Appendix 1: Linear Algebra Basics for Quantum Mechanics
This appendix provides supplementary explanations from a beginner’s perspective on key concepts from Chapter 1: linear operators, Hermitian operators, and bra-ket notation.
1. Linearity: Vectors, Functions, and Operators
The core of linear algebra is “Linearity”. This means that the transformation of “what is added” is the same as “the sum of the transformations of each individual part”.
- \(L(a\vec{v} + b\vec{w}) = a L(\vec{v}) + b L(\vec{w})\)
We usually learn \(L\) as a matrix and \(\vec{v}\) as a column vector, but in quantum mechanics, this concept is extended.
- Vector (\(|\psi\rangle\)): State. (Example: \(\begin{pmatrix} a \\ b \end{pmatrix}\))
- Vector space: A set of functions. (Example: \(L^2[0,1]\), square-integrable functions learned in Chapter 1)
- Linear operator (\(\hat{A}\)): A matrix or a differential operator.
Why is a differential operator linear? The differential operator \(\frac{d}{dx}\) satisfies the following for two functions \(f(x), g(x)\) and scalars \(a, b\):
\[ \frac{d}{dx}[a f(x) + b g(x)] = a \frac{d f(x)}{dx} + b \frac{d g(x)}{dx} \]
This exactly matches the definition of linearity. Therefore, the differential operator (\(\frac{d}{dx}\)) is also an “operator” acting on the vector space of functions, and can be handled within Hilbert space theory.
2. Hermitian Operators (Observables)
In the physical world, the values we measure (position, momentum, energy) are always real numbers. In quantum mechanics, these “observables” are expressed as Hermitian operators.
Definition: When the adjoint (\(\dagger\), “dagger”) of an operator \(\hat{A}\) is equal to itself \[\hat{A} = \hat{A}^\dagger\] (\(\hat{A}^\dagger\) is obtained by taking the complex conjugate of all elements of the matrix (\(\overline{a+ib} = a-ib\)) and then transposing the matrix.)
Key Properties:
- Eigenvalues are always real: This property ensures that the measurement values of Hermitian operators are always real.
- Different eigenvectors are always orthogonal: This provides the mathematical foundation (orthogonal basis) that allows us to clearly distinguish between “this” and “that” when measuring the world.
Examples:
- Real symmetric: \(A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \implies A^\dagger = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}^\mathsf{T} = A\) (Hermitian)
- Complex: \(B = \begin{pmatrix} 1 & i \\ -i & 4 \end{pmatrix} \implies B^\dagger = \overline{\begin{pmatrix} 1 & -i \\ i & 4 \end{pmatrix}}^\mathsf{T} = \begin{pmatrix} 1 & i \\ -i & 4 \end{pmatrix} = B\) (Hermitian)
- Non-Hermitian: \(C = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \implies C^\dagger = \overline{\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}}^\mathsf{T} = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} \neq C\)
3. Bra-Ket Notation and Expectation Value
Dirac notation makes linear algebra calculations very intuitive. * Ket (Ket) \(|\psi\rangle\): Column vector (state)
\[|\psi\rangle = \begin{pmatrix} a \\ b \end{pmatrix}\]
* Bra (Bra) \(\langle\psi|\): Hermitian adjoint of Ket (row vector)
\[\langle\psi| = |\psi\rangle^\dagger = \overline{\begin{pmatrix} a \\ b \end{pmatrix}}^\mathsf{T} = \begin{pmatrix} \bar{a} & \bar{b} \end{pmatrix}\]
Key Operations:
Inner Product (Overlap): \(\langle\phi|\psi\rangle\)
Multiplying “Bra” and “Ket” gives a scalar (complex number).
\[\langle\phi|\psi\rangle = \begin{pmatrix} \bar{c} & \bar{d} \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \bar{c}a + \bar{d}b\]Expectation Value: \(\langle\psi|\hat{A}|\psi\rangle\)
The average value when measuring the observable \(\hat{A}\) in the state \(|\psi\rangle\).
