A • B ≡ AB − BA, A,B are NxN matrix
|a>: column vector
<a|: row vector
L(V, W)： The vector space of linear maps from V to W
R$_λ$(T) ≡(T - λ1)^(-1)is called the 【resolvent of T at λ】 (Definition 17.7.1)
<S>: The subgroup generated subgroup generated by by S
C$_G$(x): the centralizer of x in G. C$_G$(x)={g | ∃g∈G, if xg=gx}
Z(G): center of G. Z(G)={g | ∀g∈G, if ∀x∈G && xg=gx}
[a,b]≡aba$^-1$b$^-1$
f$_{,i}$ ≡ ∂f/∂xi ≡ ∂$_i$f
(∂i|p)f ≡ ∂f/∂xi|p
∂$_θ$ ≡ ∂/∂θ
Ψ$^i_{,jk}$ ≡ ∂$^2$Ψ$^i$/∂xj∂xk (Ψ的第i分量对xj,xk求二阶导)
Ψ$^i_{x0,jk}$ ≡ ∂$^2$Ψ$^i$/∂xj∂xk |x=x0 (Ψ的第i分量对xj,xk求二阶导,并求x=x0时的导数)
f$_{;i}$ ≡ (∇f)i
f$_{;ij}$≡ (∇$^2$ f)ij
∇f = f$_{;i}$ dx$^i$
DiS: The total derivative of S with respect to total derivative xi
Ey,i : L2(Ω)→R, Ey,i(f) ≡ ∂i f(y) (the evaluation of the derivative of functions with respect to the i-th coordinate)
u$^(n)$: nth-order derivative of function u
ψ$_{∗P}$: differential of ψ at P. (Let t∈T$_P$(M) and f ∈F$^∞$(Q). The action of ψ$_{∗P}$(t)∈T$_Q$(N) on f is defined as ( ψ$_{∗P}$(t) )(f) ≡ t(f◦ψ) )
df ≡ f$_∗$: differential of f. The map df: T$_P$(M)→R given by df(t)=t(f)
F$_{αβ,γ}$ ≡ ∂F$_{αβ}$ /∂x$^γ$
gl(V): the Lie algebra of GL(V)
T*： dual(or pullback) of T
differentiable map ψ : M → N
differentiable map ψ$_*$ : T(M)-->T(N)
ψ$^*$: the dual of ψ$_*$(or the pullback of ψ)
ψ$^*$: It takes a 1-form on N to a 1-form on M (是否可以这么写： ψ$^*$: Λ1(N)-->Λ1(M) )
ξ$_M$|P ≡ ξ$_M$(P) ≡ dΦ(exp tξ, P)/dt |t=0
infinitesimal generator of the action induced by ξ
--------------------------------------------------
The collection of all subsets (including ∅) of a set A is denoted by 2$^A$ .
.
Although sets are at the root of modern mathematics, by themselves they
are only of formal and abstract interest. To make sets useful, it is necessary
to introduce some structures on them. There are two general procedures for the implementation of such structures.
These(two general procedures) are the abstractions of the two major branches of mathematics—algebra and analysis.
.
two general procedures(to implement a structure):
- introduce a binary operation on set. (the abstractions of algebra) 产生了各种代数结构，如(X,+), (G,*)
- vector space: set + 加法交换律 + 加法结合律 + 0-vector
- linear map: vector space(set+加法交换律+加法结合律+0-vector)+ 元素与scalar的乘法
- algebra: vector space(set+加法交换律+加法结合律+ 0-vector)+ 元素与元素的乘法
- introduce the concept of continuity.(the abstractions of analysis) 产生了topology结构
.
vector space: set + 加法交换律/加法结合律/0元
subspace： W is ~ of V, if |a>,|b>∈W then (α|a>+β|b>)∈W
.
Let W be a subspace of the vector space V
If |a>,|b>∈V and |a>−|b>∈W, then |a> ▹◃ |b>
（类比,Let W be a subgroup of V, if a, b∈V and ab{^{-1}$∈W, then a▹◃b）
.
equivalence class of |a> : [[a]] := {|b> | |a> ▹◃ |b>, |b>∈V} = {|b> | |a>−|b>∈W, |b>∈V}
factor set (quotient set) : V/W := {[[a]] | |a>∈V}
[[a]] ≡ |a> + W
.
We turn the factor set into a factor space by defining: α[[a]] + β[[b]] = [[αa + βb]]
where [[αa + βb]] is the equivalence class of α|a> + β|b>
.
2 ways of constructing a new vector space out of two vector spaces:
- Direct sum⊕ (类比dot product)
a (|u>, |v>) = (a |u>, a |v>)
(|u1>, |v1>) + (|u2>, |v2>) = (|u1>+|u2>, |v1>+ |v2>)
dim(U⊕V) = dim(U) + dim(V)
- Tensor Product ⊗ (类比cross product)
a (|u>, |v>) = (a |u>, |v>) = ( |u>, a |v>)
(a1 |u1> + a2 |u2>, |v>) = (a1 |u1>, |v>) + (a2 |u2>, |v>)
(|u>, b1 |v1> + b2 |v2>) = (|u>, b1 |v1>) + (|u>, b2 |v2>)
U⊗V里的元素记为|u>⊗|v>(或简记为|uv>)
dim(U⊗V) = dim(U) * dim(V)
.
A function that is linear in both of its arguments is called a 【bilinear function】
。
The question of the existence of an inner product on a vector space is a deep problem in higher analysis. Generally, if an inner product exists, there may be many ways to introduce one on a vector space. A finite-dimensional vector space always has an inner product and this inner product is unique.
。
inner product:
1.<a|b> = <b|a>* ∈ C (sesquilinear or hermitian(为了保证 positive definite，舍弃了bilinear))
2.<a|(β|b> + γ |c>) = β<a|b> + γ <a|c> (We write<a|(β|b> + γ |c>) as <a|βb + γ c> )
3.<a|a> ≥ 0, and <a|a> = 0 iff |a> = |0> (positive definite)
(1+2可得： <βb + γ c|a> = β*<b|a> + γ*<c|a>)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
(在C域里 positive definite和bilinear是冲突的。因为g(i|a>, i|a>) = i^2 g(|a>, |a>) = −g(|a>, |a>).
在inner product 的定义里保留positive definite而抛弃bilinear。
但如果保留bilinear而抛弃positive definite，会怎样？数学上有这个定义吗？Definition 2.4.2讨论的就是这个问题。
两者的对比：
Theorem 2.3.8 (operator T is positive definite)
A linear operator T on an inner product space is 0 iff <x|T|x> = 0 for all |x>
Proposition 2.4.4 (operator A is bilinear)
An operator A ∈ End(V) is skew (A^T = −A) iff <x|Ax>F ≡ <x|A|x>F = 0 for all |x>
Theorem 2.3.8 shows how strong a restriction the positive definiteness imposes on the inner product.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
。
natural inner product for Cn：
Let |a>, |b>∈ Cn,
<a|b> := ∑aj* bj
。
Any function on a vector space satisfying the four properties above is
called a norm, One does not need an inner product to have a norm.
。
Norm:
1.The norm of the zero vector is zero: ||0|| = 0.
2.||a|| ≥ 0, and ||a|| = 0 if and only if |a> = |0>.
3.||αa|| = |α| ||a|| for any complex α.
4.||a + b|| ≤ ||a|| + ||b||. ( triangle inequality.)
。
Inner product spaces are automatically normed spaces, but the converse
is not, in general, true
Theorem 2.2.9 A normed linear space is an inner product space if and only
if the norm satisfies the parallelogram law:
||a + b||^ 2 + ||a − b||^ 2 = 2||a||^2 + 2||b||^2
.
Theorem 2.2.10 Every finite-dimensional vector space can be turned into
an inner product space.
.
linear map (or transformation) :
T(α|a> + β|b>) = αT(|a>) + βT(|b>)
The set of linear maps from V to W is denoted by L(V, W)
L(V, V) 简记为L(V) or End(V) (endomorphism)
.
