The study of quantum mechanics has been built on a Hilbert space, the complex vector space with inner product <.|.>:VxV--> C with C is a complex vector space dan is complex numbers. Hilbert module is a generalization of Hilbert space, by changing the scalar set become ring with certain conditions so that a Hilbert module. This ring is called the number ring. Hilbert modules that will be used to build the modular foundations of quantum mechanics. Keywords : Scalar, Rings, Hilbert Module, Quantum Mechanics
For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.
Vector spaces are the mathematical structures in which the state vectors of quantum computation live. When endowed with an inner product, a vector space becomes an inner product space. Of particular relevance are complex inner product spaces. An objective of this chapter is to provide a formal definition of a vector space and give several examples. You will see that, in this context, a ‘vector’ is a much more general object than vectors such as those in https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 3 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_2.tif "/> . However, the already familiar operations of vector addition and scalar multiplication are essential ingredients of all vector spaces. We demonstrate how vectors in a vector space can be constructed using a set of vectors called a basis and see how vectors can be written in terms of different bases. Of particular importance are orthonormal bases in which the basis vectors are mutually orthogonal and normalised. Finally, we introduce cosets and explain how a quotient space is constructed.
The quantum analogue of a finite set X (in its role as a configuration space in classical mechanics) is the finite-dimensional Hilbert space
ℓ2(X) , by which we mean the vector space of functions
ψ:X→C , equipped with the inner product
⟨ψ,φ⟩=∑x∈Xψ(x)ϕ(x).This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space
X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of
X under the semi-inner product, and then the state space
S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space
X equipped with a fixed semi-inner product, and then define a physical event at a quantum state
ω∈S(X) to be a projection
P (bounded operator such that
P2=P) in
X satisfying the positivity condition
0≤ω(P)≤1, and a physical quantity at a quantum state
ω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at
ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .