimension of 1 if q₁ ≠ q₂, a dimension of n if q₁ = q₂ and ξ₁ ≠ ξ₂, and a dimension of 2n when (q₁, ξ₁) equals (q₂, ξ₂).
In the study of vector fields, we can assign a tangent vector to each point in space and use metrics to determine the length of these vectors Similarly, we can explore fields of tangents between two points and extend the concept of metrics to measure the lengths of these connections.
If we do, we find very nice models for treating the notion of identical particles, and interaction between fields, see the Examples (5.1), (5.2).
In our exploration of Alternative Histories, we aim to formally define this concept using our language, as referenced in [6] and discussed in paper [7] This endeavor represents a significant application of noncommutative deformation theory, which shows potential as a valuable tool in the field of mathematical physics.
Apology
In his historical introduction to "The Quantum Theory of Fields," Weinberg references Heisenberg's 1925 manifesto, highlighting a philosophical stance I must admit that this paper aligns with the same positivistic philosophy that Weinberg ultimately dismisses.
Although I am not a physicist, this paper explores the geometry of specific finitely generated non-commutative algebraic schemes, utilizing my interpretation of physicist terminology to enhance the clarity of the results.
Even though I see a lot of difficulties in the interpretation of the math- ematical notions of my models, in a physics context, I hope that the model
I propose may help other mathematicians to gain faith in their jugend- traums; sometime, somehow, to be able to understand some physics.
An attentive reader will also see that, if mymodelist philosophy about
Nature should be approached with seriousness, as it simplifies physicists' tasks in defining mathematical models of objects Collaborating with mathematicians, they can explore the moduli space of these models, establish an infinite phase space, and hypothesize a metric for time along with a corresponding dynamical structure Ultimately, they would submit their findings to the computer algebra group in Kaiserslautern, hoping for positive outcomes.
Phase Spaces and the Dirac Derivation 17
Phase Spaces
Given ak-algebraA, denote byA/k−algthe category where the objects are homomorphisms ofk-algebrasκ:A→R, and the morphisms, ψ:κ→κ 0 are commutative diagrams,
It is representable by a k-algebra-morphism, ι:A−→P h(A), with auniversal familygiven by a universal derivation, d:A−→P h(A).
Ph (A) is relatively easy to compute It can be constructed as the non- commutative versal base of the deformation functor of the morphism ρ :
Clearly we have the identities, d∗:Derk(A, A) =M orA(P h(A), A), and, d ∗ :Derk(A, P h(A)) =EndA(P h(A)), the last one associating d to the identity endomorphism of P h Let now
V be a right A-module, with structure morphism ρ:A →Endk(V) We obtain a universal derivation, c:A−→Homk(V, V ⊗ A P h(A)), defined by, c(a)(v) = v⊗d(a) Using the long exact sequence, see the introduction,
The non-commutative Kodaira-Spencer class, denoted as c(V) := κ(c) ∈ Ext^1_A(V, V ⊗_A P h(A)), is derived from the exact sequence ι Derk(A, Hom_A(V, V ⊗_A P h(A))) → κ Ext^1_A(V, V ⊗_A P h(A)) → 0 This class induces the Kodaira-Spencer morphism g: Θ_A → Ext^1_A(V, V) through the identity ˜δ∗ If c(V) = 0, the exact sequence indicates the existence of a connection ∇ ∈ Hom_k(V, V ⊗_A P h(A)) such that ˜δ = ι(∇), confirming that ˜δ is represented by a connection.
As is well known, in the commutative case, the Kodaira-Spencer class gives rise to aChern character by putting, ch i (V) := 1/i!c i (V)∈Ext i A (V, V ⊗AP h(A)), and ifc(V) = 0, the curvatureR(V) induces a curvature class,
AnyP h(A)-module W, given by its structure map, ρW :P h(A)−→Endk(W) corresponds bijectively to an induced A-module structure on W, and a derivationδρ∈Derk(A, Endk(W)), defining an element,
[δρ]∈Ext 1 A (W, W), see the introduction Fixing this element we find that the set of P h(A)- module structures on the A-module W is in one to one correspondence with,
Phase Spaces and the Dirac Derivation 19
Conversely, starting with an A-module V and an element δ ∈
Derk(A, Endk(V)), we obtain a P h(A)-module Vδ It is then easy to see that the kernel of the natural map,
Ext 1 P h(A) (Vδ, Vδ)→Ext 1 A (V, V), induced by the linear map,
Derk(P h(A), Endk(Vδ))→Derk(A, Endk(V)) is the quotient,
Remark 2.1 SinceExt 1 A (V, V) is the tangent space of the miniversal de- formation space of V as an A-module, see e.g [18], or the next chapter, we see that the non-commutative spaceP h(A) also parametrizes the set of generalized momenta, i.e the set of pairs of a simple moduleV ∈Simp(A), and a tangent vector ofSimp(A) at that point.