The dimension of ker(T) is also called the 【nullity】 of V.
。
Let T : V → W be a linear transformation， then dim V = dim ker(T)+ dim T(V)
.
<x|Ax>F ≡ <x|A|x>F
.
.
γ$_a$ |ai> = αi, <a|ai> = αi ===> (|a>)† ≡ <a| ≡ γ$_a$ ( the dual of |a> )
(α |a> + β |b>)† = α*<a| + β*<b|
|a> = (α1, α2, ..., αn ), <a|=(α1*, α2*, ..., αn* )T,
.
ad(A): classical adjoint of A
.
Definition 2.5.4 Let T : V → U be a linear map. Define T* : U*→ V* by
[T*(γ)] |a>= γ(T|a>) , ∀|a>∈V, γ∈U* ,
T* is called the 【dual(or pullback) of T】.
=================================
CH3
algebra: vector space(set+加法交换律+加法结合律+ 0-vector)+ 元素与元素的乘法
associative
commutative
identity/unit/1/e
left inverse
right inverse
subalgebra: be closed under multiplication. (If a,b∈A, then ab∈A)
an algebra with identity is also called a unital algebra
.
.
center of A : Z(A)={x | ∀a∈A, ax∈A && xa∈A}
.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
为什么研究sub-space：
the significance of subspaces resulting from the fact that physics frequently takes place not inside the
whole vector space, but in one of its subspaces. For instance, the study of projectile motion teaches us that it is very convenient to “project” the motion onto the horizontal and vertical axes and to study these projections separately.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
为什么研究subalgebra：
the product of elements of a subalgebra do not leave the subalgebra.
the product of elements of a subalgebra do not leave the entire algebra.
.
Given an associative algebra A, x ∈ A
a left ideal of A generated by x: Ax ≡ {ax | a ∈ A}.
a right ideal of A generated by x: xA ≡ {xa | a ∈ A}.
an ideal of A generated by x: AxA ≡ {axb | a, b ∈ A}
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
left ideal的用途 ：if L is left ideal of A, then AL ⊂= L (A的元素与L的元素作用，结果仍在L里)
right ideal 的用途：if L is right ideal of A, then LA ⊂= L
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
.
derivation operator D: D(|a>|b>) = D(|a>)|b> + |a>D(|b>)
If D1 and D2 are derivations, then (αD1+βD2) is also a derivation.
但D1D2 is not derivation
但D1D2−D2D1≡D1•D2 is derivation
(我感觉类似上面这两点性质，可能意味着乘法和加法本质的区别)
radical of A ： Rad(A)： the unique nilpotent ideal in A which contains every nilpotent left, right, and two-sided ideal of A.
.
Theorem 3.5.25 An algebra is semi-simple iff it is the direct sum of simple algebras.
(semi-simple algebra can be built up from simple algebra)
Definition 3.5.17 An algebra whose radical is zero is called semi-simple.
Proposition 3.5.18 A simple algebra is semi-simple.
e^tS * e^tT = e^t (S+T) iff [S, T]=0
anticommutator {S, T}≡ ST + TS
adjoint of T, or hermitian conjugate of T: T† : (T|b>)†= <b| T†
(<a|T|b>)* = <b|T† |a>
如何求T†：P114， Example 4.3.4
hermitian, or self-adjoint：T†=T
anti-hermitian：T†=-T
.
Definition 4.3.6 The 【expectation value of an operator T in the “state” |a> 】 <T>$_a$ is a complex number defined by <T>$_a$ = <a|T|a>
<T>* = <a|T|a>* = <a|T† |a>
Theorem 4.3.7 A linear map T on a complex inner product space is hermitian if and only if <a|T|a> is real for all |a>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
|a>∈C, <a|H|a>is real. 这就是H的意义，因为<a|H|a>是实数，所以可以在物理实验中测量<a|H|a>的值。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
T = 1/2 (T+T†) +1/2(T-T†) ≡ X+A ≡X+iY
X≡1/2 (T+T†)
A≡1/2 (T-T†)
Y≡ -iA ≡ -i/2 (T-T†)
.
positive operator: A>=0, if <a|A|a> ≥ 0 for all |a> != |0>
strictly positive(or positive definite) operator: A>0, if <a|A|a> >0 for all |a> != |0>
.
strictly positive operator is invertible
hermitian operators 类似于 real numbers
H^2 is positive
Unitary Operators: U† U = 1 (<==>U U†=1) 类似于 complex numbers of unit magnitude e^(iθ)
.
Definition 4.4.2 A hermitian idempotent of End(V) is called a projection operator.
.
P$_y$|x> is the component of |x> along |y>
.
Proposition 4.4.6 由坐标基，计算subspace的投影矩阵
combine two vector spaces ----> new vector space类比：
combine two representations ----> a new representation
orthogonal. matrix: A A^t = A^t A =1 例如rotation
hermitian conjugation matrix: (α ij)† = (αji)*
hermitian matrix: H† = H 即 ηji* = ηij， 例如Pauli spin matrice
unitary matrix: U† U = UU† = 1
.
M ij： minor of order
cofactor of (A) ij： 矩阵A的代数余子式， 记为(cof A)ij
。
O(n): orthogonal transformations whose determinants are −1: reflection through the origin
SO(n): orthogonal transformations whose determinants are +1: rotation
U(n): The set of unitary matrices in n dimensions. det U = e^{iα}
SU (n): a group who has the set of U(n) with α = 0
det(1 + εA) = 1 + ε tr(A)
det(A(t)) ' |t=0 = tr( A'(t)|t=0 )
ln(det D ) = tr(ln D)
det D = e^{tr(ln D)}.
det A = exp{ tr(R(ln A)R^-1) }= exp{tr(ln A )}
.
A(M)⊂M <==>A†(M⊥)⊂M⊥
.
To make the upper half also zero, additional properties will be required for
the operator. Thus, for a general operator on a
complex vector space, upper-triangular representation is the best we can ac-
complish.
。
Simultaneous Diagonalization的意义：
For instance, if there exists a
basis of a Hilbert space of a quantum-mechanical system consisting of si-
multaneous eigenvectors of two operators, then one can measure those two
operators at the same time（测量的是operator）. In particular, they are not restricted by an uncer-
tainty relation。
。
Theorem 6.4.18 Two normal operators A and B are simultaneously
diagonalizable iff [A, B] = 0
.
A normal operator on a real vector space does not have a
real eigenvalue in general. However, if the operator is self-adjoint (hermi-
tian, symmetric), then it will have a real eigenvalue.
.
Theorem 6.6.6 Let V be a real inner product space and T a self-
adjoint operator on V. Then there exists an orthonormal basis in V
with respect to which T is represented by a diagonal matrix.
。
Theorem 6.6.6的应用：
- Theorem 6.6.6 says that it is always
possible to choose coordinate systems in which the moment of inertia matrix
is diagonal
。
- a1*x^2 + a2*y^2 + a3 xy + a4 x + a5 y + a6 = 0, 旋转坐标系，消除xy项
- 多元函数求极值
Theorem 6.4.8 (Complex Spectral Decomposition)
Theorem 6.6.11 (Real Spectral Decomposition)
Theorem 6.6.12 A real orthogonal operator on a real inner product space
V cannot, in general, be completely diagonalized. The closest it can get to a
diagonal form is
Theorem 6.6.12 的应用：
Example 6.6.14 An interesting application of Theorem 6.6.12 occurs in
classical mechanics, where it is shown that the motion of a rigid body con-
sists of a translation and a rotation. The rotation is represented by a 3 × 3
orthogonal matrix. Theorem 6.6.12 states that by an appropriate choice of
coordinate systems (i.e., by applying the same orthogonal transformation
that diagonalizes the rotation matrix of the rigid body), one can “diagonal-
ize” the 3 × 3 orthogonal matrix
.
Theorem 6.6.15 (Euler) The general motion of a rigid body consists of
the translation of one point of that body and a rotation about a single axis
through that point.
.
unitary operators can be written as exp(iH): U = e^(iH)
e.g.
