Free Algebras and Tensor Products Joe Lesnefsky Section 1 Introduction Introduction ∗ Physicists use tensors all the time, and I have yet to see a physicist actually explain what a tensor really is ∗ Have you ever used ∗ = , ∗ , 4 ∗ | =| | ∗ = + ↑↓⟩ ↑⟩ ↓⟩ ∗ 2 2 = 3 1 � � � ∗ And been confused what you were actually doing? ⊗ ⊕ Introduction (cont.) ∗ I want to answer: ∗ What is a tensor ∗ How do you build tensors ∗ How do you calculate with tensors ∗ Tensors hiding and overlooked in everyday physics ∗ To do this properly, we need to talk about category theory, multi-linearity, and free algebra presentations Introduction (cont.) ∗ If you carefully examine the example equations, you will see they are different “things” built up from smaller “things” ∗ : Hom , Hom , ℳ ⊗ ℳ → ℝ ⇒ ∗ | = | | | ∈ ℳ ℝ ⊗ ℳ ℝ ∗ = + = 2 + 2 4 End ↑↓⟩ ↑⟩ ↓⟩ ⇒ �↑⟩ ⊗ ↓⟩ ∈ ℂ ⊗ ℂ ≈ ℂ End End 2 � � ∗ We see� that2 � this symbol⇒4 joins� ⊗ these different⊗ � ∈ “things”ℂ ⊗ togetherℂ somehow≈ ℂ to make another “thing” ∗ How do we mathematically⊗ write down “thing”? Section 2 Category Theory Category Theory Definition 1 : A Category, , consists of 3 properties I. A collection of objects Ob II. Sets of morphismsℭ between these objects, including a distinct identity morphism : ℭ III. A composition of morphisms function ° which allows you to combine morphisms → Example Object Category Morphisms Hom , rings ring homomorphisms ℳ ⊗ ℳ ℝ | - vector spaces Continuous maps, 4 often restricted to �↑⟩ ⊗ ↓⟩ ℂ linear maps End rings ring 4 homomorphisms � ℂ Functors and Monoidal Categories Definition 2 : A functor, , maps category s.t. I. for Ob , associates for some Ob II. for Mor associates ℭ for⇝ some Mor III. for ∀ ∈ Morℭ , Mor ⇝ obeys II. and∈ III. From Defn 1. ∀ ∈ ℭ ⇝ ∈ ∀ ∈ ℭ ∈ Definition 3 : A monoidal category is a category s.t. functor : × where, up to natural isomorphism I. functor is associative : ℭ ∃ ⊗ forℭ , ℭ, ⇝ ℭOb II. there an identity object 1 Ob s.t. 1 1 ⊗ ⊗ ⊗ ≈ ⊗ ⊗ ∈ ℭ N.B. – The ∃ functor does not specify∈ HOWℭ we ⊗a linking ≈ these⊗ ≈ categories together just that we can do it somehow ⊗ Given the definition of a monoidal category, mathematician John Baez has made the conjecture . The (strong) Baez Conjecture Conjecture 1 : All physical phenomena can be modeled by a monoidal category Products ( ) vs. Co-Products ( ) ⊗ ⊕ ∗ The tensor functor does not say HOW objects are to be combined ∗ There are 2 ways to⊗ do this – Let = for , Ob for a monoidal category ∈ℑ ∗ Product Structure ⊗ ∗Focuses∈ ℭ on projecting down from ℭ with projections ⊗ ∗ Co-Product Structure ⟼ ∗ Focuses on including with inclusions ⊕ ⟼ Category Theory - Review ∗ So, given category theory and the (strong) Baez conjecture, we see that given the functor on a monoidal category we CAN link all mathematical objects useful for physics together⊗ ∗ Now, the question is HOW do we explicitly construct a s.t. it is useful for our physics calculations ∗ To do this we need to talk about multi-linearity . ⊗ Section 3 Multi-Linearity Why Do We Care About Linearity? ∗ Somewhat surprisingly, after some research I was not able to find a lot of literature about this fact ∗ Conjecture: Because it works!! ∗ Linear things are easy to calculate! ∗ QM is built on linear algebra: linear maps / eigenvalues / eigenvectors ∗ Superposition of waves implies linearity, lots of physics consists of non-interacting waves ∗ Often times, perturbative calculations are only carried out to linear order Multi-Linearity ∗ Given the functor and the desire to make compound objects out of linear primitives, we want to create an object linear in each⊗ primitive ⊗ Definition 4: For vector spaces , over field , a map 2 is multi-linear iff, for , and I. the vector sum ∶ ⊗ ⋯ − ⋯ ⊗ → , , + ∈, , =∈ , , , , + , , , , II. The scalar multiplication 1 1 1 , ⋯ , , ⋯, = , ⋯, ⋯= , , ⋯, , ⋯ 1 ⋯ ⋯ 1 ⋯ 1 ⋯ ⋯ ∗ How do we construct a functor which imposes multi- linearity? ⊗ Section 4 Free Algebras Groups - Review Definition 5: A group is a set, , with a binary operation × I. Operation is associative: = , , II. There an identity element 1 s.t. 1 = ∙∶1= → III. For ∙ there s.t.