In the context of the algebra A = k[t], we observe that P h(A) is represented as k < t, dt >, where the derivation d is defined by d(t) = dt For any polynomial f in k[t], the non-commutative derivation d(f) corresponds to Jt(f) as noted in [19], which also relates to the non-commutative Taylor formula When considering V as an A-module with a matrix X in M2(k), and δ in Derk(A, Endk(V)), we define the P h(A)-module Vδ through the matrices X and Y in M2(k) The dimensions are calculated as follows: e 1 V = dimkExt 1 A (V, V) = dimkEndA(V) = dimk{Z ∈ M2(k) | [X, Z] = 0} and e 1 V δ = dimkExt 1 P h(A) (Vδ, Vδ) = 8−4 + dim{Z ∈ M2(k) | [X, Z] = [Y, Z] = 0}.
We have the following inequalities,
Notice thatP h(A) just has 2 points, i.e simple representations, given by, k(r) :x=r, dx= 0, k(−r) :x=−r, dx= 0.
An easy computation shows that,
Ext 1 P h(A) (k(α), k(α)) = 0, α=r,−r, Ext 1 P h(A) (k(α), k(−α)) =kãω, whereω is represented by the derivation given byω(x) = 2r, ω(dx) =t∈k where t is the tension of this string of dimension −1, see end of §2, and end of§3 Notice also that this is an example of the existence oftangents between different points, in non-commutative algebraic geometry.
(iii) Now, letA=k[x] :=k[x1, x2, x3] and consider,
P h(A) =k < x1, x2, x3, dx1, dx2, dx3> /([xi, xj], d([xi, xj])).
Any rank 1 representation ofAis represented by a pair of a closed point q of Spec(k[x]), and a tangent p at that point Given two such points,
(qi, pi), i= 1,2, an easy calculation proves, dimkExt 1 P hA (k(q1, p1), k(q2, p2)) = 1,for q16=q2 dimkExt 1 P hA (k(q1, p1), k(q2, p2)) = 3,for q1=q2, , p16=p2 dimkExt 1 P hA (k(q1, p1), k(q2, p2)) = 6,for(q1, p1) = (q2, p2)
In the context of algebraic structures, we define the mappings Putxj(qi, pi) as qi,j and dxj((qi, pi) as pi,j, where αj is the difference between q1,j and q2,j, and βj represents the difference between p1,j and p2,j For any element α in Homk(k((q1, p1)), k((q2, p2))), the following relationships hold: xjα equals q1,jα, αxj equals q2,jα, dxjα equals p1,jα, and αdxj equals p2,jα, demonstrating a clear identification Additionally, any derivation δ within Derk(P hA, Homk(k((q1, p1)), k((q2, p2)))) must adhere to specific relations, including δ([xi, xj]) = [δ(xi), xj] + [xi, δ(xj)] = 0 and δ([dxi, xj] + [xi, dxj]) = [δ(dxi), xj] + [dxi, δ(xj)] + [δ(xi), dxj] + [xi, δ(dxj)] = 0, ensuring the consistency of derivations within the framework.
By applying the left-right action rules, we derive the results from the long exact sequence that computes Ext 1 P hA The two sets of relations lead to the formulation of two systems of linear equations.
The first, in the variablesδ(x1), δ(x2), δ(x3), δ(dx1), δ(dx2), δ(dx3), with matrix,
and the second, in the variables,δ(x1), δ(x2), δ(x3), with matrix,
Phase Spaces and the Dirac Derivation 21
In particular we see that thetrivialderivation given by, δ(xi) =αi, δ(dxj) =βj, satisfies the relations, and the generator of Ext 1 P hA (k(q1, p1), k(q2, p2)) is represented by, δ(xi) = 0, δ(dxj) =αi.
