It appears from the description in Sec. 15.3 that a global state containing a single photon in each of the modes i = (q, s) and j = (q′, s′) has the form |ψ〉 = |ψij〉 ⊗ ∏ k 6=i,j |0k〉, (33.1) where |ψij〉 = |1i〉 ⊗ |1j〉 = aˆ†i (0)aˆ†j(0)|0i〉 ⊗ |0j〉. (33.2) A Fock state containing precisely two photons also can be constructed by putting the two quanta into the same mode (say i), that is |ψij〉 = |2i〉 ⊗ |0j〉 = 1√ 2 [ aˆ†i (0) ]2 |0i〉 ⊗ |0j〉. (33.3) By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatic two-photon state, |Φ2〉, can be formed: |Φ2〉 = 1 2 L−3 ∑ i,j φij aˆ † i (0)aˆ † j(0)|0〉. (33.4) The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relation between the two-photon wave functions in direct and reciprocal space having the same structural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitude weight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq. (33.4) shows that only the sum φij + φji can play a physical role, and without loss of generality one therefore may set φji = φij . (33.5) The inner product of |Φ2〉 with itself is readily obtained utilizing that 〈0|aˆkaˆlaˆ†i aˆ†j|0〉 = δkiδlj + δkjδli. (33.6) of Thus, 〈Φ2|Φ2〉 =L −6 ∑ i,j,k,l = L−6 ∑ i,j,k,l φijφ ∗ kl (δkiδlj + δkjδli) = L−6 ∑ i,j φij ( φ∗ij + φ ) = L−6 ∑ i,j |φ∗ij |2. (33.7) The two-photon state is normalized, i.e., 〈Φ2|Φ2〉 = 1, (33.8) provided the weight factors satisfy the condition L−6 ∑ i,j |φij |2 → 1 |φs,s′(q,q′)|2 d 3q (2π)3 d3q′ (2π)3 = 1. (33.9) By acting on |Φ2〉 with the global number operator one obtains Nˆ |Φ2〉 = 1 2 L−3 ∑ i,k,l φklaˆ † i aˆiaˆ = 1 2 L−3 ∑ i,k,l ( aˆ†kaˆi + δik ) aˆ†l |0〉 = 1 2 L−3 ∑ i,k,l φkl [ aˆ†i aˆ † l δik + aˆ ( aˆ†l aˆi + δil )] |0〉 = 1 2 L−3 ∑ ∑ i,k = 1 2 L−3 ∑ i,j (φij + φji) aˆ † i aˆ † j |0〉. (33.10) In view of Eqs. (33.4) and (33.5), one finally has Nˆ |Φ2〉 = 2|Φ2〉. (33.11) The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator with eigenvalue 2, a result which of course follows from the construction in Eq. (33.4).
It appears from the description in Sec. 15.3 that a global state containing a single photon in each of the modes i = (q, s) and j = (q′, s′) has the form |ψ〉 = |ψij〉 ⊗ ∏ k 6=i,j |0k〉, (33.1) where |ψij〉 = |1i〉 ⊗ |1j〉 = aˆ†i (0)aˆ†j(0)|0i〉 ⊗ |0j〉. (33.2) A Fock state containing precisely two photons also can be constructed by putting the two quanta into the same mode (say i), that is |ψij〉 = |2i〉 ⊗ |0j〉 = 1√ 2 [ aˆ†i (0) ]2 |0i〉 ⊗ |0j〉. (33.3) By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatic two-photon state, |Φ2〉, can be formed: |Φ2〉 = 1 2 L−3 ∑ i,j φij aˆ † i (0)aˆ † j(0)|0〉. (33.4) The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relation between the two-photon wave functions in direct and reciprocal space having the same structural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitude weight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq. (33.4) shows that only the sum φij + φji can play a physical role, and without loss of generality one therefore may set φji = φij . (33.5) The inner product of |Φ2〉 with itself is readily obtained utilizing that 〈0|aˆkaˆlaˆ†i aˆ†j|0〉 = δkiδlj + δkjδli. (33.6) of Thus, 〈Φ2|Φ2〉 =L −6 ∑ i,j,k,l = L−6 ∑ i,j,k,l φijφ ∗ kl (δkiδlj + δkjδli) = L−6 ∑ i,j φij ( φ∗ij + φ ) = L−6 ∑ i,j |φ∗ij |2. (33.7) The two-photon state is normalized, i.e., 〈Φ2|Φ2〉 = 1, (33.8) provided the weight factors satisfy the condition L−6 ∑ i,j |φij |2 → 1 |φs,s′(q,q′)|2 d 3q (2π)3 d3q′ (2π)3 = 1. (33.9) By acting on |Φ2〉 with the global number operator one obtains Nˆ |Φ2〉 = 1 2 L−3 ∑ i,k,l φklaˆ † i aˆiaˆ = 1 2 L−3 ∑ i,k,l ( aˆ†kaˆi + δik ) aˆ†l |0〉 = 1 2 L−3 ∑ i,k,l φkl [ aˆ†i aˆ † l δik + aˆ ( aˆ†l aˆi + δil )] |0〉 = 1 2 L−3 ∑ ∑ i,k = 1 2 L−3 ∑ i,j (φij + φji) aˆ † i aˆ † j |0〉. (33.10) In view of Eqs. (33.4) and (33.5), one finally has Nˆ |Φ2〉 = 2|Φ2〉. (33.11) The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator with eigenvalue 2, a result which of course follows from the construction in Eq. (33.4).