\[\langle\psi|\hat{A}|\psi\rangle = \begin{pmatrix} \bar{a} & \bar{b} \end{pmatrix} \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix}\]
If \(\hat{A}\) is a Hermitian operator, the result of this calculation is always a real number.Outer Product: \(|\psi\rangle\langle\phi|\)
Multiplying “Ket” and “Bra” results in an operator (matrix).
\[|\psi\rangle\langle\phi| = \begin{pmatrix} a \\ b \end{pmatrix} \begin{pmatrix} \bar{c} & \bar{d} \end{pmatrix} = \begin{pmatrix} a\bar{c} & a\bar{d} \\ b\bar{c} & b\bar{d} \end{pmatrix}\]
In particular, \(|\psi\rangle\langle\psi|\) is the projection operator (projector) in the direction of \(|\psi\rangle\).
4. Basic Exercise Problems
Problem 1: Hermitian Check
Determine whether the following matrices are Hermitian.
\(A=\begin{pmatrix}2&1+i\\1-i&3\end{pmatrix} \quad B=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)
Solution:
\(A^\dagger = \overline{\begin{pmatrix} 2 & 1-i \\ 1+i & 3 \end{pmatrix}}^\mathsf{T} = \begin{pmatrix} 2 & 1+i \\ 1-i & 3 \end{pmatrix} = A\). (Hermitian O)
\(B^\dagger = \overline{\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}}^\mathsf{T} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = B\). (Hermitian O, real symmetric matrix)
(The previous original matrix \(C\) was not real symmetric, but the revised version \(B\) is real symmetric, hence Hermitian.) Problem 2: Expectation Value Calculation (Checking for Real Numbers)
For the state \(|\psi\rangle = \frac{1}{\sqrt{5}}\begin{pmatrix} 1 \\ 2i \end{pmatrix}\) and operator \(\hat{A} = \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}\), calculate the expectation value \(\langle\psi|\hat{A}|\psi\rangle\).
Solution:
1. Bra calculation: \(\langle\psi| = \frac{1}{\sqrt{5}}\begin{pmatrix} 1 & -2i \end{pmatrix}\)
2. \(\hat{A}|\psi\rangle\) calculation: \(\frac{1}{\sqrt{5}} \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2i \end{pmatrix} = \frac{1}{\sqrt{5}} \begin{pmatrix} 1 + (i)(2i) \\ (-i)(1) + (1)(2i) \end{pmatrix} = \frac{1}{\sqrt{5}} \begin{pmatrix} 1 - 2 \\ -i + 2i \end{pmatrix} = \frac{1}{\sqrt{5}} \begin{pmatrix} -1 \\ i \end{pmatrix}\)
3. \(\langle\psi|(\hat{A}|\psi\rangle)\) calculation: \(\frac{1}{\sqrt{5}}\begin{pmatrix} 1 & -2i \end{pmatrix} \cdot \frac{1}{\sqrt{5}} \begin{pmatrix} -1 \\ i \end{pmatrix} = \frac{1}{5} ( (1)(-1) + (-2i)(i) ) = \frac{1}{5} ( -1 - 2i^2 ) = \frac{1}{5} ( -1 + 2 ) = \frac{1}{5}\)\(\hat{A}\) is a Hermitian operator, and the expectation value \(\frac{1}{5}\) is a real number.
Appendix 2: (New) Classical Probability and Quantum Probability
One of the main reasons quantum mechanics is confusing is that the concept of ‘probability’ differs from classical probability. To understand the density matrix introduced in Chapter 2, it is essential to clearly distinguish between these two concepts.
1. Classical Probability (Dice Rolling 🎲)
Classical probability represents our ‘ignorance’.
- State: The die already has a clear state of either 1, 2, 3, 4, 5, or 6. We simply do not know which one will appear until it is rolled.
- Probability: The probability of each face is \(p_i = 1/6\).
- Expectation Value: The average (expectation value) of the outcome when rolling the die is calculated by multiplying each value by its probability and summing them up.
\[\langle \text{value} \rangle = \sum_i p_i \cdot (\text{value})_i = (1/6)·1 + (1/6)·2 + \dots + (1/6)·6 = 3.5\]
2. Quantum Probability (Quantum Coin 🪙)
Quantum probability is inherently uncertain and arises from ‘superposition’. * State: A qubit (quantum bit) does not need to be in \(|0\rangle\)(front) or \(|1\rangle\)(back). Before measurement, it can be in a state where both states are superposed simultaneously. \[|\psi\rangle = c_0 |0\rangle + c_1 |1\rangle\] * Probability amplitude: \(c_0, c_1\) are complex numbers and are referred to as probability amplitude. * Probability (basic rule): \(c_i\) itself is not probability, but the square of the amplitude is the probability. \(P(0) = |c_0|^2, \quad P(1) = |c_1|^2 \quad (\text{Note that } |c_0|^2 + |c_1|^2 = 1)\) * Expectation value: The bra-ket notation introduced in Chapter 1 is used. The expectation value of \(\hat{A}\) is as follows. \[\langle \hat{A} \rangle = \langle\psi|\hat{A}|\psi\rangle\]
3. Deterministic Difference: ‘Ignorance’ or ‘Superposition’?
The density matrix from Chapter 2 is a mathematical tool that distinguishes and integrates these two different types of ‘uncertainty’.
A. Mixed state (classical ignorance)
Suppose we have an ensemble (set) that is in the \(|0\rangle\) state with 50% probability and in the \(|1\rangle\) state with 50% probability. This is classical probability that arises from our ignorance of whether the system is in the \(|0\rangle\) or \(|1\rangle\) state.
- Density matrix: \(\rho_{\text{mix}} = \sum_i p_i |\psi_i\rangle\langle\psi_i| = 0.5 |0\rangle\langle 0| + 0.5 |1\rangle\langle 1| = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.5 \end{pmatrix}\)
- Features: Only the diagonal contains probabilities, and the off-diagonal (interference terms) are zero.
- Purity: \(\mathrm{Tr}(\rho_{\text{mix}}^2) = \mathrm{Tr}\left( \begin{pmatrix} 0.25 & 0 \\ 0 & 0.25 \end{pmatrix} \right) = 0.5 < 1\)
B. Pure state (quantum superposition)
Suppose we have a single system where \(|0\rangle\) and \(|1\rangle\) are superposed in a 50:50 ratio. \(|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\)
- Density matrix: \(\rho_{\text{pure}} = |\psi\rangle\langle\psi| = \left( \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} \right) \left( \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \end{pmatrix} \right) = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}\)
- Features: The off-diagonal components (\(\rho_{01}, \rho_{10}\)) are non-zero. This is the mathematical evidence of ‘interference’ or ‘coherence’.
- Purity: \(\mathrm{Tr}(\rho_{\text{pure}}^2) = \mathrm{Tr}\left( \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \right) = \mathrm{Tr}\left( \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \right) = 1\) Summary: \(\rho_{\text{mix}}\) and \(\rho_{\text{pure}}\) have the same \(\sigma_z\) measurement probability (both 50%), but different \(\sigma_x\) measurement probabilities. In this way, purity (\(\mathrm{Tr}(\rho^2)\)) is a key indicator to distinguish whether the system is in a pure quantum superposition state (\(=1\)) or a mixed state with classical probability mixing (\(<1\)).
Appendix 3: Notation Guide
This appendix explains the conventions for mathematical and physical notation used throughout the main text.
1. Linear Algebra Notation: \([D]_{\mathcal{P}_n}\)
- Meaning: In \([D]_{\mathcal{P}_n}\), the brackets
[ ]denote the matrix representation of the linear transformation \(D\). - Subscript below: \(\mathcal{P}_n\) indicates that the matrix is expressed with respect to a specific basis (in this case, the basis \(\{1, x, \dots, x^n\}\) of the polynomial space \(\mathcal{P}_n\)).
- Distinction: This is a conventional linear algebra notation for “writing as a matrix,” distinct from Dirac’s bra-ket \(\langle \psi |\) or commutators \([A, B]\).
2. Common Uses of Brackets [ ]
Brackets have very different meanings depending on context, so caution is required.
| Context | Notation Example | Meaning |
|---|---|---|
| Linear Transformation Representation | \([T]_{\beta}^{\gamma}\) | Matrix representation of transformation \(T\) with respect to bases \(\beta, \gamma\) |
| Matrices and Components | \(A = [a_{ij}]\) | Matrix itself or its component \(a_{ij}\) |
| Commutator | \([A, B]\) | \(AB - BA\). Measures non-commutativity of two operators |
| Lie Bracket | \([X, Y]\) | Lie bracket of two vector fields (differential geometry) |
| Antisymmetrization | \(T_{[ab]}\) | Antisymmetrization of tensor indices. \(\frac{1}{2}(T_{ab} - T_{ba})\) |
| Iverson Bracket | \([P]\) | Indicator function: 1 if condition \(P\) is true, 0 if false |
| Equivalence Class | \([a]\) | Equivalence class containing element \(a\) |
Appendix 4: Group Classification System: U, S, O, and Beyond
In Chapter 3, we encountered Lie groups such as time translation (\(U(1)\)) and rotation (\(SU(2)\)). Although there are infinitely many types of groups, several criteria exist to systematically classify them. Symbols like \(U, S, O\) represent classifications based on “what property is preserved”.
This appendix explores where this classification criterion fits within the overall classification system and examines other important classification criteria directly relevant to physics (topology, algebra).
1. Fundamental Classification: Discrete Group vs. Continuous Group
The first classification criterion is whether the group elements are ‘continuous’ or ‘discrete’.
Discrete Group (Discrete Group):
- Elements are “discrete” and countable. (Example: addition group of integers \(Z\))
- No continuous parameters.
- Physics Example: Symmetry of crystal lattices (translation, rotation), \(Z_2\) (parity/reflection symmetry), \(S_n\) (permutation of identical particles).
Continuous Group (Continuous Group) / Lie Group (Lie Group):
- Elements are “smoothly” connected and described by continuous parameters (e.g., angle \(\theta\), time \(t\)).
- Naturally possess a “smooth manifold” structure (differentiable).
- Physics Example: Time translation, space translation, rotation, Lorentz transformation, etc.
- (Subsequent classifications mainly focus on these Lie groups.)
2. Classification Based on Properties (The S, U, O Hierarchy)
The most useful classification in physics is based on “what does this transformation preserve?” This is mostly expressed as a Matrix Group and defined by a combination of the following constraints.
A. All Beginnings: “General Linear Group” GL(n)
GL(n, R) or GL(n, C)
- G (General): General
- L (Linear): Linear (Matrix)
- n: Size of the matrix (\(n \times n\))
- R / C: Whether the matrix elements are real (Real) or complex (Complex).
GL(n) is the set of all \(n \times n\) matrices that have an inverse. This represents the most general condition of “all linear transformations.” All other Lie groups are subsets (subgroups) of GL(n), and are defined by adding additional constraints to GL(n).
B. Constraint (1): Preservation of Inner Product
The most important physical constraint is that “geometric properties (length, angles) should not change even after the transformation.”
U(n) - Unitary Group
- U (Unitary): Unitary
- Constraint: \(M^\dagger M = \mathbf{1}\) ( \(M^\dagger\) is the Hermitian conjugate)
- Meaning: Preserves the inner product (probability amplitude) in complex vector spaces.
- Physics: The progenitor of all symmetries in quantum mechanics (\(U(1), SU(2), SU(3)\), etc.).
O(n) - Orthogonal Group
- O (Orthogonal): Orthogonal
- Constraint: \(M^T M = \mathbf{1}\) ( \(M^T\) is the simple transpose)
- Meaning: Preserves the inner product (length, angles) in real vector spaces.
- Physics: 3D space rotations (\(SO(3)\)) and reflections in classical mechanics.
C. Constraint (2): Preservation of Volume and Orientation (Determinant)
The second constraint is that “volume should not change even after the transformation.”
- S (Prefix) - Special
- S (Special): Special
- Constraint: \(\det(M) = 1\)
- Meaning: The set of transformations that preserve volume (\(|\det(M)|=1\)) and do not flip orientation (e.g., right-handed coordinate system) (\(\det(M) > 0\)).
D. Completion of Classification: Combination of Symbols
- GL(n): (Prototype) All matrices with an inverse.
- SL(n): Special Linear. (Linear transformations with \(\det(M)=1\))
- O(n): Orthogonal. (Rotation + Reflection).
- SO(n): Special Orthogonal. (Orthogonal transformations with \(\det(M)=1\). Applies only to “pure rotation”).
- U(n): Unitary. (Quantum symmetry + global phase included).
- SU(n): Special Unitary. (Unitary transformations with \(\det(M)=1\). Global phase excluded).
4. Classification by Algebra (Simplicity)
Finally, groups are classified by the algebraic structure of their generators (Lie algebra).
- Abelian vs. Non-Abelian (commutativity)
- Abelian (Abelian): \([G_i, G_j] = 0\). (Example: \(U(1)\), time/space translation)
Non-Abelian (Non-Abelian): \([G_i, G_j] \neq 0\). (Example: \(SU(n)\), \(SO(n)\), rotation)
Simple vs. Semi-simple (Decomposability)
- Simple (Simple): The “building blocks” of fundamental symmetries that cannot be further decomposed. (Example: \(SU(n)\), \(SO(n)\), etc.)
- Semi-simple (Semi-simple): Decomposes into a direct product of simple groups. (Example: \(SU(2) \times SU(3)\))
- Others (Abelian, etc.): \(U(1)\) is not simple.
💡 Physical Connection (Standard Model)
Modern particle physics’ Standard Model is based on a gauge group \(U(1) \times SU(2) \times SU(3)\). According to this classification system, the Standard Model is perfectly described as a “Lie group composed of the product of one Abelian group (\(U(1)\)) and two Simple Non-Abelian groups (\(SU(2), SU(3)\))”.
Appendix 5: Core Learning Roadmap and Strategy
This appendix provides an overview of how to study the book’s 12 chapters in an optimal order and strategy.
1. 4-Step Core Roadmap
The learning order starts with mathematical foundations, then deepens into quantum axioms, open systems (entanglement), and modern quantum foundations.
- Mathematical Foundations (Part 1)
- Hilbert space (inner product, completeness)
- Dirac notation (bra-ket)
- Linear operators (Hermitian, unitary, projector)
- Tensor product, density matrix, partial trace
- Dynamics and Path (Part 2)
- Variational calculus and Lagrangian mechanics (principle of least action)
- Path integral (functional integral, stationary phase approximation)
- Information and Entanglement (Part 3)
- Open systems and quantum channels (CPTP maps, Kraus representation)
- Consistent histories (decoherence function, probability condition)
- Delayed choice, quantum Darwinism (objectivity)
- Time and Reality (Part 4)
- Emergence of time (Page-Wootters, thermal time)
- Causal structure (indefinite causal order)
- Reality (Leggett-Garg inequalities)
2. Recommended Learning Strategy 💡
The content of the main text includes graduate-level advanced topics beyond standard undergraduate courses. Therefore, a three-step approach is efficient.
Step 1 (Concept Acceptance): First accept and memorize key terms, notations, and basic properties such as Hilbert space, density matrix, partial trace, CPTP, etc.
Step 2 (Example Application): Apply to computable models such as qubits (2x2 matrices) or simple Gaussian examples to develop an understanding of “operational mechanisms” and mathematical intuition.
