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 ∗ = , 𝑖𝑖 ∗ 𝜇𝜇𝜇𝜇 , 𝜇𝜇 𝜈𝜈 𝑆𝑆 4 𝛾𝛾 𝛾𝛾 ∗ | =𝜌𝜌| | 𝑔𝑔𝜇𝜇𝜇𝜇 𝑅𝑅𝜇𝜇𝜇𝜇𝜇𝜇 ∗ = + ↑↓⟩ ↑⟩ ↓⟩ ∗ 2 2 = 3 1 � �𝟏𝟏 �𝟐𝟐 ∗ And𝑺𝑺 been𝒔𝒔 confused𝒔𝒔 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 , 𝑔𝑔𝜇𝜇𝜇𝜇 ℳ ⊗ ℳ → ℝ ⇒ ∗ | = | | | 𝑔𝑔𝜇𝜇𝜇𝜇 ∈ ℳ ℝ ⊗ ℳ ℝ ∗ = + = 2 + 2 4 End ↑↓⟩ ↑⟩ ↓⟩ ⇒ �↑⟩ ⊗ ↓⟩ ∈ ℂ ⊗ ℂ ≈ ℂ End End 2 � 𝟏𝟏 𝟐𝟐 � 𝟏𝟏 𝟐𝟐 ∗ We𝑺𝑺 see𝒔𝒔� that2 𝒔𝒔� this symbol⇒4 𝑺𝑺 joins𝒔𝒔� ⊗ these𝟏𝟏 𝟏𝟏 different⊗ 𝒔𝒔� ∈ “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 𝟏𝟏 𝑋𝑋 → 𝑋𝑋 Example Object Category Morphisms

Hom , rings ring homomorphisms 𝑔𝑔𝜇𝜇𝜇𝜇 ℳ ⊗ ℳ ℝ | - vector spaces Continuous maps, 4 often restricted to �↑⟩ ⊗ ↓⟩ ℂ 𝔽𝔽 linear maps End rings ring 4 homomorphisms 𝑺𝑺� ℂ Functors and Monoidal Categories

Definition 2 : A functor, , maps category s.t. I. for Ob , associates for some Ob II. for Mor 𝔉𝔉 associates ℭ for⇝ 𝔇𝔇some Mor III. for ∀ 𝑋𝑋 ∈ Morℭ , Mor𝑋𝑋 ⇝ 𝑌𝑌 obeys II. 𝑌𝑌and∈ III. From𝔇𝔇 Defn 1. ∀ 𝑓𝑓 ∈ ℭ 𝑓𝑓 ⇝ 𝑔𝑔 𝑔𝑔 ∈ 𝔇𝔇 ∀ 𝑓𝑓 ∈ ℭ 𝔉𝔉 𝑓𝑓 ∈ 𝔇𝔇 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 ⊗ 𝐴𝐴 ⊗ 𝐵𝐵 ⊗ 𝐶𝐶 ≈ 𝐴𝐴 ⊗ 𝐵𝐵 ⊗ 𝐶𝐶 𝐴𝐴 𝐵𝐵 𝐶𝐶 ∈ ℭ N.B. – The ∃ functor does not specify∈ HOWℭ we ⊗a linking𝐴𝐴 ≈ 𝐴𝐴 these⊗ ≈ categories𝐴𝐴 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 𝑖𝑖∈ℑ 𝑖𝑖 ∗ Product Structure 𝑌𝑌 ⊗ 𝑋𝑋 𝑖𝑖 𝑋𝑋 ∗𝑌𝑌Focuses∈ ℭ on projecting down from ℭ with projections ⊗ 𝜋𝜋𝑖𝑖 𝑖𝑖 ∗ Co-Product Structure 𝑌𝑌 ⟼ 𝑋𝑋 𝜋𝜋𝑖𝑖 ∗ Focuses on including with inclusions ⊕ 𝔦𝔦𝑘𝑘 𝑋𝑋𝑗𝑗 ⟼ 𝑌𝑌 𝔦𝔦𝑘𝑘 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 𝑉𝑉 𝑊𝑊 𝔽𝔽 𝜙𝜙 ∶ 𝑉𝑉 ⊗ ⋯ 𝑛𝑛 − ⋯ ⊗ 𝑉𝑉 → 𝑊𝑊 , , 𝑎𝑎+𝑏𝑏 ∈, 𝑉𝑉 , 𝛼𝛼=∈ 𝔽𝔽 , , , , + , , , , II. The scalar multiplication 1 𝑛𝑛 1 𝑛𝑛 1 𝑛𝑛 𝜙𝜙 𝑥𝑥 , ⋯ , 𝑎𝑎 ,𝑏𝑏 ⋯, 𝑥𝑥 = 𝜙𝜙 𝑥𝑥 , ⋯, 𝑎𝑎 ⋯=𝑥𝑥 , 𝜙𝜙 ,𝑥𝑥 ⋯, 𝑏𝑏, ⋯ 𝑥𝑥

𝜙𝜙 𝑥𝑥1 ⋯ 𝛼𝛼𝑥𝑥𝑖𝑖 ⋯ 𝑥𝑥𝑛𝑛 𝛼𝛼𝛼𝛼 𝑥𝑥1 ⋯ 𝑥𝑥𝑛𝑛 𝜙𝜙 𝑥𝑥1 ⋯ 𝛼𝛼𝑥𝑥𝑗𝑗 ⋯ 𝑥𝑥𝑛𝑛 ∗ 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 element𝐺𝐺 1 s.t. 1 = ∙∶1𝐺𝐺= 𝐺𝐺 → 𝐺𝐺 III. For ∙ there 𝑎𝑎 s.t.∙ 𝑏𝑏 ∙ 𝑐𝑐 𝑎𝑎=∙ 𝑏𝑏 ∙ 𝑐𝑐 ∀=𝑎𝑎1 𝑏𝑏 𝑐𝑐 ∈ 𝐺𝐺 ∃ −1 ∈ 𝐺𝐺 −1 ∙ 𝑎𝑎−1 𝑎𝑎 ∙ 𝑎𝑎 ∀𝑎𝑎 ∈ 𝐺𝐺 ∀𝑎𝑎 ∈ 𝐺𝐺 ∃ 𝑎𝑎 ∈ 𝐺𝐺 𝑎𝑎 ∙ 𝑎𝑎 𝑎𝑎 ∙ 𝑎𝑎 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, 𝑆𝑆 , be the word 𝑆𝑆with no elements. 𝑆𝑆

