The entire quantum enterprise rests on the quantum superposition principle (QSP). The principle posits that states of a quantum system are elements of a certain set and that the sum of any pair of elements of the set is again an element of the set. The superposition principle endows the set the mathematical structure of being a vector space, the elements of which, referred to as vectors, constitute the state space of a quantum system. In simpler terms, the QSP states that if two vectors are legitimate states of a quantum system, then their linear sum is again a legitimate state of the same system. I will describe later in detail how the vector space arises naturally in the description of quantum systems. Here I give the precise mathematical definition of vector space which is the foundation of the mathematical structure of quantum mechanics.
Field
A field is a set \(F\) together with the operation of addition , denoted by \(+\), and the operation of multiplication, denoted by \(\cdot\), of every pair of elements of \(F\). The operations satisfy the following axioms:
Example: The set of real numbers, denoted by \(\mathbb{R}\), endowed with the usual addition and multiplication of real numbers is a field, called the real field. The set of complex numbers, denoted by \(\mathbb{C}\), endowed with the usual addition and multiplication of complex numbers is a field, called the complex field. When we refer to \(\mathbb{R}\) and \(\mathbb{C}\) as real and complex fields, respectively, we will mean the sets themselves together with their respective operations of addition and multiplication. \(\square\)
- For all \(a, b\) in \(F\), the sum \(a+b\) and the product \(a\cdot b\) belong to \(F\) (closure property).
- For all \(a, b, c\) in \(F\), the equalities \((a+b)+c=a+(b+c)\) and \((a\cdot b)\cdot c=a\cdot (b\cdot c)\) hold (associativity property).
- For all \(a, b\) in \(F\), it holds that \(a+b=b+a\) and \(a\cdot b=b\cdot a\) (commutativity property).
- For every \(a\) in \(F\) there exists an element of \(F\), denoted by \(0\), such that the equality \(a+0=a\) holds; the element \(0\) is called the additive identity of \(F\) or simply the zero element of \(F\).
- For every \(a\) in \(F\) there exists an element of \(F\), denoted by \(1\), such that the equality \(a\cdot 1=a\) holds; the element \(1\) is called the multiplicative identity of \(F\) or simply the identity element of \(F\).
- For every \(a\) in \(F\) there exists an element of \(F\), denoted by \((-a)\), such that \(a+(-a)=0\); the element \((-a)\) is called the additive inverse of \(a\). The existence of additive inverse for all elements of the set defines the operation of subtraction among elements of the set.
- For every \(a\) in \(F\) other than the zero element \(0\), there exists an element of \(F\), denoted by \(a^{-1}\) such that \(a\cdot a^{-1}=1\); the element \(a^{-1}\) is called the multiplicative inverse of \(a\). The existence of multiplicative inverse for every non-zero element of the set defines the operation of division among elements of the set.
- For all \(a, b, c\) in \(F\), the equality \(a\cdot (b+c)=a\cdot b+ a\cdot c\) holds (distributive property).
A field is then an ordered triplet \((F,+,\cdot)\), and elements of the set \(F\) are called scalars. When the two operations are clear from the outset, we can refer to the set \(F\) as a field itself or a scalar field.
Vector Space
A vector space has four components: (a) a scalar field \(F\), (b) a set \(V\), (c) an operation that defines the sum (denoted by \(+\)) of every pair of elements of the set \(V\) and (d)
an operation that defines the product (denoted by \(\cdot\)) of every scalar in the scalar
field \(F\) and every element of the set \(V\). The sum and the product satisfy the following axioms:
- For all \(\psi, \varphi\) in \(V\), the sum \(\psi+\varphi\) belongs to \(V\).
- For every \(\alpha\) in \(F\) and \(\psi\) in \(V\), the product \(\alpha\cdot\psi\) belongs to \(V\).
- For all \(\psi, \varphi\) in \(V\), the equality \(\psi+\varphi=\varphi+\psi\) holds (commutative property).
- For all \(\psi, \varphi, \phi\) in \(V\), the equality \((\psi+\varphi)+\phi=\psi+(\varphi+\phi)\) holds (associative property).
- For every \(\alpha\) in \(F\) and all \(\psi,\varphi\) in \(V\), the equality \(\alpha\cdot(\psi+\varphi)=\alpha\cdot\psi+\alpha\cdot\varphi\) holds.
- For all \(\alpha,\beta\) in \(F\) and every \(\psi\) in \(V\), the equality \((\alpha+\beta)\cdot\psi=\alpha\cdot\psi+\beta\cdot\psi\) holds.
- For every \(\psi\) in \(V\), the equality \(1\cdot\psi=\psi\) holds, where \(1\) is the identity of \(F\).
- For every \(\psi\) in \(V\) there exists a unique element \(\theta\) of \(V\) such that, the equality \(\psi+\theta=\psi\) holds; the element \(\theta\) is called the zero element of \(V\).