Step 3 (Proof Deepening): After becoming familiar with computations, learn the proofs and theoretical backgrounds of related theorems when curious about the general properties of concepts.
3. Extensibility: Quantum Cosmology 🌌
If you wish to extend the content of this book to Quantum Cosmology, you need to additionally study the following three key blocks:
- Differential geometry and general relativity (GR)
- Quantum field theory in curved spacetime (QFT in Curved Spacetime)
- Advanced operator algebra (deepening of C* algebra from the main text)
The “consistent histories” and “thermal time hypothesis” discussed in the main text will serve as important conceptual bridges to these advanced topics.
Appendix 6: The Book’s Topology and Next Steps
This book, “Quantum History Mathematics,” is not a standard quantum mechanics textbook. After covering the core mathematics of quantum theory (Hilbert space, density matrix) in Part I, it immediately delves into the most profound questions of modern quantum foundations: “time,” “history,” “realism,” and “objectivity.”
The position of this book is a crucial conceptual/mathematical bridge between standard quantum mechanics and modern research topics (quantum information, quantum cosmology).
1. Coverage by Academic Field
Readers who complete this book will acquire the core foundations of each academic field.
| Academic Field | Core Areas Covered by This Book (No Need for Relearning) | Areas Requiring Additional Learning (Next Steps) |
|---|---|---|
| Standard Quantum Mechanics | Entire Mathematical Axiomatic System: Hilbert space, operators, tensor product, density matrix (Chapters 1, 2). Advanced Dynamics: Lagrangian and path integral (Chapters 3, 4). | Specific Hamiltonian solutions (e.g., hydrogen atom), angular momentum addition, sophisticated perturbation theory (Perturbation Theory), etc. |
| Quantum Information/Computing (QIT/QC) | “Why” of QIT: Qubit, entanglement, mathematics of mixed states (Chapters 1, 2). Principles of Noise and Errors: CPTP maps, Kraus operators (Chapter 5). Measurement and Information: Entanglement, delayed-choice, quantum eraser (Chapters 6, 7). | “How” of QIT/QC: Specific quantum algorithms (Shor, Grover), quantum error correction codes (QECC), specific hardware implementation methods. |
| Quantum Cosmology (Quantum Cosmology) | Mathematical Approach to Core Challenges: 1. Emergence of Classicality: Consistent history (Chapter 6), quantum Darwinism (Chapter 8). 2. Problem of Time: Emergent time (Chapter 9), thermal time (Chapter 10). |
Physical Background: General Relativity (GR). Specific cosmological models such as Wheeler-DeWitt equation, Loop Quantum Gravity (LQG), String Theory. |
| Theory of Everything (ToE)/QFT | Advanced Mathematical Foundations: Path integral (Chapter 4), C* and von Neumann algebras (Chapter 10). | Quantum Field Theory (QFT), Standard Model, General Relativity (GR), and attempts to unify them (String Theory, LQG, etc.). |
2. Advancing to Quantum Cosmology
This book has already thoroughly addressed modern answers to the key questions of quantum cosmology: “What is time?” and “How does the classical world emerge?” (Chapters 9, 10 and 6, 8).
The areas that readers who have completed this book must necessarily supplement to advance to quantum cosmology or the Theory of Everything are clear.
General Relativity (GR)
- Reason: Cosmology is a theory of “gravity,” and GR is the classical theory that deals with gravity (geometry of spacetime). Quantum cosmology is an attempt to quantize this GR.
- Essential Concepts: Differential geometry, Einstein field equations, spacetime dynamics.
Quantum Field Theory (QFT)
- Reason: The “Theory of Everything” uses QFT, which views particles as excitations of “fields,” as its fundamental language.
- Essential Concepts: Field quantization, Feynman diagrams, renormalization.
This book completes the philosophical and mathematical foundations of quantum theory. Adding the two pillars of GR (gravity/spacetime) and QFT (matter/field) prepares readers to advance to the forefront of modern physics.