Definition 8: Let concatenation be ∅an operation × by appending 2 words in a particular order 𝑆𝑆 𝑆𝑆 → 𝑆𝑆 Definition 9: For define the element s.t. = = −1 −1 −1 Definition 10: Let∀ a𝑎𝑎 reduced∈ 𝑆𝑆 word be a word 𝑎𝑎with∈ all𝑆𝑆 possible𝑎𝑎𝑎𝑎 𝑎𝑎 evaluated𝑎𝑎 ∅ and removed for −1 𝑎𝑎𝑎𝑎 Definition 11:∀ A𝑎𝑎 ∈free𝑆𝑆 group over (alphabet) set is the set of all reduced words of with the group operation of concatenation. I. Concatenation is associative – proof is beyond𝑆𝑆 the scope of this lecture 𝑆𝑆 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 ∗ - reduced𝑆𝑆 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑆𝑆 𝑆𝑆 ∗ -reduced 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∗ −1 −1 −1 −1 – NOT reduced 𝑎𝑎 𝑏𝑏𝑐𝑐 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎 𝑐𝑐 ∗ = −1 −=1 −1 so = (NOT 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 reduced) and reducing−1 gives = −1 (reduced) 𝑤𝑤1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑤𝑤2 𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑤𝑤1𝑤𝑤2 𝑎𝑎𝑎𝑎𝑎𝑎𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑤𝑤1𝑤𝑤2 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 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 you𝑆𝑆 can form over ∗ All free groups have infinite𝑆𝑆 order 𝐹𝐹 𝑆𝑆 ∗ The simplest non-trivial free group𝑆𝑆 is ∗ In general, free groups are NOT abelian ∗ Starting with these most generic groups𝐹𝐹 𝑎𝑎 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 associate𝑆𝑆 2 elementsℛ of a set together.𝑆𝑆 𝑆𝑆 Consider the relations of =, , > etc . . . So = is associating elements , with each other via relation = ~ that satisfies the properties Definition≤ 13: An Equivalence𝑎𝑎 𝑏𝑏 Relation is a relation𝑎𝑎 𝑏𝑏 ∈ 𝑆𝑆 , , I. Reflexive - ~ ∀𝑥𝑥 𝑦𝑦II.𝑧𝑧 ∈Symmetry𝑆𝑆 ~ ~ III. Transitive 𝑥𝑥~ 𝑥𝑥 and ~ ~ 𝑥𝑥 𝑦𝑦 ⇒ 𝑦𝑦 𝑥𝑥 Example: Recall from𝑥𝑥 𝑦𝑦grade𝑦𝑦 school𝑧𝑧 ⇒ these𝑥𝑥 𝑧𝑧 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 𝑎𝑎 we can see that2 for a word = = 𝑎𝑎 =𝑒𝑒 ∗ So,𝐹𝐹 𝑎𝑎if there≈ ℤ are an even2 number2 of a’s we have ~ but if𝑤𝑤 there𝑎𝑎𝑎𝑎𝑎𝑎 are⋯ an𝑎𝑎𝑎𝑎𝑎𝑎 odd number𝑎𝑎 ⋯ 𝑎𝑎 of 𝑒𝑒 ’s⋯ we𝑒𝑒 have ~ ∗ This is exactly the structure of 𝑤𝑤 𝑒𝑒 𝑎𝑎 𝑤𝑤 𝑎𝑎 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 ≈ 𝑎𝑎 ∶ 𝑎𝑎 𝑒𝑒 , a presentation of a group is given by 𝑆𝑆 ∗ ℛ𝑖𝑖 𝑖𝑖∈ℑ 𝐺𝐺 𝐹𝐹 𝑆𝑆 𝐺𝐺 ≈ 𝑆𝑆 ∶ ℛ𝑖𝑖 𝑖𝑖∈ℑ ≈ �≪ ℛ𝑖𝑖 𝑖𝑖∈ℑ≫ Free Algebras

Definition 15: Given a vector space and an associative binary operation , : × an algebra is the pair , , 𝑉𝑉 ∙ ∙ 𝑉𝑉 𝑉𝑉 → 𝑉𝑉 Definition 16: Given an𝑉𝑉 algebra∙ ∙ = , , the free algebra is the algebra preserving the algebraic structure of using all elements of as generators 𝒜𝒜 𝑉𝑉 ∙ ∙ 𝐹𝐹 𝒜𝒜 ∗ Defns. 15 and 16 generalizes to𝒜𝒜 any abelian category𝒜𝒜 ∗ 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 free𝑆𝑆 algebra , given a particular𝒜𝒜 and ∗ Given a set of constraints𝐹𝐹 we𝒜𝒜 would like to impose𝑆𝑆 on 𝒜𝒜, we can write relators to impose these constraints ∗ Thm𝐹𝐹 𝒜𝒜 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 ∗ , = 𝑝𝑝 𝑐𝑐 + 𝑚𝑚 𝑐𝑐 , 𝑉𝑉 𝒙𝒙 𝑡𝑡 𝜕𝜕 2 2 2 4 ∗ 𝑖𝑖 𝜕𝜕𝜕𝜕 Ψ 𝒙𝒙 𝑡𝑡 , 𝑐𝑐= 𝑖𝑖ℏ𝛻𝛻 𝑚𝑚+𝑐𝑐 Ψ 𝒙𝒙 𝑡𝑡 , 2 𝜕𝜕 2 2 2 2 4 ∗ − 𝜕𝜕𝜕𝜕 Ψ+𝒙𝒙 𝑡𝑡 −ℏ 𝑐𝑐 𝛻𝛻 𝑚𝑚, 𝑐𝑐 =Ψ0 𝒙𝒙 𝑡𝑡 2 𝜕𝜕 2 2 2 2 4 − 𝜕𝜕𝜕𝜕 ℏ 𝑐𝑐 𝛻𝛻 − 𝑚𝑚 𝑐𝑐 Ψ 𝒙𝒙 𝑡𝑡 Origins of the Dirac Equation (cont.)