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
Starting from two Hilbert spaces H1 and H2 of dimension N1 and N2, respectively, one can construct a larger Hilbert space H = H1 ⊗H2, (32.1) a sort of union of the two. The space H is called the tensor product of H1 and H2. State vectors and operators of these spaces will be denoted by an index, (1) or (2). Associated with a pair of state vectors |φ(1)〉 and |χ(2)〉 belongs a state vector |φ(1)〉 ⊗ |χ(2)〉 ≡ |φ(1)〉|χ(2)〉 ≡ |φ(1)χ(2)〉 (32.2) in the combined Hilbert space. Let us choose basis sets {|un(1)〉} and {|uk(2)〉} for H1 and H2. In terms of these one can express the arbitrary vectors |φ(1)〉 and |χ(2)〉 as follows: |φ(1)〉 = ∑ n an|un(1)〉, (32.3) |χ(2)〉 = ∑ k bk|vk(2)〉. (32.4) In the Hilbert space H one may use a basis {|un(1)⊗ |vk(2)〉〉} for expansion of the tensor product in Eq. (32.2). Thus, |φ(1)〉 ⊗ |χ(2)〉 = ∑ n,k anbk|un(1)〉 ⊗ |vk(2)〉. (32.5) The components of a tensor product state vector in H hence are products, anbk, of components of the two state vectors in the Hilbert spaces H1 and H2. The most general state vector in H, |ψ〉, can in the basis {|un(1)〉 ⊗ |vk(2)〉} be expanded as follows: |ψ〉 = ∑ n,k cnk|un(1)〉 ⊗ |vk(2)〉, (32.6) where the double index nk on the expansion coefficient cnk specifies the vector axes in H. Although an arbitrary vector |ψ〉 in H cannot be written as a single vector product of two vectors in H1 and H2 [Eq. (32.5)], |ψ〉 can always be decomposed into a linear combination of tensor product vectors [Eq. (32.6)]. This result relates to the fact that the elements of an arbitrary matrix {cnk} cannot in general be given as the elements {anbk} of a dyadic product of two vectors with components {an} and {bk}. The scalar product in H of two arbitrary basis vectors, |un(1)〉 ⊗ |vk(2)〉 and |un′(1)〉 ⊗ |vk′(2)〉, is given by 〈un′(1)vk′(2)|un(1)vk(2)〉 = 〈un′(1)|un(1)〉〈vk′ (2)|vk(2)〉 = δnn′δkk′ . (32.7) of The last member of Eq. (32.7) is obtained provided each of the bases {|un(1)〉} and |vk(2)〉 is orthonormal in its respective Hilbert space. If this is the case, the basis in H, i.e., {|un〉⊗ |vk(2)〉}, is orthonormal, too.
In Sec. 15.4 we introduced polychromatic single-photon states in Hilbert space [Eq. (15.80)], |Φ〉 = L− 32 ∑ i φi|Pi〉, i = (q, s), (16.1) by linear superposition of monochromatic plane (P) one-photon states |Pi〉 ≡ |1i〉 = aˆ†i |0〉. (16.2) In the one-photon subspace the |Pi〉 states satisfy the orthonormality, 〈Pj |Pi〉 = δij , (16.3) and completeness ∑ i |Pi〉〈Pi| = 1 (16.4) relations. Instead of taking the |Pi〉’s as a basis set for the expansion of |Φ〉, one may use different sets of one-photon wave-packet states, as we now shall see. The generalization to wave-packet expansion is of importance from a fundamental point of view, and in this chapter we shall realize the usefulness of such an expansion for single-photon correlation studies.