- For every \(\psi\) in \(V\), there exists an element \(\psi'\) of \(V\) such that \(\psi+\psi'=\theta\); the element \(\psi'\) is denoted by \(\psi'=-\psi\).
Then a vector space is the ordered quadruple \((F,V,+,\cdot)\). If the operations \(+\) and \(\cdot\) are clear from the outset, we can simply refer to the vector space as the vector space \(V\) over the scalar field \(F\) or the set \(V\) as the vector space itself. The elements of \(V\) are referred to as vectors. If the scalar field is the real number field, the vector space is called a real vector space; on the other hand, if the scalar field is the complex number field, the vector space is called a complex vector space.
Example: The quadruple \((\mathbb{R},\mathbb{R},+,\cdot)\), where \(+\) is the usual addition of real numbers and \(\cdot\) the usual multiplication of real numbers, is clearly a real vector space. Likewise the quadruple \((\mathbb{C},\mathbb{C},+,\cdot)\), where \(+\) is the usual addition of complex numbers and \(\cdot\) the usual multiplication of complex numbers, is a complex vector space.
Define the quadruple \((\mathbb{R},\mathbb{C},+,\cdot)\). The operation of addition \(+\) is the usual addition of complex numbers. The operation of multiplication \(\cdot\) is defined as follows: for every real \(\alpha\) and complex number \(z=\mbox{Re}\, z + i \mbox{Im}\, z\), we have \(\alpha\cdot z=\alpha \mbox{Re}\, z + i \alpha \mbox{Im}\, z\), where the multiplication in the right hand side is the usual multiplication of real numbers. This quadruple is clearly a vector space; and, by definition, it is a real vector space (the real field being the scalar field) even though its vectors are complex numbers.
Now consider the quadruple \((\mathbb{C},\mathbb{R},+,\cdot)\). The operation \(+\) is the usual addition of real numbers. The operation \(\cdot\) is the usual multiplication of reals and complex numbers, as defined in the preceding paragraph. This quadruple is not a vector space, because for every complex \(z\) in the scalar field and every real \(\alpha\) in the set \(\mathbb{R}\) the number \(z\cdot\alpha\) is not in general real, i.e. it does not belong to the set \(\mathbb{R}\). \(\square\)
Example: Let \(\mathbb{R}^n\) denote the set of ordered \(n\)-tuple of real numbers. Elements of \(\mathbb{R}^n\) are denoted by \(x=\{x_1,x_2,\dots,x_n\}\), where all the \(x_k\)'s are real. We turn this set into a vector space by first defining the operation of addition among pairs of its elements, identifying an appropriate scalar field over which it is defined, and defining the operation of multiplication over every scalar of the field and every element of \(\mathbb{R}^n\), with the two operations consistent with the above enumerated properties.
For all pairs of elements \(x=\{x_1,x_2,\dots,x_n\}\) and \(y=\{y_1,y_2,\dots,y_n\}\) of \(\mathbb{R}^n\), we define their sum to be
\begin{eqnarray}
x+y&=&\{x_1,x_2,\dots,x_n\}+\{y_1,y_2,\dots,y_n\}\nonumber\\
&=&\{x_1+y_1,x_2+y_2,\dots,x_n+y_n\}, \label{sum1}
\end{eqnarray}where the addition in \((x_k+y_k)\) is the usual addition of real numbers. Since the reals form a field under their usual addition, the definition \ref{sum1} satisfies all the requirements for an operation of addition of pairs of elements of \(\mathbb{R}^n\), in particular, the sum in \ref{sum1} belongs to \(\mathbb{R}^n\).
Suppose we wish to define a vector space out of \(\mathbb{R}^n\) over the real field \(\mathbb{R}\). Then we chose our scalar field to be \(\mathbb{R}\). For every real \(r\) in \(\mathbb{R}\) and element \(x=\{x_1,x_2,\dots,x_n\}\) in \(\mathbb{R}\), we define
\begin{equation}\label{prod1}
r\cdot x=r\cdot \{x_1,x_2,\dots,x_n\} = \{r x_1,r x_2,\dots,r x_n\},
\end{equation} where the product \(r x_k\) is the usual product of real numbers. Again from the field properties of the reals under their usual multiplication, the product \ref{prod1} satisfies all the required properties. If we wanted to define a complex vector space out of \(\mathbb{R}^2\) with the \(\cdot\) operation given by \ref{prod1} and \(r\) replaced by a complex number, it would not be possible because the right hand side of \ref{prod1} would have been an \(n\)-tuple of complex numbers, which was not an element of \(\mathbb{R}^n\).
Now that we have defined the operations \(+\) and \(\cdot\), it remains to show that the last two properties are satisfied. The zero vector \(\theta\) is not an element of the given set that is usually defined from the start. It is identified and shown to exist in the set only after the operation of addition has been defined. That is the zero vector depends on the \(+\) operation. It is important to recognize that different \(+\) operations can be assigned to one and the same set; and these different operations lead to different zero elements of the set. Since \(\theta\) depends on the \(+\) operation, the solution to \(\psi+\psi'=\theta\) likewise depends on the given \(+\) operation.
For the \(+\) operation defined in \ref{sum1}, the solution to \(x+\theta=x\) for every \(x\) in \(\mathbb{R}^n\) is the element \(\theta=\{0,0,\dots,0\}\), the \(n\)-tuple of zeros. Since the \(n\)-tuple of zeros belongs to \(\mathbb{R}^n\), the zero vector exists for the given \(+\) operation. We refer to \(\theta\) as the zero vector not because it is an \(n\)-tuple of zeros; it is the zero vector because it is the solution to \(x+\theta=x\) for the given \(+\) operation. Also for every \(x=\{x_1,x_2,\dots,x_n\}\) in \(\mathbb{R}^n\), we have the unique solution \(x'=\{-x_1,-x_2,\dots,-x_n\}\) to \(x+x'=\theta\). We have thus established that all the properties of a vector space are satisfied.
Then the quadruple \((\mathbb{R},\mathbb{R}^n,+,\cdot)\), with \(+\) and \(\cdot\) defined by \ref{sum1} and \ref{prod1} respectively, is a real vector space. From now on whenever we refer to \(\mathbb{R}^n\) as a vector space we will always mean this quadruple. \(\square\)
Example: Let \(\mathbb{C}^n\) denote the ordered \(n\)-tuple of complex numbers. Its elements are denoted by \(z=\{z_1,z_2,\dots,z_n\}\). Following the steps leading to the construction of the real vector space \(\mathbb{R}^n\), it can be shown that the quadruple \((\mathbb{C},\mathbb{C}^n,+,\cdot)\), with the operations\(+\) and \(\cdot\) defined respectively by
\begin{eqnarray}
z+w&=&\{z_1,z_2,\dots,z_n\}+\{w_1,w_2,\dots,w_n\}\nonumber\\
&=&\{z_1+w_1,z_2+w_2,\dots,z_n+w_n\}
\end{eqnarray}\begin{equation}
c\cdot z=c\cdot \{z_1,z_2,\dots,z_n\}=\{c z_1,c z_2,\dots, c z_n\}
\end{equation}for all \(z,w\) in \(\mathbb{C}^n\) and every complex \(c\), is a complex vector space. From now on whenever we refer to \(\mathbb{C}^n\) as a vector space we will mean the complex vector space \((\mathbb{C},\mathbb{C}^n,+,\cdot)\). \(\square\)
Example: Let \(C(\mathbb{R})\) denote the set of continuous, complex valued functions in the real line \(\mathbb{R}\). The function \(\zeta(x)=0\) for all \(x\in\mathbb{R}\) belongs to the set \(C(\mathbb{R})\). The quadruple \((\mathbb{R},C(\mathbb{R}),+,\cdot)\), with the \(+\) and \(\cdot\) operations defined by the pointwise addition and multiplication of functions respectively, is a real vector space with \(\zeta(x)\) as the zero vector. On the other hand, the quadruple \((\mathbb{C},C(\mathbb{R}),+,\cdot)\), with the \(+\) and \(\cdot\) operations defined by the pointwise addition and multiplication of functions respectively, is a complex vector space with \(\zeta(x)\) as the zero vector. \(\square\)
Example: Let \(\mathbb{R}^n\) denote the set of ordered \(n\)-tuple of real numbers. Elements of \(\mathbb{R}^n\) are denoted by \(x=\{x_1,x_2,\dots,x_n\}\), where all the \(x_k\)'s are real. We turn this set into a vector space by first defining the operation of addition among pairs of its elements, identifying an appropriate scalar field over which it is defined, and defining the operation of multiplication over every scalar of the field and every element of \(\mathbb{R}^n\), with the two operations consistent with the above enumerated properties.
For all pairs of elements \(x=\{x_1,x_2,\dots,x_n\}\) and \(y=\{y_1,y_2,\dots,y_n\}\) of \(\mathbb{R}^n\), we define their sum to be
\begin{eqnarray}
x+y&=&\{x_1,x_2,\dots,x_n\}+\{y_1,y_2,\dots,y_n\}\nonumber\\
&=&\{x_1+y_1,x_2+y_2,\dots,x_n+y_n\}, \label{sum1}
\end{eqnarray}where the addition in \((x_k+y_k)\) is the usual addition of real numbers. Since the reals form a field under their usual addition, the definition \ref{sum1} satisfies all the requirements for an operation of addition of pairs of elements of \(\mathbb{R}^n\), in particular, the sum in \ref{sum1} belongs to \(\mathbb{R}^n\).
Suppose we wish to define a vector space out of \(\mathbb{R}^n\) over the real field \(\mathbb{R}\). Then we chose our scalar field to be \(\mathbb{R}\). For every real \(r\) in \(\mathbb{R}\) and element \(x=\{x_1,x_2,\dots,x_n\}\) in \(\mathbb{R}\), we define
\begin{equation}\label{prod1}
r\cdot x=r\cdot \{x_1,x_2,\dots,x_n\} = \{r x_1,r x_2,\dots,r x_n\},
\end{equation} where the product \(r x_k\) is the usual product of real numbers. Again from the field properties of the reals under their usual multiplication, the product \ref{prod1} satisfies all the required properties. If we wanted to define a complex vector space out of \(\mathbb{R}^2\) with the \(\cdot\) operation given by \ref{prod1} and \(r\) replaced by a complex number, it would not be possible because the right hand side of \ref{prod1} would have been an \(n\)-tuple of complex numbers, which was not an element of \(\mathbb{R}^n\).
Now that we have defined the operations \(+\) and \(\cdot\), it remains to show that the last two properties are satisfied. The zero vector \(\theta\) is not an element of the given set that is usually defined from the start. It is identified and shown to exist in the set only after the operation of addition has been defined. That is the zero vector depends on the \(+\) operation. It is important to recognize that different \(+\) operations can be assigned to one and the same set; and these different operations lead to different zero elements of the set. Since \(\theta\) depends on the \(+\) operation, the solution to \(\psi+\psi'=\theta\) likewise depends on the given \(+\) operation.
For the \(+\) operation defined in \ref{sum1}, the solution to \(x+\theta=x\) for every \(x\) in \(\mathbb{R}^n\) is the element \(\theta=\{0,0,\dots,0\}\), the \(n\)-tuple of zeros. Since the \(n\)-tuple of zeros belongs to \(\mathbb{R}^n\), the zero vector exists for the given \(+\) operation. We refer to \(\theta\) as the zero vector not because it is an \(n\)-tuple of zeros; it is the zero vector because it is the solution to \(x+\theta=x\) for the given \(+\) operation. Also for every \(x=\{x_1,x_2,\dots,x_n\}\) in \(\mathbb{R}^n\), we have the unique solution \(x'=\{-x_1,-x_2,\dots,-x_n\}\) to \(x+x'=\theta\). We have thus established that all the properties of a vector space are satisfied.
Then the quadruple \((\mathbb{R},\mathbb{R}^n,+,\cdot)\), with \(+\) and \(\cdot\) defined by \ref{sum1} and \ref{prod1} respectively, is a real vector space. From now on whenever we refer to \(\mathbb{R}^n\) as a vector space we will always mean this quadruple. \(\square\)
Example: Let \(\mathbb{C}^n\) denote the ordered \(n\)-tuple of complex numbers. Its elements are denoted by \(z=\{z_1,z_2,\dots,z_n\}\). Following the steps leading to the construction of the real vector space \(\mathbb{R}^n\), it can be shown that the quadruple \((\mathbb{C},\mathbb{C}^n,+,\cdot)\), with the operations\(+\) and \(\cdot\) defined respectively by
\begin{eqnarray}
z+w&=&\{z_1,z_2,\dots,z_n\}+\{w_1,w_2,\dots,w_n\}\nonumber\\
&=&\{z_1+w_1,z_2+w_2,\dots,z_n+w_n\}
\end{eqnarray}\begin{equation}
c\cdot z=c\cdot \{z_1,z_2,\dots,z_n\}=\{c z_1,c z_2,\dots, c z_n\}
\end{equation}for all \(z,w\) in \(\mathbb{C}^n\) and every complex \(c\), is a complex vector space. From now on whenever we refer to \(\mathbb{C}^n\) as a vector space we will mean the complex vector space \((\mathbb{C},\mathbb{C}^n,+,\cdot)\). \(\square\)
Example: Let \(C(\mathbb{R})\) denote the set of continuous, complex valued functions in the real line \(\mathbb{R}\). The function \(\zeta(x)=0\) for all \(x\in\mathbb{R}\) belongs to the set \(C(\mathbb{R})\). The quadruple \((\mathbb{R},C(\mathbb{R}),+,\cdot)\), with the \(+\) and \(\cdot\) operations defined by the pointwise addition and multiplication of functions respectively, is a real vector space with \(\zeta(x)\) as the zero vector. On the other hand, the quadruple \((\mathbb{C},C(\mathbb{R}),+,\cdot)\), with the \(+\) and \(\cdot\) operations defined by the pointwise addition and multiplication of functions respectively, is a complex vector space with \(\zeta(x)\) as the zero vector. \(\square\)
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