∗ = 0 – the Klein-Gordon Equation ∗ Assuming𝜇𝜇 Minkowski2 4 𝜇𝜇 metric 𝜕𝜕 𝜕𝜕𝜇𝜇 − 𝑚𝑚 𝑐𝑐 Ψ 𝑥𝑥 1 0 0 0 0 1 0 0 ∗ = −0 0 1 0 𝜂𝜂𝜇𝜇𝜇𝜇 0 0 0 1 ∗ So adopting the notation of the d‘Alembertian wave operator = ∗ =𝜇𝜇0 □ 𝜕𝜕 𝜕𝜕𝜇𝜇 2 4 □ − 𝑚𝑚 𝑐𝑐 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 PDE EOM would be third order ∗ The general physics consensus suggests that the EOM are second order PDEs ∗ Solutions of the Klein-Gordon⇒ equation fail to preserve the probability 3- current = ∗ We know, again𝑖𝑖 a� posteriori,∗ that∗ the Klein-Gordon equation is the EOM 𝒋𝒋⃗ − 2𝑚𝑚 Ψ 𝛻𝛻� − Ψ𝛻𝛻Ψ for the spin-0 Klein-Gordon field ∗ Would it be possible to somehow factor the Klein-Gordon equation? ∗ Would fix the first problem 𝜙𝜙 ∗ Hopefully it will fix the second problem, too Factorization of the Klein-Gordon Equation

= 0 2 4 ∗□Remember− 𝑚𝑚 𝑐𝑐 thereΨ are 4 derivative terms hiding in ∗ In fact, the Klein-Gordon equation can only be factored over in 0 + 1 dimensions □ ∗ In 3 + 1 dimensions, cannot be factored over ℂ ∗ Essentially, what we want to do is find a “square root” for □ ℂ ∗ To accomplish this: □∗ Cleverly introduce a 4-dimensional free vector field, , over with basis = , , , ∗ Extend the base𝜇𝜇 3 field of 0the1 Klein2 -Gordon3 equation, 𝒮𝒮 , to a ℂ base vector𝛾𝛾 space𝜇𝜇=0 : 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝛾𝛾 ∗ Try to factor the Klein-Gordon equation over , and seeℂ what constraints (relators)𝒮𝒮 ℂ you⇝ need𝒮𝒮 to make this work 𝒮𝒮 Factorization of the Klein-Gordon Equation (cont.)

∗ Consider a functor Diff End , , , which is a functor from R-Modules2 (differentiable2 1 endomorphisms of𝔇𝔇 2x∶ cont. diff functions𝒞𝒞 ℳ ℂ over𝒞𝒞 ℳa spacetimeℂ ⇝ 𝒮𝒮manifold ) to R-Modules ( ) ∗ In particular let = = ℳ 𝒮𝒮𝜇𝜇 𝜇𝜇 ∗ This is slightly sloppy,𝜇𝜇 but can be treated𝜇𝜇 as manifestly scalar w.r.t. (b/c𝔇𝔇 𝜕𝜕 is independent𝔇𝔇 𝜕𝜕 ↦ 𝛾𝛾 of𝜕𝜕 ) 𝜕𝜕 𝜇𝜇 ∗ N.B. – Lesson Learned:𝜇𝜇 for𝜕𝜕 typesetting purposes,𝜇𝜇 I don’t think MS Equation𝒮𝒮 2010 does𝛾𝛾 Feynman “Slash”𝑥𝑥 notation :( ∗ N.B. B. – = by construction.; so ~ w.r.t. functor 𝜇𝜇 𝜇𝜇 𝔇𝔇 𝜕𝜕 𝔇𝔇 𝜕𝜕𝜇𝜇 𝜕𝜕 𝜕𝜕𝜇𝜇 1 This definitely works for , and may possibly work for : the difference between these being that the second derivative may exist, just not2𝔇𝔇 be continuous 1 𝒞𝒞 𝒞𝒞

The Dirac (Clifford) Algebra

Claim 1: Using an appropriate relator, = 2 Fact 1: , = 0 𝜕𝜕 □

𝜕𝜕𝜇𝜇 𝜕𝜕𝜈𝜈 Proof of Claim 1: = = = = 2 +𝜇𝜇 2 = 𝜇𝜇 𝜈𝜈 + 𝜇𝜇 𝜈𝜈 𝜕𝜕 𝛾𝛾 𝜕𝜕𝜇𝜇 𝛾𝛾 𝜕𝜕𝜇𝜇 𝛾𝛾 𝜕𝜕𝜈𝜈 𝛾𝛾 𝛾𝛾 𝜕𝜕𝜇𝜇 𝜕𝜕𝜈𝜈 * -- relabeling𝜇𝜇 𝜈𝜈 1 dummy indexes1 𝜇𝜇 𝜈𝜈 for 𝜇𝜇 𝜈𝜈 𝛾𝛾 𝛾𝛾 2 𝜕𝜕𝜇𝜇𝜕𝜕𝜈𝜈 𝜕𝜕𝜈𝜈𝜕𝜕𝜇𝜇 2 𝛾𝛾 𝛾𝛾 𝜕𝜕𝜇𝜇𝜕𝜕𝜈𝜈 𝛾𝛾 𝛾𝛾 𝜕𝜕𝜈𝜈𝜕𝜕𝜇𝜇 = + 𝜇𝜇 𝜈𝜈 𝜇𝜇 ↔ 𝜈𝜈 𝛾𝛾 𝛾𝛾 𝜕𝜕𝜈𝜈𝜕𝜕𝜇𝜇 ** 1-- now𝜇𝜇 𝜈𝜈 come𝜈𝜈 the𝜇𝜇 relators for – if + 2 ~ for 2 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝜕𝜕𝜇𝜇𝜕𝜕𝜈𝜈 empty word (unity) 𝜇𝜇 𝜈𝜈 𝜈𝜈 𝜇𝜇 𝜇𝜇𝜇𝜇 = 𝒮𝒮 𝛾𝛾 𝛾𝛾 𝛾𝛾 𝛾𝛾 − 𝜂𝜂 𝟏𝟏 𝟏𝟏 = 𝜇𝜇𝜇𝜇 𝟏𝟏 ∈ 𝒮𝒮 ⇒ 𝜇𝜇 𝜈𝜈 Q.E.D𝜂𝜂 𝜕𝜕 𝜕𝜕 □ The Dirac (Clifford) Algebra (cont.)