The vector −(q1, q2) represents an acceleration in Simp1(k[x]), classified as δ(d−) This concept is further explored as an interaction in Chapter 5, and the findings can be extended from three dimensions to any dimension n.
(iv) Consider now the space of 2-dimensional representation ofP h(A).
It is an easy computation that any such is given by the actions, x1 a1 0
Theangular momentumis now given by,
, etc Andthe isospin, see (3.18) and (3.19), has the form,
(v) Let A = M2(k), and let us compute P h(A) Clearly the exis- tence of the canonical homomorphism,i:M2(k)→P h(M2(k)) shows that
P h(M2(k)) must be a matrix ring, generated, as an algebra, overM2(k) by di,j, i, j= 1,2, wherei,j is the elementary matrix A little computation shows that we have the following relations, d1,1 0 (d1,1)1,2=−(d2,2)1,2
From this follows that any section,ρ:P h(M2(k))→M2(k), ofi:M2(k)→
P h(M2(k)), is given in terms of an element φ∈M2(k), such that ρ(da) [φ, a].
The Dirac Derivation
The phase-space construction may, of course, be iterated Given the k- algebra A we may form the sequence, {P h n (A)} 1≤n , defined inductively by
Let \( i_{n}^{0} : P h_{n}(A) \rightarrow P h_{n+1}(A) \) represent the canonical embedding, while \( d_{n} : P h_{n}(A) \rightarrow P h_{n+1}(A) \) denotes the corresponding derivation The composition of \( i_{n}^{0} \) and the derivation \( d_{n+1} \) forms a derivation from \( P h_{n}(A) \) to \( P h_{n+2}(A) \) By universality, this leads to the existence of a homomorphism \( i_{n+1}^{1} : P h_{n+1}(A) \rightarrow P h_{n+2}(A) \) such that \( d_{n} \circ i_{n+1}^{1} = i_{n}^{0} \circ d_{n+1} \) It is important to note that functions and functors are composed from left to right in this context.
Clearly we may continue this process constructing new homomorphisms,
{i n j :P h n (A)→P h n+1 (A)} 0≤j≤n , with the property, dn◦i n+1 j+1 =i n j ◦dn+1.
It is easy to see, [21], that, i n p i n+1 q =i n q−1 i n+1 p , p < q i n p i n+1 p =i n p i n+1 p+1 i n p i n+1 q =i n q i n+1 p+1 , q < p,
Phase Spaces and the Dirac Derivation 23 i.e the P h ∗ A is a semi-cosimplicial algebra The system of k-algebras and homomorphisms of k-algebras{P h n (A), i n j } n,0≤j≤n has an inductive
(direct) limit,P h ∞ A, together with homomorphisms, in:P h n (A)−→P h ∞ (A) satisfying, i n j ◦in+1=in, j= 0,1, , n.
The family of derivations {dn} for 0≤n defines a unique Dirac derivation, δ: P h ∞ (A)−→P h ∞ (A), satisfying the condition in◦δ=dn◦in+1 This construction is universal, meaning that for any morphism i: A−→B and derivation ξ∈Derk(B), the factorization occurs through P h ∞ (A) and δ.
In the context of associative algebra B, the category of B-modules is denoted as Rep(B) The isomorphism classes of B-modules form a set, and the induced map from the forgetful functor, ω: Rep(P h ∞ (A)) → Rep(A), is a set-theoretical mapping, despite possessing a clearly defined tangent map.
As we shall see, assuming the algebraA of finite type, the set of sim- ple finite dimensional A-modules form an algebraic scheme, Simp6=∞(A).
Theorem 2.2.1 (Preparation) The canonical morphism i0 : A →
P h ∞ (A) parametrizes simple representations of A with fixed momentum, acceleration, and any number of higher order momenta.
This should be understood in the following way Consider, for any simple
0→EndA(V)→Endk(V)→Derk(A, Endk(V))→Ext 1 A (V, V)→0.