U =(cosθ, −sinθ, sinθ, cosθ)
U = e^(iH), H = θ(−P1 + P2 )
。
hermitian operators 类似于 real numbers
Unitary Operators 类似于 complex numbers of unit magnitude e^(iθ)
那么 任意T是否能类比于某个数？答案是Theorem 6.7.1 (Polar Decomposition) ：
An operator T on a (real or complex) finite-dimensional inner product space can be written as T = UR where theorem
R is a positive operator and U an isometry (a unitary or orthogonal operator).
.
实际上R = sqrt(T† T)， 但U is not unique。实际做的时候，当R求得后，用T=UR求得U。
It is interesting to note that the positivity of R and the nonuniqueness of U are the analogue of the positivity of r and the nonuniqueness of e^iθ: z = r e^iθ = r e^i(θ+2nπ)
.
==============================================
PART II
------------------------
CH 7 Hilbert Spaces
general Hilbert space: infinite basis
separable Hilbert space: infinite but countable basis
all continuous functions defined on an interval [a, b] forms a linear vector space, 但这个空间是不完整的。
如何把它完整化？a natural inner product for functions is defined in terms of integrals，
The space L$^2_w$(a, b) is complete
All complete inner product spaces with countable bases are isomorphic to L$^2_w$(a, b)
dθ(x−x0)/dx = δ(x−x0) 即 θ' = δ
delta function的意义：
we can define the derivative of any function, including discontinuous functions, at any point
(including points of discontinuity, where the usual definition of derivative fails) by this relation. That is, if φ(x) is a “bad” function whose derivative is not defined at some point(s), and f (x) is a “good” function, we can define the derivative of φ(x) by ∫∞ f(x)φ'(x)dx ≡ -∫∞ f'(x)φ(x)dx
.
∫∞ f(x)δ'(x-a)dx = -∫∞ f '(x)δ(x-a)dx = -f '(a)
.
An interesting application of distributions (generalized
functions) occurs when the notion of density is generalized to include not
only (smooth) volume densities, but also point-like, linear, and surface den-
sities.
-----------------------------------------
CH8 Classical Orthogonal Polynomials
compose f with a complete set of orthogonal polynomials { |Ck>}
|f> =∑ak |Ck>,
则系数ak=<Ck |f>/<Ck | Ck>
或ak= ∫ C*k(x)f (x)w(x)dx / ∫|Ck(x)|^2w(x)dx
Dirac delta function in terms of orthogonal polynomials { |Ck>}
-----------------------------------------------
CH 9 Fourier Analysis
the Dirac delta function in terms of the present orthonormal basis of Fourier expansion
Fourier integral transforms:
f(x) = 1/√2π * ∫∞ f~(x)e$^{ikx}$ dk
f~(x) = 1/√2π * ∫∞ f(x)e$^{-ikx}$ dk
if f(x) is constant b, f~(x) is Dirac function √2π b δ(x)
if f~(x) is Dirac function, f~(x) is constant 1/√2π
.
In other words, when the
function is wide, its Fourier transform is narrow. In the limit of infinite width
(a constant function), we get infinite sharpness (the delta function).
.
the Fourier transform of a Gaussian is also a gaussian
.
In quantum mechanics, for instance,
most of the time the r representation, corresponding to the position, is used,
because then the operator equations turn into differential equations that are
(in many cases) linear and easier to solve than the corresponding equations
in the k representation, which is related to the momentum.
.
The Fourier transform is very useful for solving differential equations. This
is because the derivative operator in r space turns into ordinary multipli-
cation in k space.
The real power of the Fourier transform lies in the formal analysis of differential
equations.
======================================
Part III Complex Analysis
-----------------------------
CH10 Complex Calculus
Cauchy-Riemann conditions（differentiability） <==> ∂u/∂x=∂v/∂y, ∂u/∂y=-∂v/∂x <==> ∂f/∂z* =0
。
【homographic transformations】 (z'=(az + b)/(cz + d)) can map an infinite region of the z-plane onto a finite region of the z' -plane
。
Path-independence of a line integral of a vector F
<==> the integral along a closed path is 0
<==> ∇ × F = 0 at every point of the region bordered by the closed path
。
Cauchy integral formula是非常基本的，可以证明代数基本定理，导出泰勒级数
。
primitive: 原函数
。
In general, the integer Nε may be dependent on z; that is, for different values of z, we may be forced to pick different Nε ’s. When Nε is independent of z, we say that the convergence is 【uniform】一致收敛
。
A Taylor series consists of terms with only positive powers.
A Laurent series allows for negative powers as well.
harmonic function is preserved under conformal mapping
Laplace’s equation+the boundary conditions ==> unique electrostatic potential Φ.
the value of an analytic function on a boundary (contour)
determines the function at all points inside the boundary.
.----------------------------------------------------
11 Calculus of Residues
Laurent series
analytic function 可以被表示成Laurent series
A Laurent series can give information about the integral of a function
around a closed contour in whose interior the function may not be analytic.(即是拓扑性质)
b1 ≡ Res[f(z0)]
It is important to note that the residue is independent of the contour C as long as z0 is the only isolated singular point within C.
--------------------------------
The gamma function is a generalization of the factorial function—which is defined only for positive integers—to the system of complex numbers.
gamma (or factorial) function：Γ(z)≡∫$_0^∞$ t$^{z-1}$ e$^{-t}$ dt (Re(z)>0)
(The restriction Re(z) > 0 assures the convergence of the integral.)
Γ(z + 1) = zΓ(z)
Γ(1/2) = sqrt(π)
beta function: B(a, b) ≡ ∫$_0^1$ t$^{-a-b}$ (1 − t)$^{b-1}$ dt (Re(a), Re(b) > 0)
Euler’s integral of the first kind: beta function
Euler’s integral of the second kind: gamma (or factorial) function
Γ(a) Γ(b) = Γ(a + b) B(a, b) ==> B(a, b) = B(b, a)
Γ(z) Γ(1 − z) = π/sin(π z)
Γ(z) Γ(−z) = -π/(z * sin(π z))
=================================
Part IV Differential Equations
CH 13 Separation of Variables in Spherical Coordinates
=================================
Part IV Differential Equations
CH13 Separation of Variables in Spherical Coordinates
|p>= -i∇
L = |r>⨯ |p>
εijk εimn = δjm δkn − δjn δkm
L+ ≡ Lx + iLy (这里的Lx表示L的x分量，与Sturm-Liouville (S-L) operators Lx不同)
L− ≡ Lx − iLy
----------------------------------------
CH 14
A prediction is not a prediction unless it is unique. This expectation for
linear equations is borne out in the language of mathematics in the form of
an existence theorem and a uniqueness theorem.
In general, the Wronskian of any two linearly independent solutions of y'' + q(x)y = 0 is constant.
Self-adjoint(or hermitian) differential operators are equally important because their
“eigenfunctions” also form complete orthogonal sets
- Analysis is one of the richest branches of mathematics, focusing on the endless variety of objects we call functions.
- The simplest kind of function is a polynomial, which is obtained by performing the simple algebraic operations of addition and multiplication on the independent variable x.
- The next in complexity are the trigonometric functions, which are obtained by taking
ratios of geometric objects.
- If we demand a simplistic, intuitive approach to functions, the list ends there.
- It was only with the advent of derivatives, integrals, and differential equations that a vastly rich variety of functions exploded into existence in the eighteenth and nineteenth centuries. For instance, e^x , nonexistent before the invention of calculus, can be thought of as the function that solves dy/dx = y.
-Although the definition of a function in terms of DEs and integrals seems
a bit artificial, for most applications it is the only way to define a function.
-An effective way of studying such functions is to study the differential equations they satisfy.(学习微分方程的意义)
14.6.2 Quantum Harmonic Oscillator
The Hamiltonian of a one-dimensional harmonic oscillator is
H = (p^2+(mωx)^2)/(2m)
p = -i h d/dx
a ≡ sqrt(mω/(2h)) + i p/sqrt(2mhω)
a† ≡ sqrt(mω/(2h)) - i p/sqrt(2mhω)
[H, a] = −hωa
[H, a†] = hωa†
Ha |ψ$_E$> = (aH – hωa )|ψ$_E$> = (E – hω)a |ψ$_E$>
Ha†|ψ$_E$> = (aH + hωa†)|ψ$_E$> = (E + hω)a†|ψ$_E$>
a†: raising/creation operators
a : lowering/annihilation operators
a|ψ$_E$> = c$_E$ |ψ$_{E-hω}$>
|ψ0>: ground state
a|ψ0> = 0
对“Schrödinger equation在经典力学下的近似”的解释：
Schrödinger equation describes a classical statistical mixture when h → 0
In the classical limit, the solution of the Schrödinger
equation describes a fluid (statistical mixture) of noninteracting classical
particles of mass m subject to the potential V (r). The density and the current
density of this fluid are, respectively, the probability density ρ = |ψ|^2 and
the probability current density J of the quantum particle.
-------------------------------------------------------
CH15 Complex Analysis of SOLDEs
CH17 Introductory Operator Theory
Definition 17.3.1
Let T∈L(H). A complex number λ is called a 【regular point】of T if the operator (T- λ1)^(-1) exists and is bounded.
The set of all regular points of T is called the 【resolvent set】 of T, and is denoted by ρ(T). The complement of ρ(T) in the complex plane is called the 【spectrum】 of T and is denoted by σ(T).
An intuitive way of imagining denseness is that the (necessarily)
infinite subset is equal to almost all of the set, and its members are scattered
“densely” everywhere in the set.
normal operator: [X, Y] = 0
Corollary 17.6.6
If K is a compact hermitian operator on a Hilbert space H, then the eigenvectors of K constitute an orthonormal basis for H.
Corollary 17.6.9
If T is a compact normal operator on a Hilbert space H, then the eigenvectors of T constitute an orthonormal basis for H.
Definition 17.7.1
Let T be an operator and λ ∈ ρ(T). The operator R$_λ$(T) ≡(T - λ1)^(-1)is called the 【resolvent of T at λ】
von Neumann showed that mathematical rigor
could be restored by taking as basic axioms the assumptions that the states of a physical
system were points of a Hilbert space and that the measurable quantities were Hermitian
(generally unbounded) operators densely defined in that space.
Sturm-Liouville (S-L) operators: Lx ≡ d^2/dx^2 – q(x)
Theorem 19.4.1 The eigenfunctions {u$_n$ (x)} (n=1-->∞) of an S-L system consisting of the S-L equation (pu')' + (λw − q)u = 0 and the BCs of (19.24) form a complete basis of the subspace U of L 2 w (a, b) described in (19.25). The eigenvalues are real and countably infinite and each one has a multiplicity of at most 2. They can be ordered according to size λ 1 ≤ λ 2 ≤ · · · , and their only limit point is +∞.
Theorem 19.4.1 is the important link between the algebraic and the analytic machinery of differential equation theory.
Such a decomposition of plane waves into components with definite orbital
angular momenta is extremely useful when working with scattering theory
for waves and particles。(spherical harmonics的作用)
===================================
CH20 Green’s Functions in One Dimension
GFs are inverses of differential operators
Green’s Functions ---> inhomogeneous differential equations
The secret of this success is the generalized Green’s identity. (20.11)
an inhomogeneous DE with inhomogeneous BCs can be separated into two DEs:
- homogeneous with inhomogeneous BCs and
- inhomogeneous with homogeneous BCs, which is appropriate for the GF.
Green’s functions的物理意义（p629）
An inhomogeneous DE such as Lx [u] = f(x) can be interpreted as a black box (Lx) that determines a physical quantity (u, e.g. electrostatic potential) when there is a source (f, e.g. charge) of that physical quantity。
To be more precise, let us say that the strength of the source is S1 and it is located at y1 ; then the source becomes (S1 δ(x - y1)). The physical quantity, the Green’s function, is then (S1 G(x, y1)).
physical interpretation of the Neumann series
Lx: free operator
V(x): interacting potential
Feynman’s idea is to consider G(x, y) as an interacting propagator between points x and y and G0(x, y) as a free propagator.
The main task of perturbation theory is to find the eigenvalues and eigen-
vectors of the perturbed Hamiltonian in terms of a series in powers of λ
of the corresponding unperturbed quantities.
------------------------------------------------------
∫$_D$ Δu = ∫$_{∂D}$ |e$_n$>∇u = ∫$_{∂D}$ ∂u/∂n (P672)
(|e$_n$> is an m-dimensional unit vector normal to ∂D)
Equation (22.24) lends itself nicely to a physor propagatorical interpretation. The RHS can be thought of as an integral operator with
kernel G(x, y; t). This integral operator acts on u(y, 0) and gives u(x, t);
that is, given the shape of the solution at t = 0, the integral operator pro-
duces the shape for all subsequent time. That is why G(x, y; t) is called the
evolution operator, or propagator.
G(ret)s(r, t) is capable of propagating signals only for positive times.
G(adv)s (r, t) can propagate only in the negative time direction.
In relativistic quantum field theory antiparticles are interpreted mathematically as moving
in the negative time direction!
Feynman propagator
Green’s functions for second-order differential operators in 1-D and in 2+-D.
1-D Green’s functions are continuous functions.
2+-D Green’s functions are not only discontinuous, but that they are not even functions in the ordinary sense; they contain a delta function. Thus, in general, Green’s functions in higher dimensions ought to be treated as distributions (generalized functions).
-----------------------------------------------
对coset的引入比较容易理解。
given two subspaces U and W of a vector space V, we denote by U+W all vectors of V that can be written as the sum of a vector in U and a vector in W.
类比
Theorem 23.2.12
Any two right (left) cosets of a subgroup are either disjoint or identical.
通俗理解：
- a and b belong to the same right coset of S if and only if ab$^{-1}$∈S.
- A coset can be represented by any one of its elements（某个coset里的元素是等价的，不同的coset是不同的划分）
coset的例子：
- The rational number 1/2 represents 1/2 , 2/4 , 3/6 , （1/2 , 2/4 , 3/6是等价的）
- a given magnetic field represents an infinitude of vector potentials each differing by a gradient from the others,
- a physical state in quantum mechanics is an infinite number of wave functions differing from one another by a phase.（虽然a physical state 有不同的phase不同，但认为它们是等价的）
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
factor group是核心概念，围绕factor group定义了coset，subgroup，normal等概念。
为什么要定义factor group？
为了把group G分解为相对简单的subgroup Si, 用Si的组合来表示G。
类比：把vector space V分解为若干线性无关的basis ei, 用ei的线性组合来表示V。
类比：
Let G=Z and S=Zm(e.g. m=3,S={0, 3, 6, ...}),
then coset k+mZ, (0≤k≤m-1) (e.g. coset 0+mZ={0,3,6, …}, coset 1+mZ={1,4,7, …})
and quotient group G/S=Z/Zm={0+mZ, 1+mZ, …, (m-1)+mZ}
Example 23.2.15讲的不错：
Let G=R3 and let S be a plane through the origin。
Then t+S is S if t∈S; otherwise, it is a plane parallel to S. In fact, t+S is simply the translationof all points of S by t.
Since G is abelian, S is automatically normal, and G/S is the set of planes parallel to S.
Let |e> be a normal to S. Then G/S = {r|e>+S | r∈R}
factor group为什么记为G/S？
参见Example 23.2.15。factor group G/S is isomorphic to R. Identifying S with R^2 , we can write R^3/R^2 ≃ R.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
coset ~ factor space ~ factor algebra
the subalgebra must be an ideal of the algebra
the factor group must be normal subgroup of the group
==================================
CH24 Representation of Groups
Characters can be thought of as vectors in a |G|-dimensional inner product space. According to Eq. (24.11), the characters of inequivalent irreducible representations are orthogonal.
