### 物理代写|量子场论代写Quantum field theory代考|PHYSICS 3544

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子场论代写Quantum field theory代考|Dirac’s Formalism

It is regrettable that neither the position nor the momentum operator has a basis of eigenvectors, for this would indeed be very convenient. Paul Dirac invented a remarkable formalism to deal with this problem. It is used in almost every physics textbook, where it is typically considered as self-evident. We try to explain some of the basic features and meaning of this formalism here, striving as usual to explain what this means, but with no serious attempt to make matters rigorous.

Dirac’s formalism works beautifully. It allows a great economy of thought and notation. It is however unfriendly to mathematicians, and the mathematically inclined reader must brace for a kind of cold shower, as things are likely to look horrendously confusing at first. Let us stress that it is not necessary to master this formalism to follow most of the rest of this book. Still, we will use it at times, if only to formulate results in the same language as they are found in physics textbooks. The reader who finds the present section overwhelming is encouraged to move on and to come back when the need arises.

As a consequence of the use of Dirac’s formalism, if one looks at a physics textbook discussing (say) particles on the real line, one may find it difficult to recognize any of the previous material. First, one is likely to find very early the sentence “let $|x\rangle$ denote the state of a particle located at $x . . . “$. An element of the position state space $\mathcal{H}=L^{2}(\mathbb{R}$, $\mathrm{d} x)$ that could be said to be located at $x$ would have to be an eigenvector of the position operator $X$ with eigenvalue $x$ and these do not exist. The expression $|x\rangle$ can however be given a meaning in a kind of “distributional sense”. It makes sense only when integrated against a Schwartz function $f$, that is for such a function the integral $\int \mathrm{d} x|x\rangle f(x)$ makes sense as an element of $\mathcal{H}$. The value of this integral is simply the function $f$ seen as an element of $\mathcal{H}$. Quite naturally, we denote by
$|f\rangle$ the function $f \in \mathcal{S}$ seen as an element of $\mathcal{H}$,
so that
$$|f\rangle=\int \mathrm{d} x|x\rangle f(x) .$$
This should certainly remind us of a basis expansion $f=\sum_{i}\left(f, e_{i}\right) e_{i}=\sum_{i} e_{i}\left(f, e_{i}\right)$, and physicists think of $|x\rangle$ as a “continuous basis”. In this manner we have given a meaning to the quantity $|x\rangle$ as “an element of $\mathcal{H}$ in the distributional sense”. The principle at work here is important enough to be stated clearly: ${ }^{41}$

## 物理代写|量子场论代写Quantum field theory代考|Why Are Unitary Transformations Ubiquitous

Let us go back to the setting of a general Hilbert space, whose elements are denoted $x, y, z, \ldots$. To each observable is associated a self-adjoint operator. Conversely, to each self-adjoint operator is associated an observable (although it is another matter in concrete situations to design an experiment that actually measures it). We now describe a fundamental class of such observables. Given $x \in \mathcal{H}$ of norm $1,(x, x)=1$, the projector $P_{x}(y):=(x, y) x$ is Hermitian since
$$\left(z, P_{x}(y)\right)=(z, x)(x, y)=(x, z)^{*}(x, y)=((x, z) x, y)=\left(P_{x}(z), y\right)$$
Exercise 2.9.1 In Dirac’s formalism for a norm-1 vector $|\alpha\rangle$ one writes the projector $P_{\alpha}=|\alpha\rangle\langle\alpha|$. Why does it seem to require no proof that $P_{\alpha}$ is Hermitian?

Thus the operator $P_{x}$ corresponds to an observable $\mathcal{O}$. The possible values of $\mathcal{O}$ are the eigenvalues of $P$, namely 0 and 1 . We may describe $\mathcal{O}$ as asking the question: Is the state of the system equal to $x ?^{48}$ The average value of this operator in state $y$ is given by $\left(y, P_{x}(y)\right)=(y, x)(x, y)=|(x, y)|^{2}$ and is the probability to obtain the answer “yes” to your question. It is called the transition probability between $x$ and $y^{49}$ In physics, the inner product $(x, y)$ is often called an amplitude, so that the transition probability is the square of the modulus of the amplitude. ${ }^{50}$ The transition probability does not change if one multiplies $x$ and $y$ by complex numbers of modulus 1 , as expected from the fact that this multiplication does not change the state represented by either $x$ or $y$.

We should expect that any transformation ${ }^{51}$ which preserves the physical properties of a system preserves the transition probabilities. Transition probabilities are preserved by unitary transformations, since for such a transformation $|(U x, U y)|^{2}=|(x, y)|^{2}$. They are also preserved under anti-unitary transformations, i.e. anti-linear operators which preserve the inner product. Conversely, which are the transformations of $\mathcal{H}$ which preserve transition probabilities? A deep theorem of Eugene Wigner [92, appendix to Chapter 20] shows that this is the case only for unitary and anti-unitary transformations. ${ }^{52}$ A fundamental consequence is that any transformation which preserves the physics of the system corresponds to a unitary or anti-unitary transformation. As we explain in the next section, there are many of these, corresponding to the symmetries of Nature. The symmetries of Nature of a certain type naturally form a group, bringing us to group theory. Furthermore time-evolution will also be represented by a unitary transformation.

## 物理代写|量子场论代写Quantum field theory代考|Unitary Representations of Groups

Certain types of invariance in Nature are among the most important guiding principles in developing physical theories about the real world. This will be a recurring theme in this book. It forces us to choose models which satisfy certain symmetries and this implies extremely strict restrictions on the possible forms of physical theories.

In this section we start to use this principle in the simplest case, translation invariance. In physics each observer uses a reference frame to describe the positions of points in space (or in space-time). These reference frames need not have the same origin, may use different privileged directions and may even move with respect to each other. ${ }^{53}$ Here we just consider the situation of different origins. If you study the motion of an object using a different origin for your reference frame than mine, we may disagree on the coordinates of the object, but we should agree that it follows the same laws of physics. Mathematically, the space $\mathbb{R}^{3}$ acts on itself by translations, and we examine first the effect of these translations at a purely classical level. Suppose that the system you are studying is translated by a vector $a^{54}$ The object at position $\boldsymbol{x}$ that you were studying has now been moved to position $\boldsymbol{x}+\boldsymbol{a}$. Say that you use a function $f$ on $\mathbb{R}^{3}$ to measure e.g. the electrical potential at a point of space. Before you translate the system, the value $f(\boldsymbol{x})$ measures the potential at the point $\boldsymbol{x}$. After the translation this value of the potential occurs at the point $\boldsymbol{x}+\boldsymbol{a}$. Thus the value of the new function $U(a)(f)$ you use to measure the potential at this point $x+a$ equals the value of the old function $f$ at the point $x$ :
$$U(a)(f)(x+a)=f(x)$$
i.e.
$$U(a)(f)(x)=f(x-a) .$$
Observe the all-important minus sign and note the fundamental property $U(\boldsymbol{a}+\boldsymbol{b})=$ $U(\boldsymbol{a}) U(\boldsymbol{b})$

Suppose now that more generally we study a system whose state is described by a vector $x$ in a Hilbert space $\mathcal{H}$. If the system is translated by a vector $a$ we expect that the system will be described by a new state $U(a)(x)$. This new description should not change the physics. The transition probability between $x$ and $y$ should be the same as the transition probability between $U(a)(x)$ and $U(a)(y)$, i.e. $|(x, y)|^{2}=|(U(a)(x), U(a)(y))|^{2}$. According to the discussion of the previous section, $U(a)$ is either unitary or anti-unitary. Moreover, it is obvious that $U(0)$ should be the identity. What becomes very interesting is when we perform two such translations in succession, first by a vector $a$ and then by a vector $b$. The state $x$ is transformed first in $U(a)(x)$ and then in $U(b)(U(a)(x))$. This also amounts to perform the translation by the vector $\boldsymbol{a}+\boldsymbol{b}$, and this transforms the state $x$ into the state $U(\boldsymbol{a}+\boldsymbol{b})(x)$. Therefore $U(b) U(\boldsymbol{a})$ and $U(\boldsymbol{a}+\boldsymbol{b})$ should represent the same transformation of the system.

## 物理代写|量子场论代写Quantum field theory代考|Dirac’s Formalism

|F⟩功能F∈小号被视为一个元素H,

|F⟩=∫dX|X⟩F(X).

## 物理代写|量子场论代写Quantum field theory代考|Why Are Unitary Transformations Ubiquitous

(和,磷X(是))=(和,X)(X,是)=(X,和)∗(X,是)=((X,和)X,是)=(磷X(和),是)

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