Let ρ:A →Endk(V) define the structure ofV, then any morphism ρ1 :
P h(A) → Endk(V) extending ρ, corresponds to a derivation, ξ1 : A →
Endk(V), and therefore, via the maps in the exact sequence above, to a tangent vector, also called ξ1, in the tangent space of the point v ∈
In the context of Simp(A) corresponding to V, any element ξ1 is associated with a pair (v, ξ) representing a point in Simp(A) and an infinitesimal deformation, or momentum Consequently, a morphism ρ2: Ph2(A) → Endk(V) that extends ρ1 relates to the triple (v, ξ1, ξ2), which encompasses a point, a momentum, and an infinitesimal deformation Given the canonical morphisms from Phr(A) to Ph∞(A), it follows that any morphism ξ: Ph∞(A) → Endk(V) that extends ρ generates a sequence of such tuples, regardless of order This illustrates a fundamental consequence of the definitions involved.
P h ∞ (A), that we identify all i n q , q = 1, , n, shows that the set of such morphismsξ, extending a given structure-morphism, parametrizes the set offormal curvesin Simp(A) throughv.
A key challenge in our understanding of physics is determining the future behavior of an object when its momentum and higher-order moments are known up to a specific order.
In our mathematical model, we examine the A-module preparation of the object V by establishing its structure as a P h ∞ (A)-module This setup necessitates a transformation in V, where the Dirac derivation δ, belonging to Derk(P h ∞ (A)), is mapped through the module's structure homomorphism, ρV: P h ∞ (A) → Endk(V) Consequently, this results in the element δV within Derk(P h ∞ (A), Endk(V)), achieved through the composition of the canonical linear maps.
Derk(P h ∞ (A), Endk(V))−→Ext 1 P h ∞ (A) (V, V)−→Ext 1 A (V, V) to the element δ(V)∈ Ext 1 A (V, V), i.e to a tangent vector ofSimpn(A) at the point, v, corresponding to V, see [18] Suppose first that, δ(V) is
0 This means that δV in Derk(P h ∞ (A), Endk(V)) is an inner derivation given by an endomorphismQ∈Endk(V), such that for everyf ∈P h ∞ (A), we findδ(f)(v) = (Qf−f Q)(v) ThisQis the correspondingHamiltonian,
(orDirac operator in the terminology of Connes, see [29]), and we have a situation that is very much like classical quantum mechanics, i.e a set-up
In the context of phase spaces, the Dirac derivation involves representing physical objects through a fixed Hilbert space, denoted as V This framework utilizes an algebra of observables, P h ∞ (A), which operates within this space Additionally, the concept of time and energy is encapsulated by a specific Hamiltonian operator, Q, highlighting its crucial role in the dynamics of the system.
A system defined by a P h ∞ (A)-module V, where δ(V) = 0, is termed stable or singular The system is in state ψ when an element ψ ∈ V is selected The Dirac derivation δ leads to the Hamiltonian operator Q, also known as the Dirac operator Time, represented by δ, transforms the state ψ into exp(τ Q)(ψ) ∈ V, which aligns with the isomorphism of module V established by the inner isomorphism of the algebra of observables, P h ∞ (A), defined by U := exp(τ δ), provided it is well-defined This scenario is analogous to classical quantum mechanics, reflecting the equivalence between the Schrödinger and Heisenberg frameworks.
To treat the situation when [δ]6= 0, we first have to take a new look at non-commutative algebraic geometry, as developed in [17], [18], [19].
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Non-commutative Deformations and the Structure of the Moduli Space of
Non-commutative Deformations
In recent studies, we explored non-commutative deformations of module families associated with non-commutative k-algebras and introduced the concept of swarms of right modules within k-linear abelian categories For any associative k-algebra S, we defined the category of S-valued associative k-algebras, characterized by diagrams of k-algebras.
S → ι R→ π S such that the composition of ι and π is the identity In particular, a r denotes the category of r-pointed k-algebras, i.e a s , with S = k r Any such r-pointed k-algebra R is isomorphic to a k-algebra of r×r-matrices
The radical of R is the bilateral ideal Rad(R) := kerπ, such that
R/Rad(R) ' k r The dual k-vector space of Rad(R)/Rad(R) 2 is called the tangent space ofR.
Forr= 1, there is an obvious inclusion of categories l⊆a 1 where l, as usual, denotes the category of commutative local Artinian k- algebras with residue fieldk.