χ(g) ≡ tr Tg = ∑ Tii(g),
24.5 Group Algebra
Kronecker product=tensor product=direct product
(|v>, |w>) abbv. |v, w> or |vw>
A common situation in quantum mechanics is to combine two simple sys-
tems into a composite system and see which properties of the original sim-
ple systems the composite system retains. For example, combining the an-
gular momenta of two particles gives a new total angular momentum op-
erator. The question of what single-particle angular momentum states are
included in the states of the total angular momentum operator is the con-
tent of selection rules and is of great physical interest:
------------------------------------
finite group of order n ≃ a subgroup of Sn
representation of Sn ---> representation of many of the Lie groups
===============================
Part VIII Tensors and Manifolds
--------------------------------------
CH26 Tensors
Klein’s Erlanger Program was seen to be inadequate as a general description of geometry by Weyl and Veblen, and Cartan was to play a major role.
--------------------------------------
CH27 Clifford Algebras
n-vector ⊕ n-vector ==>2 n -dimensional algebra (exterior algebra)
exterior product & inner product ==> Clifford algebra
interior product(or contraction) of 1-form and p-vector:(θ: 1-form, A: p-vectors)
iθ A(θ1 , . . . ,θp−1 ) = A(θ,θ1 , . . . ,θp−1 ).
iθ is an antiderivation with respect to the wedge product:
iθ(A∧B) = (iθA)∧B + (−1)^p A∧(iθB)
the interior product of a 1-vector v and a p-vector A: i$_v$ A ≡ i$_{g∗(v)}$A (g∗:V→V∗)
Clifford product ∨:
V × Λ$^p$ (V*) → Λ$^{p+1}$(V*) ⊕ Λ$^{p−1}$(V*)
v ∨ A = v ∧ A + iv A
if p = 1, then v ∨ u = v ∧ u + iv u = v ∧ u + g(u, v) <==> v ∨ u + u ∨ v = 2g(u, v)
Almost 50 years before the advent of Einstein’s general theory of relativity, William Clifford wrote On the space theory of matter in which he argued that energy and matter are different aspects of the curvature of space.
Λ(V∗) is called a Clifford algebra and denoted by Cv
The collection of even and odd elements all even elements of Cv is a subalgebra of Cv and is denoted by C0v.
The odd elements are denoted by C1v, and although they form a subspace of Cv, obviously, they do not form a subalgebra.
Cv = C0v ⊕ C1v
The center of the Clifford algebra Cv , denoted by Zv ,
The anticenter of the Clifford algebra Cv , denoted by Z̅v
Zv = Z0v ⊕v Z1v
C10(R) ∼= C
C01(R) ∼= R ⊕ R
C20(R) ∼= H （algebra of quaternions）
Suppose that μ=ν+4k，k∈Z， then Cνμ(R) ∼= Cμν(R)
Cμμ(R) ~= L(R2^u)
ωv: degree involution
---------------------------------------------
Ch 28 Analysis of Tensors
Space is one of the undefinables in elementary physics. Length and time
intervals are concepts that are “God given”, and any definitions of these
concepts will be circular. This is true as long as we are confined within a
single space. In classical physics, this space is the three-dimensional Eu-
clidean space in which every motion takes place. In special relativity, space
is changed to Minkowski space-time. In nonrelativistic quantum mechanics,
the underlying space is the (infinite-dimensional) Hilbert space, and time is
the only dynamical parameter. In the general theory of relativity, gravitation
and space-time are intertwined through the concept of curvature.
.
Mathematicians have invented a unifying theme that brings the common
features of all spaces together. This unifying theme is the theory of differ-
entiable manifolds.
.
The set of diffeomorphisms of a manifold M onto itself also forms a group, which is denoted by
Diff(M).(类比GL(V))
T(M): tangent bundle of M. The union of all tangent spaces of M. it is spanned by {∂i |P }
类比：T*(M): cotangent bundle of M. it is spanned {dxi |p}
.
X(M): The set of vector fields on M
X*(M)： The set of one-forms on M
。
Let |t>∈Tp(M) we can write |t> = αi ∂i|p, then |t>(f)= αi ∂if|p
。
the binary operation X(M)×X(M) → X(M) as [X, Y] ≡ X ◦ Y − Y ◦ X,
X(M) becomes an algebra, called the 【Lie algebra】 of vector fields of M.
。
Tp(M) is spanned by {∂i |p}
T*p(M) is spanned by {dxi |p}
。
dxi is one-form
.
At each point P of M, there is a unique local integral curve γp of X starting at P defined on an open subset U of M。The map Ft :U-->M defined by Ft(p) = γp(t). The collection of such maps with different t’s is called the 【flow of the vector field X】.
.
A 【tensor field T】 of type (r, s) over a subset U of M is a mapping T:U-->Trs(M) such that T(p) ≡ Tp ≡ T| p∈ Trs,p(M)
.
Box 28.4.11 A tensor is linear in vector fields and 1-forms, even when the coefficients of linear expansion are functions.
.
In all the above cases, the objects being differentiated reside in the same
space。When we try to define derivatives of tensor fields, however, we run immediately into trouble: Tp and Tp' cannot be compared because they belong to two different spaces。To make comparisons, we need first to establish a “connection” between the two spaces. This connection has to be a vector space isomorphism so that there is one and only one vector in the second space that is to be compared with a given vector in the first space. The problem is that there are infinitely many isomorphisms between any given wo vector spaces. No “natural” isomorphism exists between T s,P rT s,P # (M); thus the diversity of tensor “derivatives!” 。 We narrow down this r (M) and seeking a natural diversity by choosing a specific vector at T s,P way of defining the derivative along that vector by associating a “natural” isomorphism corresponding to the vector.
.
The 【Lie derivative】 of a tensor field T at p with respect to vector field X is denoted by (L$_X$T)p
.
(L$_X$T)p(f) = [X,Y]$_p$(f) ==> L$_X$T= [X,Y]
.
Lie derivative's properties:
- L$_X$<θ, Y> = <L$_X$θ , Y> + <θ, L$_X$Y>
- L$_X$f = Xf
- L$_X$Y = [X, Y]
.
L$_{[X,Y]}$ T = L$_X$L$_Y$ T − L$_Y$L$_X$ T, (tensor field T, ∀T∈Trs(M), vector field X,Y)
.
d: exterior derivative operator d : Λ p (U) → Λ p+1 (U)
dω: exterior derivative of p-form ω
.
df = (∂i f )dxi for any real-valued function f
.
应用：Example 28.5.5
In relativistic electromagnetic theory， the electric and magnetic fields are combined to form the electromagnetic field tensor：
F = −Ex dt ∧ dx − Ey dt ∧ dy − Ez dt ∧ dz + Bz dx ∧ dy − By dx ∧ dz + Bx dy ∧ dz, (28.38)
then dF：
dF =
[ (∇ × E +∂B/∂t)$_z$ ]dt ∧ dx ∧ dy +
[ (∇ × E +∂B/∂t)$_y$ ]dt ∧ dz ∧ dx +
[ (∇ × E +∂B/∂t)$_x$ ]dt ∧ dy ∧ dz + (∇ · B)dx ∧ dy ∧ dz
.
Box 28.5.6 The two homogeneous Maxwell’s equations can be written as dF = 0, where F is defined by Eq. (28.38). （如果电磁场表示成tensor F，则由dF可得出齐次maxwell方程组。）
Box 28.5.14 In the language of forms, the inhomogeneous pair of Maxwell’s equations has the simple appearance d(∗F) = 4π(∗J).
.
d ◦ Lx = Lx ◦ d
.
X⎦ ω: ix ω: interior product of a vector field X and a p-form(or differential form) ω.
.
Theorem 28.5.9 Let ω be a p-form and η a q-form on a manifold M. Then
ix (ω ∧ η) = (ix ω) ∧ η + (−1)^p ω ∧ (ix η ).
.
three types of derivation on the algebra of differential forms:
- exterior derivative: d ( Λ p (M) → Λ p+1 (M) )
- Lie derivative: Lx ( Λ p (M) → Λ p (M) )
- interior product: ix ( Λ p (M) → Λ p−1 (M) ) (这货是导数？)
.