∗ Thus given Claim 1, if then the Klein Gordon Equation can be factored𝒮𝒮 𝜇𝜇 𝜈𝜈 over𝜈𝜈 𝜇𝜇 𝜇𝜇𝜇𝜇 ≪𝛾𝛾 𝛾𝛾 +𝛾𝛾 𝛾𝛾 −2𝜂𝜂 𝟏𝟏≫ ∗ = + 𝒮𝒮̃ ≈ � ̃ ∗ Recall from2 4 Martin’s first2 IN YOUR 2FACE colloquium𝒮𝒮 on Clifford Algebras□ − 𝑚𝑚 𝑐𝑐 that for𝜕𝜕 some𝑚𝑚𝑐𝑐 𝜕𝜕 − 𝑚𝑚𝑐𝑐 (free vector field, non-degenerate metric), , the presentation of modulo the relators ∗ + ~ , constitute a Clifford Algebraℱ 𝑑𝑑 ∗ Thus,ℱ as we were hoping, we can form a 1st-order PDE 𝒗𝒗𝒘𝒘 𝒘𝒘𝒗𝒗 2𝑑𝑑 𝒗𝒗 𝒘𝒘 ∗ = 0 ∗ This equation2 is known as the Dirac Equation for the Dirac Field ∗ Trying𝑖𝑖𝑖𝑖 − 𝑚𝑚 an𝑐𝑐 appropriate𝜓𝜓 variant of the Dirac Equation as Field Lagrangian fixes Problem (1) and (trust me) also fixes Problem (2)!𝜓𝜓 ∗ is the familiar 2 2 Lorentz spin rep 𝜇𝜇𝜇𝜇 𝒮𝒮̃ 𝔰𝔰𝔰𝔰 ⊗ 𝔰𝔰𝔰𝔰 𝑆𝑆

Application : The Dirac Equation - Review

∗ Attempting to find a SR compatible Schrödinger equation, we arrived at the Klein-Gordon Equation = 0 ∗ The Klein-Gordon Equation is not a good Hamiltonian2 4 / Lagrangian □ − 𝑚𝑚 𝑐𝑐 Ψ ∗ We want to try to factor the Klein-Gordon Equation, but this is impossible over in 3 + 1 dimensions ∗ By introducing the appropriate free vector field and trying to factor Klein-Gordonℂ over , and then a posteriori imposing some relators , we find a free algebra𝒮𝒮 , which the Klein-Gordon Equation can be factored𝒮𝒮 over! ℛ 𝑆𝑆̃ For you Sleeping Bastards . . . Section 6 The “usual multi-linear” Tensor Product

⊗ The Multi-Linear Tensor Functor

⊗

∗ Recall, from Section 2 and Section 3 that, via the (strong) Baez Conjecture all physics can be modeled via monoidal categories ∗ We also want these categories to be multi-linear ∗ Q: How do we explicitly construct such a functor ? ∗ A: Free category “presentations” !!!

Construction of for Free Vector Spaces ⊗

∗ There are various ways to explicitly construct the multi- linear functor for various categories ∗ over different categories and coefficients acts quite differently⊗ ⊗∗ for example A A for R-modules ∗ Because we are physicists,ℂ WLOG,ℤ I will𝑅𝑅 construct a bi- linear for the category⊗ 𝐵𝐵 ≠ of⊗ vector𝐵𝐵 ≠ 𝐴𝐴 spaces⊗ 𝐵𝐵 ∗ To accomplish this, we will need to write appropriate relators⊗ on a free vector field which enforces Defn. 4 ∗ N.B. – Defn. 4 deals specifically with multi-linear MAPS. Because there is a natural isomorphism from Hom , the maps case is in essence a vector field – so the defn is still∗ valid. ⊗𝑖𝑖 𝑉𝑉 𝔽𝔽 →⊗𝑖𝑖 𝑉𝑉 Construction of for Vector Spaces (cont.) - Relators ⊗

Let , be 2 vector spaces over field , , be the free vector space on , and , , , , , 𝑉𝑉 𝑊𝑊 𝔽𝔽 𝐹𝐹 𝑉𝑉 𝑊𝑊 𝑉𝑉 𝑊𝑊 ∀𝑣𝑣 𝑣𝑣1 𝑣𝑣2 ∈ 𝑉𝑉 ∀𝑤𝑤 𝑤𝑤1 𝑤𝑤2 ∈ 𝑊𝑊 ∀𝛼𝛼 ∈ 𝔽𝔽