Fix a not necessarily commutative k-algebra A and consider a right
A-moduleM The ordinary deformation functor
DefM :l→Sets is defined under the assumption that Ext i A (M, M) has finite k-dimension for i = 1, 2 It is well established in the literature, as referenced in [28] and [16], that Def M possesses a pro-representing hull H, which serves as the formal moduli of M Additionally, the tangent space of H is isomorphic to
Ext 1 A (M, M), andH can be computed in terms ofExt i A (M, M), i= 1,2 and theirmatric Massey products, see [15], [16], [21].
In the general case, consider a finite family V = {Vi} r i=1 of right A- modules Assume that, dimkExt 1 A (Vi, Vj) be the free k-algebra on dsym- bols, and let V ∈Simpn(A) Then
This should be compared with the results of [24], see also [4].
In general, the natural morphism, η(n) :A(n)→ Y
H A(n) (V)⊗ k Endk(V) is not an injection, as it follows from the following,
The idealI(2) is generated by [e1,1, e1,2, e2,2, e2,3] =e1,3 So
LetO(n), be the image ofη(n), then,
H A(n) (V)⊗kEndk(V) and for everyV ∈Simpn(A),
V ∈Simp n (A)H A(n) (V) Choosing bases in all V ∈ Simpn(A), then
Let xi ∈ A, i= 1, , d be generators of A, and consider their images
(x i p,q )∈Mn(B) Now,B is commutative, so thek-sub-algebra C(n)⊂B generated by the elements{x i p,q }i=1, ,d; p,q=1, ,nis commutative We have an injection,
O(n)→Mn(C(n)), and for allV ∈Simpn(A), with a chosen basis, there is a natural compo- sition of homomorphisms ofk-algebras, α:Mn(C(n))→Mn(H A(n) (V))→Endk(V), inducing a corresponding composition of homomorphisms of the centers,
Z(α) :C(n)→H A(n) (V)→k This sets up a set theoretical injective map, t:Simpn(A)−→M ax(C(n)),defined byt(V) :=kerZ(α).
Since A(n) → H A(n) (V) ⊗k Endk(V) is topologically surjective,
H A(n) (V)⊗ k Endk(V) is topologically generated by the images ofxi, i1, , d It follows that we have a surjective homomorphism,
Categorical properties implies, that there is another natural morphism,
H A(n) (V)→C(n)ˆ t(V ), which composed with the former is an automorphism ofH A(n) (V) Since
H A(n) (V)⊗kEndk(V), it follows that for mv ∈ M ax(C(n)), corresponding to V ∈ Simpn(A), the finite dimensionalk-algebraMn(C(n)/mv 2) sits in a finite dimensional quotient of some,
H A(n) (V)⊗kEndk(V). whereV⊂Simpn(A) is finite However, by Lemma (2.5), the morphism,
H A(n) (V)⊗kEndk(V) is topologically surjectiv Therefore the morphism,
A(n)−→Mn(C(n)/mv 2) is surjectiv, implying that the map
We now have the following theorem, see Chapter VIII,§2, of the book
[25], where part of this theorem is proved.
Theorem 3.4.8 Let V ∈ Simpn(A), correspond to the point mv ∈
(i) There exist a Zariski neighborhoodUv ofv inSimp1(C(n))such that any closed point m 0 v ∈U corresponds to a unique pointV 0 ∈Simpn(A).
Let U(n) be the open subset of Simp1(C(n)), the union of all Uv for
(ii) O(n) defines a non-commutative structure sheafO(n) :=OU (n) of
Azumaya algebras on the topological spaceU(n)(Jacobson topology).
(iii) The centerS(n)ofO(n), defines a scheme structure onSimpn(A).
(iv) The versal family ofn-dimensional simple modules,V˜ :=C(n)⊗ k V, over Simpn(A), is defined by the morphism, ˜ ρ:A→O(n)⊆EndC(n)(C(n))⊗kV)'Mn(C(n)).
(v) The trace ring T r˜ρ ⊆ S(n) ⊆ C(n) defines a commutative affine scheme structure onSimpn(A) Moreover, there is a morphism of schemes, κ:U(n)−→Simpn(A), such that for any v∈U(n),
Let ρ: A → Endk(V) be a surjective homomorphism of k-algebras that defines V in Simpn(A) Consider the elementary matrices ei,j in Endk(V) and select elements yi,j in A such that ρ(yi,j) = ei,j Define σ as the cyclical permutation of the integers {1, 2, , n} and construct sk as [y σ k (1), σ k (2), y σ k (2), σ k (2), y σ k (2), σ k (3), , y σ k (n), σ k (n)] The overall expression is given by s = Σ k=0 to n-1 sk ∈ A.
The matrix ρ(s) in Endk(V) is defined as having non-zero elements equal to 1 exclusively in the (σ k (1), σ k (n)) position, resulting in a determinant of either +1 or -1 Consequently, the determinant det(s) in C(n) remains non-zero at the point v in Spec(C(n)) that corresponds to V.
U = D(det(s)) ⊂ Spec(C(n)), and consider the localization O(n){s} ⊆
The inclusion Mn(C(n){det(s)}) arises from the general properties of localization Each closed point v₀ in U corresponds to an n-dimensional representation of A, where elements in I(n−1) are invertible Consequently, this representation cannot possess an m < n dimensional quotient, indicating that it must be simple.
Sinces∈I(n−1), the localizedk-algebraO(n){s}does not have any sim- ple modules of dimension less thann, and no simple modules of dimension
For any finite-dimensional O(n){s}-module V with dimension m, the image of s in Endk(V) must be invertible, with the inverse being the image of a polynomial of degree m-1 in s If V is simple over O(n){s}, the homomorphism from O(n){s} to Endk(V) is surjective, indicating that V is also simple over A Given that s belongs to I(n-1), it follows that m is at least n If m exceeds n, we can similarly construct an element in I(n) that maps to a nonzero element of Endk(V) Since I(n) equals zero in A(n) and consequently in O(n){s}, we have established our proof According to M Artin's theorem, O(n){s} is an Azumaya algebra with center S(n){s} = Z(O(n){s}) Thus, O(n) defines a presheaf O(n) on U(n), consisting of Azumaya algebras with center S(n) = Z(O(n)).
Clearly, anyV ∈Simpn(A), corresponding tomv∈M ax(C(n)) maps to a point κ(v)∈Simp(O(n)) Let mκ(v) be the corresponding maximal ideal ofS(n) SinceO(n) is locally Azumaya, it follows that,
Spec(C(n)) is, in a sense, a compactification ofSimpn(A) It is, how- ever, not the correct completion of Simpn(A) In fact, the points of
The expression Spec(C(n))−Simpn(A) relates to semi-simple modules that cannot be transformed into simple n-dimensional modules We will revisit the concept of completion and examine the degeneration processes that take place at infinity in Simpn(A).
Example 3.3 Let us check the case of A = k < x1, x2 >, the free non-commutative k-algebra on two symbols First, we shall compute
Ext 1 A (V, V) for a particularV ∈Simp2(A), and find a basis{t ∗ i } 5 i=1, repre- sented by derivations ∂i :=∂i(V)∈Derk(A, Endk(V)), i=1,2,3,4,5 This is easy, since for any twoA-modulesV1, V2, we have the exact sequence,
→Ext 1 A (V1, V2)→0 proving that, Ext 1 A (V1, V2) k(A, Homk(V1, V2))/T riv, whereT riv is the sub-vector space of trivial derivations Pick V ∈ Simp2(A) defined by the homomorphism A → M2(k) mapping the generators x1, x2 to the matrices
=:e2,2=e2, and recall also that for any 2×2-matrix (ap,q)∈M2(k),ei(ap,q)ej=ai,jei,j.
The trivial derivations are generated by the derivations {δp,q}p,q=1.2, de- fined by, δp,q(xi) =Xiep,q−ep,qXi.
Clearlyδ1,1+δ2,2= 0 Now, compute and show that the derivations∂i, i1,2,3,4,5, defined by,
∂3(x2) =e2,1, form a basis for Ext 1 A (V, V) = Derk(A, Endk(V))/T riv Since
Ext 2 A (V, V) = 0 we find H(V) = k > and so
H(V) com 'k[[t1, t2, t3, t4, t5]] The formal versal family ˜V, is defined by the actions ofx1, x2, given by,
One checks that there are polynomials ofX1, X2 which are equal totiep,q, modulo the ideal (t1, , t5) 2 ⊂H(V), for alli, p, q= 1,2 This proves that
C(2)ˆ v must be isomorphic toH(V), and that the composition,
A−→A(2)−→M2(C(2))⊂M2(H(V))) is topologically surjective By the construction ofC(n) this also proves that
C(2)'k[t1, t2, t3, t4, t5]. locally in a Zariski neighborhood of the origin Moreover, the Formanek center, see [4], in this case is cut out by the single equation: f :[X1, X2] =−((1 +t3) 2 −t2t5) 2 + (t1(1 +t3) +t2t4)(t4(1 +t3) +t1t5).