∙ ∙ =∙ ∙ ∀=1 ∈ ∃ −1 ∈ −1 ∙ −1 ∙ ∀ ∈ ∀ ∈ ∃ ∈ ∙ ∙ Free Groups Definition 6: Given a (alphabet) set , let a word in be a finite string of elements of Definition 7: Let the empty word, , be the word with no elements. Definition 8: Let concatenation be ∅an operation × by appending 2 words in a particular order → Definition 9: For define the element s.t. = = −1 −1 −1 Definition 10: Let∀ a reduced∈ word be a word with∈ all possible evaluated ∅ and removed for −1 Definition 11:∀ A ∈free group over (alphabet) set is the set of all reduced words of with the group operation of concatenation. I. Concatenation is associative – proof is beyond the scope of this lecture II. Identity element is III. Inverses are given by Defn. 9 ∅ Free Groups (cont.) ∗ Example: Let = , , . Words of are strings of elements of . For example ∗ - reduced ∗ -reduced ∗ −1 −1 −1 −1 – NOT reduced ∗ = −1 −=1 −1 so = (NOT reduced) and reducing−1 gives = −1 (reduced) 1 2 12 12 Properties of Free Groups ∗ Free groups allow you to impart a group structure to any (alphabet) set ∗ Given the (alphabet) set , the free group , is the most generic group you can form over ∗ All free groups have infinite order ∗ The simplest non-trivial free group is ∗ In general, free groups are NOT abelian ∗ Starting with these most generic groups over≈ aℤ set, we can introduce constraints called relators to customize the groups to our requirements (constraints) Equivalence Relations and Relators Definition 12: Given a set , a relation is a subset of × Example: Relations associate 2 elementsℛ of a set together. Consider the relations of =, , > etc . So = is associating elements , with each other via relation = ~ that satisfies the properties Definition≤ 13: An Equivalence Relation is a relation ∈ , , I. Reflexive - ~ ∀ II. ∈Symmetry ~ ~ III. Transitive ~ and ~ ~ ⇒ Example: Recall from grade school ⇒ these are all the properties of =. Thus an equivalence relation is a generalization of the idea of equality Definition 14: A relator is an equivalence relation on a free group Relators (cont.) ∗ Example: Construction of from ∗ The group is an abelian group of 2 elements with 2 every element having orderℤ 2 ( =ℤ1). 2 ∗ If we applyℤ the relator = to2 the free group we can see that2 for a word = = = ∗ So, if there≈ ℤ are an even2 number2 of a’s we have ~ but if there are⋯ an odd number ⋯ of ’s⋯ we have ~ ∗ This is exactly the structure of ℤ2 Group Presentations ∗ The previous example is that of a group presentation, which is a free group modulo (the normal closure of) some number of equivalence relations. ∗ For the previous example it is written: ∗ = ∗ In general, for2 an (alphabet) set and a set of relators ℤ2 ≈ ∶ , a presentation of a group is given by ∗ ℛ ∈ℑ ≈ ∶ ℛ ∈ℑ ≈ �≪ ℛ ∈ℑ≫ Free Algebras Definition 15: Given a vector space and an associative binary operation , : × an algebra is the pair , , ∙ ∙ → Definition 16: Given an algebra∙ ∙ = , , the free algebra is the algebra preserving the algebraic structure of using all elements of as generators ∙ ∙ ∗ Defns. 15 and 16 generalizes to any abelian category ∗ This includes virtually anything we would want for physics ∗ Examples of algebras are ∗ Cross Products / Outer Products ∗ Clifford Algebras N.B. – My defn for Free Algebras differs from the accepted defn written in terms of a universal property. Given the accepted defn I conjecture that these are equivalent, however I have not proven this rigorously Free Algebra “Presentations” Theorem 1: All groups are isomorphic to a presentation of a free group Theorem 2: All algebras have a projective resolution ∗ All the aforementioned properties of free groups carry over to other categories of algebraic objects mainly ∗ Free Rings ∗ Free R-Modules ∗ Free Vector Spaces ∗ If we want to “custom build” a mathematical object for our physics purposes all we need to do is the following: ∗ Cleverly choose an appropriate free algebra ∗ Figure out what physical constraints we want to impose ∗ Write relators for these constraints ∗ There is an object isomorphic to the free category modulo the relators (Thm 2) and should solve our problem Free Algebras - Review ∗ Given some set and some algebraic structure, , on that set, it is possible to create the “most generic” algebra, the free algebra , given a particular and ∗ Given a set of constraints we would like to impose on , we can write relators to impose these constraints ∗ Thm 2. guarantees that there will be an appropriate algebra isomorphic to this “presentation” which satisfies these constraints Section 5 Application : The Dirac Equation Origins of the Dirac Equation ∗ In an attempt to find a SR QM equation, we can look at our friend the Schrödinger Equation with a (classical) SR Hamiltonian modivated by E = + (for a free particle , = 0) 2 2 2 4 ∗ , = + , 2 2 2 4 ∗ Ψ , = ℏ + Ψ , 2 2 2 2 2 4 ∗ − Ψ+ −ℏ , =Ψ0 2 2 2 2 2 4 − ℏ − Ψ Origins of the Dirac Equation (cont.) ∗ = 0 – the Klein-Gordon Equation ∗ Assuming Minkowski2 4 metric − Ψ 1 0 0 0 0 1 0 0 ∗ = −0 0 1 0 0 0 0 1 ∗ So adopting the notation of the d‘Alembertian wave operator = ∗ =0 □ 2 4 □ − Ψ Problems with the Klein-Gordon Equation ∗ A posteriori we know the Klein-Gordon Equation is a poor choice for a Hamilton / Lagrangian because: ∗ It is a second order