Relator Defn 4. Property Relator in , Name 𝟏𝟏 𝑭𝑭 𝑽𝑽 𝑾𝑾 + = + + ~ + = + + ~ ℛ1 𝑣𝑣1 𝑣𝑣2 ⊗ 𝑤𝑤 𝑣𝑣1 ⊗ 𝑤𝑤 𝑣𝑣2 ⊗ 𝑤𝑤 𝑣𝑣1 𝑣𝑣2 𝑤𝑤 − 𝑣𝑣1𝑤𝑤 − 𝑣𝑣2𝑤𝑤 𝟏𝟏 = = ~ ℛ2 𝑣𝑣 ⊗ 𝑤𝑤1 𝑤𝑤2 𝑣𝑣 ⊗ 𝑤𝑤1 𝑣𝑣 ⊗ 𝑤𝑤2 𝑣𝑣 𝑤𝑤1 𝑤𝑤2 − 𝑣𝑣𝑤𝑤1 − 𝑣𝑣𝑤𝑤2 𝟏𝟏 ℛ3 𝛼𝛼 𝑣𝑣 ⊗ 𝑤𝑤 𝛼𝛼𝛼𝛼 ⊗ 𝑤𝑤 𝑣𝑣 ⊗ 𝛼𝛼𝛼𝛼 2𝛼𝛼 𝑣𝑣𝑣𝑣 − 𝛼𝛼𝛼𝛼 𝑤𝑤 − 𝑣𝑣 𝛼𝛼𝛼𝛼 𝟏𝟏

1 – To write this as a bona-fide free vector space I should be writing , . Here I am writing this as an -module. This is sloppy in some subtleties, but works for our scope / purposes 𝑣𝑣 𝑤𝑤 𝔽𝔽 Construction of for Vector Spaces (cont.) ⊗

Definition 15: The bi-linear tensor product of 2 vector spaces , is the free vector space , mod the relators , , : 𝑉𝑉 𝑊𝑊 𝐹𝐹 𝑉𝑉 𝑊𝑊 ℛ1 ℛ2 ℛ3 , , , 𝐹𝐹 𝑉𝑉 𝑊𝑊 𝑉𝑉 ⊗ 𝑊𝑊 ≡ �≪ℛ1 ℛ2 ℛ3≫ ∗ Properties ∗ dim = dim × dim ∗ 𝑉𝑉 ⊗ 𝑊𝑊 𝑉𝑉 𝑊𝑊 ∗ Elements of are of the form for and 𝑉𝑉 ⊗ 𝑊𝑊 ≅ 𝑊𝑊 ⊗ 𝑉𝑉 ∗ A basis for is of the form for a basis for and a 𝑉𝑉 ⊗ 𝑊𝑊 𝑎𝑎 ⊗ 𝑏𝑏 ∀𝑎𝑎 ∈ 𝑉𝑉 ∀𝑏𝑏 ∈ 𝑊𝑊 basis for 𝑉𝑉 ⊗ 𝑊𝑊 𝑒𝑒𝑖𝑖 ⊗ 𝑓𝑓𝑗𝑗 𝑒𝑒𝑖𝑖 𝑉𝑉 𝑓𝑓𝑗𝑗 𝑊𝑊 Pet Peeve : Tensor Algebra

∗ Let be a tensor algebra over with OK ∗ + = + - linearity ∗ 𝑉𝑉 ⊗ 𝑊𝑊= – scalars 𝔽𝔽 𝛼𝛼 ∈ 𝔽𝔽 OK 𝑎𝑎 ⊗ 𝑏𝑏 𝑎𝑎 ⊗ 𝑐𝑐 𝑎𝑎 ⊗ 𝑏𝑏 𝑐𝑐 ∗ + + + 𝛼𝛼𝛼𝛼 ⊗ 𝑏𝑏 𝑎𝑎 ⊗ 𝛼𝛼𝛼𝛼 ∗ No relator for this !!! 𝑎𝑎∗ ⊗ 𝑏𝑏 and𝑐𝑐 ⊗ 𝑑𝑑 ≠ 𝑎𝑎SEPARATE𝑐𝑐 ⊗ VECTORS𝑏𝑏 𝑑𝑑 WRONG

𝑎𝑎 ⊗ 𝑏𝑏 𝑐𝑐 ⊗ 𝑑𝑑 ∗ In fact, this mathematical fact accounts for most cool QM phenomena

Tensor as a Change of Coefficients

∗ The multi-linearity of makes it ideal to change to coordinates of an object. ∗ In fact, in many mathematical⊗ calculations (homology and cohomology in particular) is used to change the base ring calculations are performed over ∗ The Universal Coefficient theorems⊗ concerning give rise to Tor , - the failure of to be exact ⊗

𝑀𝑀 𝐺𝐺 ⊗ The Kronecker Product

∗ Consider SL 2; all invertible 2x2 matrixes over

∗ = andℂ = , SL 2; ℂ 𝑎𝑎 𝑏𝑏 𝑒𝑒 𝑓𝑓 ∗ 𝐴𝐴The elements , 𝐵𝐵, , , , , , , 𝐴𝐴 𝐵𝐵 ∈ ℂ 𝑐𝑐 𝑑𝑑 𝑔𝑔 ℎ ∗ Now, consider SL 2; SL 2; 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 𝑒𝑒 𝑓𝑓 𝑔𝑔 ℎ ∈ ℂ ℂ ⊗ ℂ ∗ = 𝑒𝑒 𝑓𝑓 𝑒𝑒 𝑓𝑓 SL 2; SL 2; 𝑎𝑎 𝑏𝑏 𝑔𝑔 ℎ 𝑔𝑔 ℎ 𝐴𝐴 ⊗ 𝐵𝐵 𝐴𝐴 ⊗ 𝐵𝐵 ∈ ℂ ⊗ ℂ 𝑒𝑒 𝑓𝑓 𝑒𝑒 𝑓𝑓 𝑐𝑐 𝑑𝑑 𝑔𝑔 ℎ 𝑔𝑔 ℎ The Kronecker Product (cont.)