In the realm of computing, we encounter specific formulas such as trX1=t4 and trX2=t1, with determinants given by detX1=−t5−t3t5 and detX2=−t2−t2t3 The trace of the product X1X2 is expressed as tr(X1X2) = (1 +t3)² +t2t5, leading to the trace ring represented as k[t1, t2+t2t3, 1 + 2t3+t²3 +t2t5] =:k[u1, u2, , u5] Here, the variables are defined as u1=t1, u2=(1 +t3)t2, u3=(1 +t3)² +t2t5, u4=t4, and u5=(1 +t3)t5 Additionally, the function f is formulated as f =−u²3 +4u2u5 +u1u3u4 +u²1u5 +u2u²4 Notably, k[t] is algebraic over k[u], with a discriminant defined as ∆ := 4u2u5(u²3 −4u2u5) = 4(1 +t3)²t2t5((1 + t3)² −t2t5)², indicating the existence of an étale covering.
Notice that if we putt1=t4= 0, thenf = ∆ See the Example (3.7)
In the affine scheme Spec(C(n)), elements of the complement of U(n) can be represented by both indecomposable and decomposable representations, highlighting the significance of decomposable representations in this context.
In general, W cannot be transformed into a simple representation due to the requirement that effective deformations preserve EndA(W) Although we refer to Spec(C(n)) as a compactification of Simpn(A), it is not an effective completion The points at infinity missing from Simpn(A) should be represented as indecomposable representations where EndA(W) equals k.
Any such is an iterated extension of simple representations {Vi}i=1,2, s, with representation graph Γ (corresponding to an extension type, see
[18]), and Ps i=1dim(Vi) = n To simplify the notations we shall write,
|Γ|:={Vi} i=1,2, s In [16], we treat the problem of classifying all such in- decomposable representations, up to isomorphisms Let us recall the main ideas.
In a scenario where we have a family of modules {Vi} with finite-dimensional Ext^1 A (Vi, Vj) as ask-vector spaces, we can represent this relationship using an ordered graph Γ, where the nodes correspond to the modules |Γ|={Vi} By initiating the process from a chosen node in Γ, we can explore various methods to construct an extension of the module Vi 1 alongside the corresponding module.
Vi 2 corresponding to the end point of the first arrow of Γ, then continue, choosing an extension of the result with the module corresponding to the endpoint of the second arrow of Γ, etc untill we have reached the endpoint of the last arrow Any finite length module can be made in this way, the oppositely ordered Γ corresponding to a decomposition of the module into simple constituencies, by peeling off one simple sub-module at a time, i.e by picking one simple sub-module and forming the quotient, picking a second simple sub-module of the quotient and taking the quotient, and repeating the procedure until it stops.
Morphisms, Hilbert Schemes, Fields and Strings
In our exploration of moduli spaces, we examined representations of finitely generated k-algebras Additionally, we could have investigated the Hilbert functor, HA r, which pertains to subschemes of length r within the spectrum of the algebra A Furthermore, we could have analyzed the moduli space F(A;R), which focuses on morphisms κ: A → R for specified algebras A and R.
The key distinction lies in the fact that for finite n, the set Simpn(A) exhibits a well-defined finite dimensional scheme structure However, this property does not generally extend to the set H r A or the set of fields F(A; R), often referred to by physicists, unless additional conditions, known as decorations, are imposed on the fields.
Artinian of length n, then the corresponding F(A;R) does exist and has a nice structure, both as commutative and as non-commutative scheme.
The toy model of relativity theory is based on the set of surjective homomorphisms from k[x1, x2, x3] to R = k^2 In general, the space F(A;R) possesses a tangent structure Depending on the perspective, the tangent space of a morphism φ: A → R can be defined accordingly.
In the study of Deformations and Moduli Spaces, the role of the inner derivations induced by R is crucial, even though F(A;R) lacks a clear algebraic structure This general framework is significant as it underpins our approach to Quantum Field Theory, which will be explored further in the next chapter, particularly in relation to the concept of time.