Theorem 28.5.10. d, Lx, ix三者的联系：(ω∈Λ p (M), f∈Λ 0 (M), X∈X(M))
- ix df = Lx f
- Lx = ix ◦ d + d ◦ ix
- L$_{fX}$ ω = f Lx ω + df ∧ (ix ω)
δαβ γμ = δαγ δβμ − δαμ δβγ
。
if dω=0 ==> ω is a closed form
if ω1 = dω0 ==> ω1 is called an exact form
Poincaré lemma: every exact form is automatically closed
,
Aα,β ≡ ∂Aα /∂x$^β$, Aαβ,γ ≡ ∂Aαβ /∂x$^γ$
Aαβ dxμ ∧ dxν = Aαβ,γ dxγ ∧ dxμ ∧ dxν
.
Definition 28.6.4 The 【codifferential δ】 is a map δ : Λ p (M) → Λ p−1 (M) given by:
δω = (−1) ^(ν+1) (−1)^ (n(p+1)) ∗d∗ω
。
δf = 0, if f is a 0-form
since ∗∗= ±1, δδ = 0
。
symplectic form (or symplectic structure) : M + closed 2-form ω
symplectic manifold (M,ω): M + symplectic form ω
。
b : X(M) → X*(M): b(X) ≡ X^b = ix ω = ω^b (X)
# : X*(M) → X(M): the inverse of b
。
[v^b(X)](Y) = v(X,Y) ?
。
Lagrangian is a function on the tangent bundle, L : T(N ) → R.
。
The effect of the Legendre transformation is to replace q ̇i by pi as the second set of independent variables. This has the effect of replacing T(N) with T*(N).
。
Hamiltonian system (M, ω , X$_H$)
Hamiltonian vector field X$_H$≡ω# (dH)≡(dH)#, energy function H
。
The significance of the Hamiltonian vector field lies in its integral curve
which turns out to be the path of evolution of the system in the phase space.
。
Poisson bracket of f and g:
{f, g} ≡ ω(Xf, Xg) = i$_{Xg}$ i$_{Xf}$ ω = −i$_{Xg}$ i$_{Xf}$ ω
------------------------------------------------
CH29 Lie Groups and Lie Algebras
Definition 29.1.1 A Lie group G is a differentiable manifold endowed with a group structure such that the group operation G×G→G and the map G→ G given by g↦g$^{-1}$ are differentiable. If the dimension of the underlying manifold is r, we say that G is an r-parameter Lie group
。
The set of left-invariant vector fields on G is denoted by g .
。
gl(V): the Lie algebra of GL(V): gl(V) ∼= L(V)
sl(V): the Lie algebra of SL(V): the set of all traceless operators
u(V): the Lie algebra of U(V): the set of all anti-hermitian operators
su(V): the Lie algebra of SU(V): anti-hermitian traceless operators
。
The special linear group SL(V) is characterized by the fact that all its elements have unit determinant.
。
unitary group: the set of unitary operators
special unitary group: the set of special unitary operators
。
U(V): unitary group on complex vector space V
SU(V): special unitary group on complex vector space: The intersection of SL(V) and U(V)
U(n): U(Cn)
SU(n): SU(Cn)
su(n):su(Cn)
。
O(V): orthogonal group : unitary groups on real vector space
SO(V): special orthogonal group : special unitary groups on real vector space
O(n): O(Cn)
SO(n): SO(Cn)
。
This theorem states that in a neighborhood of the identity element, a Lie
group, as a manifold, “looks like” its tangent space there.
.
Box 29.1.22 Two Lie groups that have identical Lie algebras are locally diffeomorphic.
。
exp : g → G: exponential map
。
φ : G → H
φ$_⁎$: g → h
。
Definition 29.1.26 The adjoint representation of a Lie group G is Ad :G→GL(g) given by Ad(g)=Adg≡I$_{g*}$ .
。
A differential form ω on G is called left-invariant if L$^*_g$(ω) = ω for every g ∈ G
。
Box 29.1.36 Angular momentum operators are the infinitesimal generators of rotation.
。
The structure constants completely determine the Lie algebra:
P(p, n − p)： Poincaré group，It contains the Lorentz, the rotation, and the translation groups as
its proper subgroups
。
x∈Rn， x · x ≡ x^t η x， η = diag(−1, −1, −1 , 1)
the vector x is called 【timelike】 if x · x > 0
the vector x is called 【spacelike】 if x · x < 0
the vector x is called 【null】 if x · x = 0
In the special theory of relativity R4 becomes the set of 【events】. At every event x the set R 4 is divided into 5 regions:
.
(X | Y) ≡ tr(ad$_X$ ◦ ad$_Y$) : Killing form
.
The study of the structure of Lie algebras boils down to the study of the
“simplest” kind of Lie algebras in terms of which other Lie algebras can
be decomposed
d^r g: the ordinary Euclidean volume element of Rr evaluated at the parameters corresponding to g
dμ(g): The volume element at g
-------------------------------------------
CH 30 Representation of Lie Groups and Lie Algebras
Definition 30.0.1
A 【representation of a Lie group G】 on a Hilbert space H is a Lie group homomorphism T : G → GL(H). Similarly,
a 【representation of the Lie algebra g】 is a Lie algebra homomorphism T : g → gl(H).
Box 30.1.2 U (n), O(n), SU(n), SO(n) groups are all compact.
irreducible representations：
Theorem 30.1.7 Every irreducible unitary representation of a compact Lie group is finite-dimensional.
In the general representation theory of Lie algebras, it is desirable to label
each irreducible representation with a quantity made out of the basis vectors
of the Lie algebra. An example is the labeling of the energy states of a quan-
tum mechanical system with angular momentum.
the irreducible representations of the rotation group are labeled by the (half) integers j , and the j th irreducible representation has dimension 2j + 1.
When j is an integer l and the carrier space is L2 (S2), the square-integrable functions on the unit sphere, then L2 becomes a differential operator, and the spherical harmonics Ylm (θ, φ), with
a fixed value of l, provide a basis for the lth irreducible invariant subspace.
A symmetry operation of mathematical physics is expressed in terms of
the action of a Lie group on an underlying manifold M, i.e., as a group
of transformations of M. The Lie algebra of such a Lie group consists of the
infinitesimal generators of the corresponding transformation. These genera-
tors can be expressed as first-order differential operators as in Eq. (29.25).
The Casimir operators {Cα } rα=1 are polynomials in the infinitesimal generators, i.e., differential operators of higher order. On the irreducible invariant subspaces of L2 (M), each Cα acts
as a multiple of the identity, so if f (r) belongs to such an invariant subspace,
we have
Cα f(r) = λ(α)f(r), α = 1, 2, . . . , r. (30.14)
This is a set of differential equations that are invariant under the symmetry
of the physical system, i.e., its solutions transform among themselves under
the action of the group of symmetries.
It is a stunning reality and a fact of profound significance that many of
the differential equations of mathematical physics are, as in Eq. (30.14), ex-
pressions of the invariance of the Casimir operators of some Lie algebra in a
particular representation. Moreover, all the standard functions of mathemat-
ical physics, such as Bessel, hypergeometric, and confluent hypergeomet-
ric functions, are related to matrix elements in the representations of a few
of the simplest Lie groups
Since we are dealing with a single particle, the total angular momentum can
only be spin. Therefore, we have the following theorem.
Theorem 30.3.11 In the absence of any interactions, a massive rel-
ativistic particle is specified by its mass m and its spin s, the former
being any positive number, the latter taking on integer or half-odd-
integer values.
H 0 is recognized as an angular momentum operator whose
eigenvalues are integer (for bosons) and half-odd integer (for fermions).
A natural axis for the projection of spin is the direction of motion of the （massless ） particle. Then the projection of spin is called 【 helicity.】
Theorem 30.3.12 In the absence of any interactions, a massless rel-
ativistic particle is specified by its spin and its helicity. The former
taking on integer or half-odd-integer values s, the latter having val-
ues +s and −s.
Theorems 30.3.11 and 30.3.12 are beautiful examples of the fruitfulness
of the interplay between mathematics and physics. Physics has provided
mathematics with a group, the Poincaré group, and mathematics, through
its theory of group representation, has provided physics with the deep result
that all particles must have a spin that takes on a specific value, and none
other; that massive particles are allowed to have 2s + 1 different values for
the projection of their spin; and that massless particles are allowed to have
only two values for their spin projection. Such far-reaching results that are
both universal and specific makes physics unique among all other sciences.
It also provides impetus for the development of mathematics as the only di-
alect through which nature seems to communicate to us her deepest secrets.
If α > 0, then the resulting representations will have continuous spin
variables. Such representations do not correspond to particles found in na-
ture; therefore, we shall not pursue them any further.
----------------------------------------------------------
CH 31 Representation of Clifford Algebras
This significance is doubled in the case of the Clifford algebras because of their relation with the Dirac equation, which describes a relativistic spin-1/2 fundamental particle such as a lepton or a quark.
C$^×_v$: the set of invertible elements of Cv
Clifford group V of V: Γv ={a | a∈C$^×_v$ and ad(a)v∈V for all v∈V }
ad(a)x = ωv(a) ∨ x ∨ a^(−1)： twisted adjoint representation
use e12 for e1 ∨ e2
Proposition 31.1.9 The map λv : Γv → C$^×$ given by λv(a) = λ$_a$ is a homomorphism from the Clifford group to C$^×$ (the multiplicative group of complex numbers).
C(μ, ν): Clifford algebra Cνμ(R):
Γ(μ, ν): Clifford group of Rnν
Definition 31.2.1 The【group Pin(μ,ν)】is the subgroup of Γ(μ,ν) consisting of elements a satisfying λv(a)=±1.
Proposition 31.2.2 The group Pin(μ,ν) is generated by x∈V for which g(x, x)=±1.
Theorem 31.2.3 The map Φ : Pin(μ, ν) → O(μ, ν) defined by Φ(a)=τ$_a$ is a surjective group homomorphism with ker Φ = {1, −1}.
Spin(μ, ν) ≡ Pin(μ, ν) ∩ C0 (μ, ν). (even elements of Pin(μ, ν))
Theorem 31.2.6 The map Ψ : Spin(μ, ν) → SO(μ, ν) defined by Ψ(a)=τ$_a$ is a surjective group homomorphism. Furthermore, ker Ψ = {1, −1}.
Those quaternions that are a linear combination of 1 and only one of the other three basis elements can be identified with C. We therefore conclude that Spin(0, 2) is the set of complex numbers of unit length, i.e., Spin(0, 2) = U(1) = {e$^{iφ}$ | φ ∈ R}
relation between the Clifford algebra C 03 (R) and a Lie algebra:
Spin(3, 0) ∼= SO(3) ∼= SU(2).
---------------------------------------------------
Ch 32 Lie Groups and Differential Equations
The crucial property of Lie group theory is that locally the group and its algebra “look alike”. This allows the complicated nonlinear conditions of invariance of subsets and functions to be replaced by the simpler linear conditions of invariance under infinitesimal actions.
The importance of knowing the symmetry group of a system of DEs lies
in the property that from one solution we may be able to obtain a family
of other solutions by applying the group elements to the given solution. To
find such symmetry groups, we have to be able to “prolong” the action of
a group to derivatives of the dependent variables as well. This is obvious
because to test a symmetry, we have to substitute not only the transformed
function u ̃=f ̃(x), but also its derivatives in the DE to verify that it satisfies
the DE.
∂$_{j^(k)}$ f(x) ≡ ∂$^k$ / ∂x$^{j1}$ ∂x$^{j2}$ · · · ∂x$^{jk}$ f(x)
j$^n_a$ f: all functions n-equivalent to f at point a by
j$^n$ (X×U) : n-th 【prolongation】 (or 【jet space】) of U. collect all j$^n_a$ f for all a and f.
pr$^{(n)}$ f : Ω→U(n): n-th prolongation of f
Prolongation allows us to turn a system of DEs into a system of algebraic equations
Just as we identified a function with its graph, we can identify the solution
of a system of DEs with the graph of its prolongation pr$^{(n)}$f
Γ$^{(n)}_f$≡{ (x, pr$^{(n)}$ f(x) ) }: the graph of its prolongation pr$^{(n)}$f
Example 32.2.10 对这些概念解释的很好
Ψ$^i_{,jk}$ ≡ ∂$^2$Ψ$^i$/∂xj∂xk (Ψ的第i分量对xj,xk求二阶导)
Ψ$^i_{x0,jk}$ ≡ ∂$^2$Ψ$^i$/∂xj∂xk |x=x0 (Ψ的第i分量对xj,xk求二阶导,并求x=x0时的导数)
DiS: The total derivative of S with respect to total derivative xi
I = (i1, i2, . . . , ik ), DI S = Di1 Di2 · · · Dik S
The symmetry groups G1 and G2 reflect the invariance of the heat equation under space and time translations. G3 and Gβ demonstrate the linearity of the heat equation: We can multiply solutions by constants and add solutions to get new solutions.
Lie group对于DEs的意义：
Since each group Gi is a one-parameter group of symmetries, if f is a solution of
the heat equation, so are the functions fi ≡ Gi · f for all i。
the fundamental solution of the heat equation
The theory of Lie groups finds one of its most rewarding applications in the
integration of ODEs. Lie’s fundamental observation was that if one could
come up with a sufficiently large group of symmetries of a system of ODEs,
then one could integrate the system.
--------------------------------------------------
CH 33 Calculus of Variations, Symmetries, and Conservation Laws
The requirement of linearity is due to our
desire for generalization of differentiation to Hilbert spaces, on which linear
maps are the most natural objects.
Box 33.1.3 For f : R n ⊃ Ω → R m , the matrix of Df (x) in the standard basis of R n and R m is the Jacobian matrix of f
Ey,i : L2(Ω)→R, Ey,i(f) ≡ ∂i f(y) (the evaluation of the derivative of functions with respect to the i-th coordinate)
The fundamental theme of the calculus of variations is to find functions
that extremize an integral and are fixed on the boundary of the integration
region.
Theorem 33.1.12 If u is an extremal of the variational problem (33.12),then it must be a solution of the Euler-Lagrange equations.
Three greatest mathematicians of modern times :Gauss , Euler and Riemann
Euler was the Shakespeare of mathematics
functional ~function: u ~ x, f ~ x0
Definition 33.3.1 A 【conservation law】 for a system of differential equations Δ(x, u (n) ) = 0 is a divergence expression D · J = 0 valid for all solutions u = f (x) of the system. Here,
J ≡ (J1(x, u^(n)) , J 2( x, u (n)) , . . . , J p (x, u (n)) )
is called 【current density】.
φ$^j_ν$ ≡ ∂φ$^j$ /∂x$_ν$
Lorentz metric η$^{ ij}$
T $^{μν}$ is called the energy momentum current density.
-----------------------------------------------
CH 34 Fiber Bundles and Connections
P (M, G, π), or P (M, G) or P: principal fiber bundle
Remark 34.1.1 Just as a fiber sprouts from a single point of the earth
(a spherical 2-manifold), so does a fiber π^ −1 (x) sprout out of a single
point x of the manifold M. And just as you can collect a bunch of fibers
and make a bundle out of them, so can you collect a bunch of π^ −1 (x)’s
and make P = ∪x π ^−1 (x). Furthermore, fibers sprout vertically from the
ground. Similarly, in a sense to be elaborated in our discussion of connec-
tions, π^ −1 (x) are “vertical” manifolds, while M is “horizontal.”
A 【linear frame p】 at x∈M is an ordered basis (X1 , X2 , . . . , Xn ) of the tangent space Tx(M).
Lx(M) : the set of all linear frames at x，
L(M): the set of all Lx(M) for all x∈M，
L(M)(M, GL(n, R)), or simply L(M)： bundle of linear frames (one type of PFB)
L(M) consists of fibers which include all ordered bases of Tx(M) and a right action by GL(n, R), which is the group of invertible linear transformation of Rn.