∗ Now, we see that the elements of SL 2; SL 2; are 2x2 matrices ∗ Also we see that, naturally, SL 2; SLℂ 2⊗; SLℂ 4; ∗ Elements of SL 2; SL 2; act on vectors ℂ ⊗ ℂ ⊂ ℂ ∗ = = , 2 2 ℂ ⊗ ℂ ℂ ⊗ ℂ ∗ So 𝑣𝑣1 𝑤𝑤1 2 𝑣𝑣 𝑣𝑣2 𝑤𝑤 𝑤𝑤2 𝑣𝑣 𝑤𝑤 ∈ ℂ ∗ = 𝑤𝑤1 𝑣𝑣1 𝑤𝑤 2 2 2 4 𝑤𝑤1 ∗ 𝑣𝑣The⊗ elements𝑤𝑤 𝑣𝑣2 𝑤𝑤of2 𝑣𝑣 ⊗are𝑤𝑤 vectors∈ ℂ ⊗ withℂ ⊂vectorsℂ as elements ∗ Multiplication and vector2 2 operations for SL 2; SL 2; and are exactly ℂ“regular”⊗ ℂ operations for SL 4; and 2 2 ℂ ⊗ 4ℂ ℂ ⊗ ℂ ℂ ℂ The Lorentz Spin-1/2 Rep 2 2

𝔰𝔰𝔰𝔰 ⊗ 𝔰𝔰𝔰𝔰

∗ Spin rep is just a change of coefficients from the usual QM spin ½ rep 2 to𝜇𝜇𝜇𝜇 include anti-particles (Dirac spinor) or opposite helicities (Weyl spinor𝑆𝑆 ) ∗ Hence,𝔰𝔰𝔰𝔰 2 2 2 ∗ Because 2 2 generates SU 2 SU 2 SL 4; the algebraic 𝔰𝔰𝔰𝔰and vector⇝ 𝔰𝔰𝔰𝔰 structure⊗ 𝔰𝔰𝔰𝔰 is a unitary version of the previous example!𝔰𝔰𝔰𝔰 ⊗ 𝔰𝔰𝔰𝔰 ⊗ ⊂ ℂ ∗ Example 0 𝜇𝜇 ∗ = for = , with SU 2 thus SU 2 SU 2 𝛾𝛾 0𝜇𝜇 𝜇𝜇 𝜎𝜎 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝛾𝛾 𝜇𝜇 𝜎𝜎 𝟏𝟏 𝝈𝝈 𝜎𝜎 ∈ 𝛾𝛾 ∈ ⊗ 𝜎𝜎� The “usual multi-linear” Tensor Product - Review

⊗ ∗ Using the free category method applied to vector spaces (which are linear by definition), we would like to : ∗ take 2 vector spaces (perhaps Hilbert spaces of quantum states) ∗ “squish” them together s.t. the result is linear in every variable (bi- linear) ∗ To do this, we need to find the most generic vector space you can make, , ∗ Next find relators which enforce the desired bi-linearity, , , ∗ Find an𝐹𝐹 appropriate𝑉𝑉 𝑊𝑊 presentation 1 2 3 , ℛ ℛ ℛ ∗ , , 𝐹𝐹 𝑉𝑉 𝑊𝑊 ∗ Additionally, can be1 used2 3 as a change of coefficient operator if we𝑉𝑉 want⊗ 𝑊𝑊 to≡ change� ≪numericalℛ ℛ ℛ ≫ coefficients to other objects ⊗ Section 7 Physics Applications of

⊗ Physics Applications of

⊗ ∗ The tensor product is used quite often in physics, but the notation does not reflect this ∗ I hope to explicitly and verbosely elucidate a few examples: 1. Lorentz Tensors , , 2. Complete Set of States𝜇𝜇 𝜌𝜌 Λ𝜈𝜈 𝑔𝑔𝜇𝜇𝜇𝜇 𝑅𝑅𝜇𝜇𝜇𝜇𝜇𝜇 3. System of spin ½ particles ∗ | = | | ∗ = + ↑↓⟩ ↑⟩ ↓⟩ ∗ 2 2 = 3 1 𝑺𝑺� 𝒔𝒔�𝟏𝟏 𝒔𝒔�𝟐𝟐

⊗ ⊕

Lorentz Tensors

∗ Recall from the Introduction ∗ : Hom , Hom , 𝑔𝑔𝜇𝜇𝜇𝜇 ℳ ⊗ ℳ → ℝ ⇒ ∗ This maps 2 4-vectors and to a distance, a real number 𝑔𝑔𝜇𝜇𝜇𝜇 ∈ ℳ ℝ ⊗ ℳ ℝ ∗ 𝜇𝜇 𝜈𝜈 You can𝜇𝜇𝜇𝜇 see it is possible to build up Lorentz tensors from any number of linear𝑔𝑔 objects, multi-linearly by𝑥𝑥 simply joining𝑥𝑥 things together

∗ (vectors) 𝜇𝜇 ∗ ∗ 𝑎𝑎𝑖𝑖 𝑖𝑖∈ℑ: ∶ ℳ → ℝ(covectors) 𝑗𝑗 𝜇𝜇 ∗ We 𝑏𝑏can𝑗𝑗 ∈construct𝔍𝔍 ℳ → ℝ some Lorentz tensor , ∗ , : 2 2 𝜇𝜇1 ⋯𝜇𝜇𝑛𝑛 ∗ ∗ 𝑇𝑇 𝜈𝜈1 ⋯𝜈𝜈𝑚𝑚 ℳ ⊗ ⋯ 𝑛𝑛 − ⊗ ℳ ⊗ ℳ ⊗ 𝑚𝑚 − ⊗ ℳ → ℝ Structure Groups and Lorentz Tensors

∗ When we do physics in SR / GR / QFT what we really care about is that our mathematics is Lorentz Invariant , ∗ Given a Lorentz tensor , , we want it to transform nicely via 𝜇𝜇1 ⋯𝜇𝜇𝑛𝑛 𝑇𝑇 𝜈𝜈1 ⋯𝜈𝜈𝑚𝑚 , , ∗ , = , 𝑗𝑗 𝜎𝜎1 ⋯𝜎𝜎𝑛𝑛 𝑛𝑛 𝜎𝜎𝑖𝑖 𝜈𝜈 𝜇𝜇1 ⋯𝜇𝜇𝑛𝑛 ∗ Each𝑇𝑇� tensor𝜆𝜆1 ⋯ index𝜆𝜆𝑚𝑚 transforms∏𝑖𝑖 Λ𝜇𝜇𝑖𝑖 ∏ withΛ𝜆𝜆𝑗𝑗 𝑇𝑇1 𝜈𝜈1 ⋯𝜈𝜈𝑚𝑚 ∗ If tensors transform nicely w.r.t. a particular group we say that they have structure group Λ ∗ For Lorentz tensors the structure group is well 𝐺𝐺 ∗ The Lorentz Group! 𝐺𝐺 BUT, THERES A PROBLEM . . . Problem: the Christoffel Connection