We may also consider the notion of string, in the same language as above Let us, for the fun of it, make the following :
Definition 3.5.1 Ageneral string, or a g-string, is an algebraRtogether with a pair of Ph-points, i.e a pair of homomorphismsi:P h(R)→k(pi), corresponding to two pointsk(pi)∈Simp1(R) each outfitted with a tangent ξi.
In our analysis, we focus on the non-commutative tangent space of a pair of points k(pi) within the framework of g-strings, specifically restricting our discussion to the case where n = 1 This simplification allows for a clearer understanding of the essential properties of g-strings, and we will explore the broader implications of this concept in future work.
We shall call it thespace of tensions, between the two points of the string.
The space of g-strings in A, denoted as spaceStringg(A), consists of isomorphism classes of algebra homomorphisms κ: A → R, where R is a g-string Isomorphisms must align with the isomorphisms of the g-string, preserving the two PhR-points Each g-string in A, represented by κ: A → R, generates a unique commutative diagram of algebras The von Neumann condition for a string κ is defined as i◦Phκ◦d = κ∗ξi = ξi = 0 for i = 1 ∨ i = 2, where xj (for j = 1, , n) and σl (for l = 1, , p) are parameters of A.
R, is equivalent to the condition,
Any derivation ξ in Derk(A, R) can be naturally lifted to Derk(P hA, P hR) by defining ξ(a) = d(ξ(a)) Utilizing the general framework of diagram deformations, as referenced in [14], we observe that a family of morphisms κ generates a corresponding family of diagrams For parameters τk, where k = 1, , d, and M = Spec(M), the differential dτi in P hM corresponds to a derivation τi in Derk(A, R), leading to tangents ξi, where i = 1, 2, at the points k(pi) of Simp1(A) The Dirichlet condition for the string is expressed as ξi = 0 for i = 1 ∨ 2, which is equivalent to the stated condition.
These conditions will define new moduli spaces which we shall call
In the affine case, the structure of the spaces String R vN (A) and String R D (A) presents challenges; however, by replacing A and R with projective schemes, we can establish the existence of all moduli spaces as classical schemes The volume form of the space, which the string is expanding into, will yield an action functional, S, defined on String R (A), as discussed in the next chapter.
In conclusion, there exists a non-commutative deformation theory for fields, analogous to that of representations of associative algebras Given a finite family of fields {κi} for i = 1, , r, we can define the A-bimodule Ri,j for each pair (i, j) with 1 ≤ i, j ≤ r, where κi serves as the left module structure and κj as the right By adapting the definition of the non-commutative deformation functor for representations and substituting Homk(Vi, Vj) with Ri,j, we can demonstrate many results discussed earlier in this chapter This development may also relate to the interaction problems explored in the previous chapter of this book.
In Example 3.4, we revisit Example (1.1)(ii), which demonstrates that the string of dimension 0, represented as R=k² and P h(R) = k < x, dx > /((x² - r²), (xdx + dxx)), has unique points at k(±r) The tension space has a dimension of 1, satisfying the von Neumann condition, while the moduli space of k²-strings in A=k[x₁, x₂, x₃] simplifies to H := Speck[t₁, , t₆]−∆ Analyzing the string with R=k[x]/(x²) and P hR=k[x, dx]/(x², (xdx + dxx)), we find a single point in R but a line of points in P hR, all corresponding to x=0 in R This leads to a 2-dimensional space of strings sharing the same R, contrasting with the blow-up ˜H, as referenced in [20].
In dimension 1, the simplest closed string is represented as R = k[x, y]/(f), where f = x² + y² - r² This leads to the definition of P hR = k < x, y, dx, dy > /(f, [x, y], d[x, y], df) The two points, i: P hR → k(pi), act on k(pi) := k, resulting in the mappings xi, yi, (dx)i, and (dy)i for i = 1, 2 It is evident that the vectors ξi := ((dx)i, (dy)i) serve as tangent vectors to the circle.
Deformations and Moduli Spaces 49 points pi, and if p1 6=p2 we find thatExt 1 P hR (k(p1), k(p2) =k The von
Neumann condition is, ξi = 0, i= 1 ∨i= 2, and this clearly means that
∂σ = ∂y ∂σ = 0 at one of the pointspi The 1-dimensionalopen stringis now left as an exercise.
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