E(M, F, G, P ): associated bundle (base, standard fiber, structure group, principal fiber bundle)
Fx: fiber over x in E, π $^{-1}_E$(x)
Tx(M): the fiber at x of the tangent bundle T(M)
tangent bundle T(M): the set of all Tx(M) for all x∈M
p: F→Fx, (pg)ξ = p(gξ ) for p∈P , g∈G, ξ∈F. Then p is diffeomorphic.
tensor bundle T$^r_s$(M) of type (r, s)
tensor field of type (r, s) T$^r_s$: M → T$^r_s$(M)
34.2 Connections in a PFB
A*: fundamental vector field
The points of each
fiber are naturally connected through the action of G. In fact, given a point of
the fiber, we can construct the entire fiber by applying all g ∈ G to that point.
This is by construction and the fact that G acts freely on each fiber. Because
each fiber is an orbit of G, and because G acts freely on the fiber, each
fiber is diffeomorphic to G. However, there is no natural diffeomorphism
connecting one fiber to its neighbor. Such a connection requires an extra
structure on the principal fiber bundle,
ω u ≡ σ u ∗ ω
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CH 35 Gauge Theories
Λ ̄ $^k$ (P, V) : the set of tensorial forms of degree k of type (ρ, V)
local section σu: choice of gauge
ωu ≡ σ*u ω : gauge potential. (ω is 1-form)
derivative of the gauge potential : gauge field (curvature form)
Box 35.1.2 Electromagnetic interaction is a principal fiber bundle
P (M, G) with M a Minkowski space and G = U (1).
Gau(P): The set of all gauge transformations of P(M, G)
S(E, M, F): the set of all sections(associated bundle E, base manifold M, standard fiber F)
S(E, M, G) ∼= Π 12 (P, G) ∼= Gau(P)
π$_{12}$: P → F with the property π$_{12}$(pg) = g^−1 · π$_{12}$(p)
a Lagrangian L is 【G-invariant】 if L(p, g · v, g · θp) = L(p, v, θp)
Theorem 35.2.3 If L in Eq. (35.5) is G-invariant, then L is welldefined and L(f* ψ , f* ω) = L(ψ ,ω), i.e., L is gauge-invariant.
∗ ̄ ≡ π*(∗)
--------------------------------------------
CH36 Differential Geometry
A natural directional derivative of the section would be to move
along x t and see how φ(x t ) changes. When t changes to t + h, the sec-
tion changes from φ(x t ) to φ(x t+h ). But we cannot compute the difference
φ(x t+h ) − φ(x t ), because φ(x t+h ) ∈ π E −1 (x t+h ) while φ(x t ) ∈ π E −1 (x t ), and
we don’t know how to subtract two vectors from two different vector spaces. That is why we need to transfer φ(x t+h ) to π E −1 (x t ) via the parallelism γ t t+h
X*∈TpP be the horizontal lift of X∈TxM.
The uniqueness of the geodesics establishes a bijection (in fact, a diffeo-
morphism) between a neighborhood of the origin of T P (M) and a neighbor-
hood of P in M.
Riemann normal coordinates are very useful in establishing tensor equa-
tions. This is because two tensors are identical if and only if their com-
ponents are the same in any coordinate frame. Therefore, to show that two
tensors fields are equal, we pick an arbitrary point in M, erect a set of RNCs,
and show that the components of the tensors are equal.
Definition 36.2.14 Given a tensor field T of type (r, s) the covariant
differential ∇T of T is a tensor of type (r, s + 1) given by
(∇T)(X1 , . . . , Xs ; X) = (∇$_X$ T)(X1 , . . . , Xs ), Xi , X ∈ T (M)
Definition 36.2.7 The 【torsion form】 of a linear connection ω is defined by Θ = D$^ω$ θ
torsion form is a tensorial 2-form.
-----------------------------------------
CH 37 Riemannian Geometry
fiber metric g: M→T$ ^0_2$(E) such that g$_x$ ≡ g(x) is an inner product in the fiber π$_E ^{−1}$ (x) which is differentiable in x.
∇g: covariant derivative of g, ∇$_u$g(v, w) ≡ ∇$_u$ <g, v⊗w> = g(∇$_u$v, w) + g(v, ∇$_u$w)
Definition 37.1.3 A 【Riemannian manifold】 is a differentiable manifold M
with a metric g∈T$_2^0$(M), such that at each point x∈M, g|x is a positive
definite inner product. A manifold with an indefinite inner product at each
point is called a 【pseudo/semi-Riemannian manifold】。
Theorem 37.1.4 Every (semi-)Riemannian manifold admits a unique
metric connection, called Levi-Civita connection, whose torsion is
zero
Definition 37.1.5 A (semi-)Riemannian manifold is called flat if its Levi-Civita connection is flat.
Proposition 37.1.6 A (semi-)Riemannian manifold is flat iff its curvature R
vanishes identically.
Equation (37.12) gives us a recipe for finding the curvature from the arc
length. Given the arc length, construct the orthogonal 1-forms as in Exam-
ples 37.1.8 and 37.1.9. Then take the exterior derivative of a typical one and
read off ω k i from the right-hand side of the equation.
The conformal group of R2 has important applications in string theory and statistical
mechanics
differentiation of the geodesic equation with respect to s will give zero.
covariant differentiation of the geodesic equation will yield zero
∇u ∇u n + R(n, u)u = 0. (37.28)
The first term can be interpreted as the relative acceleration of two geodesic
curves (or free particles), because ∇u is the generalization of the derivative
with respect to t, and (∇u n) is interpreted as relative velocity.
Gravity becomes part of the fabric of space-time
If the gravitational field were homogeneous, one could eliminate it—and the matter that produces it as well, but no such gravitational field exists.
The inhomogeneity of gravitational fields has indeed an observable effect.
Box 37.3.1 Gravity manifests itself by giving space-time a curvature
Newtonian gravity can be translated into the language
of differential geometry by identifying the gravitational tidal effects with
the curvature of space-time (p1163). This straightforward interpretation of New-
tonian gravity, in particular the retention of the Euclidean metric and the
universality of time, leads to no new physical effect.
∇ · G = 0, where Gij ≡ Rij − gij R
The automatic vanishing of the divergence of the symmetric Einstein ten-
sor has an important consequence in the field equation of GTR. It is remi-
niscent of a similar situation in electromagnetism, in which the vanishing of
the divergence of the fields leads to the conservation of the electric charge,
the source of electromagnetic fields.
Einstein’s GTR is the generalization
of Newtonian static gravity to a dynamical theory. As this generalization
ought to agree with the successes of the Newtonian gravity, Eq. (37.37) must
agree with (37.33). The bold step taken by Einstein was to generalize this
relation involving only a single component of the Ricci tensor to a full tensor
equation. The natural tensor to be used as the source of gravitation is the
stress energy tensor
T = (ρ + p)u ⊗ u + pg,
where the source is treated as a fluid with density ρ, four-velocity u, and
pressure p. So, Einstein suggested the equation G = κT as the generaliza-
tion of Newton’s universal law of gravitation.
Einstein’s equation of the general theory of relativity：
G = 8πT, or R − Rg/2 = 8π (ρ + p)u ⊗ u + pg .
Λ：cosmological constant
G + Λg = 8πT
Definition 37.4.1 A spacetime is 【stationary】 if there exists a one-parameter
group of isometries F t , called【 time translation isometries】, whose Killing
vector fields ξ are timelike for all t: g(ξξ ,ξξ ) > 0. If in addition, there ex-
ists a spacelike hypersurface + that is orthogonal to orbits (curves) of the
isometries, we say that the spacetime is 【static】
如何验证：水星进动，太阳光线弯曲，引力红移的数据
REF
Benn, I.M., Tucker, R.W.: An Introduction to Spinors and Geometry with
Applications in Physics. Adam Hilger, Bristol (1987)
Bleecker, D.: Gauge Theory and Variational Principles. Addison-Wesley,
Reading (1981)
Richtmyer, R.: Principles of Advanced Mathematical Physics. Springer,
Berlin (1978)
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