∗ Consider our favorite metric compatible connection from GR, the Christoffel connection ∗ = + 𝜎𝜎 1 𝜎𝜎𝜎𝜎 ∗ ThisΓ𝜇𝜇𝜇𝜇 looks2 𝑔𝑔 like 𝜕𝜕a𝜈𝜈 good𝑔𝑔𝜇𝜇𝜇𝜇 Lorentz𝜕𝜕𝜇𝜇𝑔𝑔𝜈𝜈𝜈𝜈 − tensor𝜕𝜕𝜌𝜌𝑔𝑔𝜇𝜇𝜇𝜇 right ? ∗ WRONG!!!! ∗ 𝜆𝜆 𝜆𝜆 𝜇𝜇 𝜈𝜈 𝜎𝜎 ∗ ThisΓ�𝛼𝛼𝛼𝛼 is ≠whyΛ𝜎𝜎 Λ𝛼𝛼Λ 𝛽𝛽isΓ 𝜇𝜇𝜇𝜇known as the Christoffel symbol, not the Christoffel tensor𝜎𝜎 because it does not transform via the 𝜇𝜇𝜇𝜇 structure groupΓ the Lorentz group Lorentz Tensors - Review

∗ Thus given a set of primitives

∗ (vectors) 𝜇𝜇 ∗ ∗ 𝑎𝑎𝑖𝑖 𝑖𝑖∈ℑ:∶ ℳ → ℝ (covectors) 𝑗𝑗 We can𝜇𝜇 use the functor to create multi-linear maps , 𝑏𝑏 𝑗𝑗∈𝔍𝔍 ℳ → ℝ , ∗ 𝜇𝜇Additionally,1 ⋯𝜇𝜇𝑛𝑛 we impose⊗ the requirement that 𝑇𝑇 , 𝜈𝜈1 ⋯𝜈𝜈𝑚𝑚 , transform via the structure group the Lorentz𝜇𝜇1 ⋯𝜇𝜇𝑛𝑛 group ∗ This𝑇𝑇 is precisely𝜈𝜈1 ⋯𝜈𝜈𝑚𝑚 what a Lorentz tensor is! Complete Sets of States

Definition 16: Given a vector space , a set of vectors spans iff can be written as some linear combination of the 𝑉𝑉 𝑣𝑣𝑖𝑖 𝑖𝑖∈ℑ 𝑉𝑉 ∀𝑤𝑤 ∈ 𝑉𝑉 𝑤𝑤 ∗ Also, recall from QM that 𝑣𝑣𝑖𝑖 𝑖𝑖∈ℑ

Definition 17: Given a vector space , a set of vectors is complete iff = , for End the identity , 𝑉𝑉 𝑣𝑣𝑖𝑖 𝑖𝑖∈ℑ 𝑖𝑖 𝑗𝑗∈ℑ 𝟏𝟏 ∑ ∑𝑖𝑖 𝑗𝑗 �𝑣𝑣𝑖𝑖⟩�𝑣𝑣𝑗𝑗� 𝟏𝟏 ∈ 𝑉𝑉 ∗ How is Defn 17. different from saying that spans ?

𝑣𝑣𝑖𝑖 𝑖𝑖∈ℑ 𝑉𝑉 The Outer Product

�𝒗𝒗𝒊𝒊⟩�𝒗𝒗𝒋𝒋� ∗ Lets closely examine what is ∗ | - vector 𝑖𝑖 𝑗𝑗 ∗ | - covector �𝑣𝑣 ⟩�𝑣𝑣 � 𝑣𝑣𝑖𝑖⟩ ∈ 𝑉𝑉 ∗ Thus ∗ ⟨𝑣𝑣𝑖𝑖 ∈ 𝑉𝑉 ∗ Lets closely examine what∗ End is �𝑣𝑣𝑖𝑖⟩�𝑣𝑣𝑗𝑗� ∈ 𝑉𝑉 ⊗ 𝑉𝑉 ∗ : or equivalently : ∗ You could write : 𝟏𝟏 ∈ (think∗ 𝑉𝑉∗ ) 𝟏𝟏 𝑉𝑉 → 𝑉𝑉 𝟏𝟏 𝑉𝑉 → 𝑉𝑉 ∗ How fortunate! Both ∗ 𝜇𝜇 𝟏𝟏 𝑉𝑉 ⊗ 𝑉𝑉 → 𝔽𝔽 𝛿𝛿𝜈𝜈 ∗ and ∗ ∗ 𝑖𝑖⟩ 𝑗𝑗 � 𝑣𝑣 �𝑣𝑣 � ∈ 𝑉𝑉 ⊗ 𝑉𝑉 𝟏𝟏 ∶ 𝑉𝑉 ⊗ 𝑉𝑉 Complete Sets of States - Review

∗ Now you see why complete sets of states are stronger than simply spanning sets of states ∗ Defn 17. (complete sets of states) says that your spanning set of quantum states span your space of operators End ∗ As you all know, this property is VERY useful for QM 𝑉𝑉

Question 1: Does, for a vector space the fact that spans spans ? (think Riesz Representation Thm) 𝑖𝑖 𝑖𝑖∈ℑ 𝑖𝑖 𝑖𝑖∈ℑ ∗ 𝑉𝑉 𝑣𝑣 𝑉𝑉 ⇒ 𝑣𝑣 𝑉𝑉 ⊗ 𝑉𝑉 Section 8 Application : System of Spin ½ Particles System of Spin ½ Particles

∗ We understand a single QM spin ½ particle quite well via the Pauli Algebra ∗ Assume this particle has a spin state | , some Hilbert space ∗ If we would like to look at an ensemble𝑠𝑠𝑠𝑠 ⟩of∈ spinℋ ½ particles, the axioms of QM demand that we look at the Hilbert space ∗ WLOG I will examine this for the familiar 2 particle 𝑖𝑖∈ℑ case ⊗ ℋ System of Spin ½ Particles (cont.)

∗ Why do we even want to look at ? | | ∗ = and = ℋ then⊗ ℋ 𝜒𝜒↑ 𝜓𝜓↑ 𝑠𝑠1𝑚𝑚1⟩ 𝜒𝜒↓ 𝑠𝑠2𝑚𝑚2⟩ 𝜓𝜓↓

∗ | | = 𝜓𝜓↑ ↑ 𝜒𝜒 𝜓𝜓↓ 1 1 2 2 𝜓𝜓↑ 𝑠𝑠 𝑚𝑚 ⟩ ⊗ 𝑠𝑠 𝑚𝑚 ⟩ ↓ ∈ ℋ ⊗ ℋ ∗ which gives us the desired𝜒𝜒 𝜓𝜓↓ combinatorics of the 4 conditional probabilities ∗ Given the probabilistic interpretation of QM, this is precisely what we ↑↓ ↑↓ need – the bi-𝜒𝜒linear𝜓𝜓 works! ∗ N.B. –MS Equation 2010 is HORRIBLE for bra / ket notation ⊗

System of Spin ½ Particles (cont.)

∗ What is a basis for ? ∗ Referring back to properties of a bi-linear we see a basis ℋ ⊗ ℋ | | | | | | ∗ , , , ⊗ ∗ Because↑⟩ ⊗ ↑⟩ ↑⟩ ⊗ has�↓⟩ a↓ ⟩finite⊗�↑⟩ number↓⟩ ⊗ ↓⟩ of s, ∗ How do we measure the net spin for this system? 2 ℋ ⊗ ℋ ℋ ℋ ⊗ ℋ ≈ ℋ

Total Spin Observable

𝑺𝑺� ∗ How do we want to measure the spin of a state | | | | ∗ We know how to measure = 𝑠𝑠1𝑚𝑚1⟩ ⊗ 𝑠𝑠2𝑚𝑚2⟩ ℏ ∗ Lets create an observable 𝒔𝒔�𝒊𝒊 𝑠𝑠𝑖𝑖Hom𝑚𝑚𝑖𝑖⟩ 2 𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑚𝑚𝑖𝑖,⟩ ∗ = + 𝑺𝑺� ∈ ℋ ⊗ ℋ ℝ 1 0 ∗ Lets𝑺𝑺� explicitly𝒔𝒔�𝟏𝟏 ⊗ 𝟏𝟏 apply𝟏𝟏 ⊗ 𝒔𝒔�𝟐𝟐 to | | - Recall = 0 1 1 0 1 0 1 0 1 0 ∗ | | = � + 𝑧𝑧 0 1𝑺𝑺 ↑0⟩ ⊗1 ↑⟩ 0 1 𝜎𝜎0 1 ℏ 1− 1 𝑺𝑺� ↑⟩ ⊗ ↑⟩ 2 ⊗ ⊗ 0 ⊗ 0 − − Total Spin Observable (cont.)

1 0 0 0 1 0 0 0 0 1 0 0 0 1 �0 0 1 ∗ | | = + 𝑺𝑺 1 ℏ 0 0 1 0 0 0 1 0 0 − 0 𝑺𝑺� ↑⟩ ⊗ ↑⟩ 2 0 0 0 1 0 0 0 1 1 − 0 2 0 0 0 1 −1 − 0 0 0 0 0 0 ∗ = = ℏ 0 0 0 0 0 0 2 0 0 0 2 0 ℏ 0 ∗ Wow – what a lot of work for what we already knew + = − ℏ ℏ ∗ This is explicitly mathematically what you are doing when you compute 2 2 ℏ the sophomoric = + ∗ Doing all this work is a waste for 2 spin ½ particles, not a waste for the 8 � 𝟏𝟏 𝟐𝟐 different gluons 𝑺𝑺. . . 𝒔𝒔� 𝒔𝒔� Section 9 Conclusion Conclusion

∗ In physics, we often divide an conquer ∗ Simplify problems, solve and understand solutions ∗ Build up large systems from these simple solutions ∗ To solve a problem which is pathological, one may need to think outside the complex field ∗ To do this, once can cleverly introduce a free category ∗ Introduce appropriate relatorsℂ for the physical constraints we desire ∗ Find a “presentation” (projective resolution) for this free category, this will yield an object which, will solve the problem ∗ If this method works, by studying the resulting object new physics could possibly be predicted!

Conclusion (cont.)

∗ Once we have these simple solutions, complex systems can be built up from them ∗ The mathematics describing this is that of monoidal categories and functors ∗ In fact, if further constraints are required for this complex system, the previous method can be reused! Conclusion (cont.)

∗ Multi-linearity is of great importance for physics, so the multi- linear tensor functor is often used in physics ∗ Many of the calculations that you have performed utilize tensors, but the common⊗ notation that physicists use try to conceal this fact ∗ The next time you are doing a calculation, there could, either extantly or tacitly, be a tensor IN YOUR FACE. ∗ Now you know what to do to that bastard! Works Cited

1. Baez Quantum Gravity Lectures http://math.ucr.edu/home/baez/qg-spring2003/ 2. Hatcher – Algebraic Topology 3. Peskin and Schroeder – Introduction to Quantum Field Theory 4. Warner – Graduate Texts in Mathematics : Foundations of Differentiable Manifolds and Lie Groups 5. The Unapologetic Mathematician - Monoidal categories - http://unapologetic.wordpress.com/2007/06/28/monoidal -categories/

Fin.