Abstract The present work develops a self-consistent quantum framework based on the algebraC3={1,j,j2∣j3=−1}C_{3} = \{1, j, j^{2} \mid j^{3} = -1\}C3={1,j,j2∣j3=−1}, in which the conventional Schrödinger dynamics, the probabilistic interpretation, and the space–time metric emerge from a single cyclic phase geometry. In this construction the wavefunctionψ=a+jb+j2c\psi = a + jb + j^{2}cψ=a+jb+j2cis not a mere three-component field but the cyclic sum of quantities that are mutually related through a derivative chain: aaa corresponds to configuration, bbb to temporal rate (phase velocity), and ccc to spatial curvature. The differential operator DDD acting as Dψ=jψD\psi = j\psiDψ=jψ closes as D3=−1D^{3} = -1D3=−1, producing the canonical cubic wave equation (D3+1)ψ=0.(D^{3} + 1)\psi = 0.(D3+1)ψ=0. The physical norm is defined by the real trace of the C3C_{3}C3 inner product, ∥ψ∥phys2=a2+b2+c2,\|\psi\|^{2}_{\mathrm{phys}} = a^{2} + b^{2} + c^{2},∥ψ∥phys2=a2+b2+c2, which remains positive definite and conserved under the first-order evolution j ∂tψ+j2κ∇2ψ+(ω0+V)ψ=0.j\,\partial_{t}\psi + j^{2}\kappa\nabla^{2}\psi + (\omega_{0} + V)\psi = 0.j∂tψ+j2κ∇2ψ+(ω0+V)ψ=0. This equation generalizes Schrödinger dynamics by embedding both time and spatial propagation in the derivative cycle generated by jjj and j2j^{2}j2. Its plane-wave solutions satisfy a cubic dispersion relation Ω3=κ3k6+ω03+3κω0k2,\Omega^{3} = \kappa^{3}k^{6} + \omega_{0}^{3} + 3\kappa\omega_{0}k^{2},Ω3=κ3k6+ω03+3κω0k2, which reduces to the standard quadratic form in the weak-phase limit but reveals a new E∝k2/3E \propto k^{2/3}E∝k2/3 regime at strong curvature. Hence, the classical and quantum regimes appear as two asymptotic sectors of the same geometric equation. Hermiticity, measurement, and probability are reformulated within the C3C_{3}C3-Hilbert space: an operator A^=A0+(j−j2)Avis+(j+j2)Ahid\hat{A} = A_{0} + (j - j^{2})A_{\mathrm{vis}} + (j + j^{2})A_{\mathrm{hid}}A^=A0+(j−j2)Avis+(j+j2)Ahid is C3C_{3}C3-Hermitian if ⟨ψ∣A^ϕ⟩3=⟨A^ψ∣ϕ⟩3.\langle \psi | \hat{A}\phi \rangle_{3} = \langle \hat{A}\psi | \phi \rangle_{3}.⟨ψ∣A^ϕ⟩3=⟨A^ψ∣ϕ⟩3. The visible channel j−j2j - j^{2}j−j2 encodes measurable phase differences (time–space curvature contrast), whereas the hidden channel j+j2j + j^{2}j+j2 contains the unobservable but dynamically effective phase sum. The probability measure is then derived from the real projection of the C3C_{3}C3 inner product, ensuring consistency with classical Born statistics in the b,c→0b, c \to 0b,c→0 limit. From the composite fields (a,b,c)(a, b, c)(a,b,c) one can further define emergent metric components N,Ni,γijN, N^{i}, \gamma_{ij}N,Ni,γij, yielding a 3+13+13+1 line element ds2=−N2dt2+γij(dxi+Nidt)(dxj+Njdt),ds^{2} = -N^{2}dt^{2} + \gamma_{ij}(dx^{i} + N^{i}dt)(dx^{j} + N^{j}dt),ds2=−N2dt2+γij(dxi+Nidt)(dxj+Njdt), where temporal lapse and spatial conformal factors follow directly from the visible and hidden curvature densities. Consequently, both the wave dynamics and the geometry of space–time arise from the same algebraic phase structure. The model closes the long-standing gap between the probabilistic and geometric foundations of quantum mechanics by replacing the external metric with an intrinsic C3C_{3}C3 phase metric. Potential experimental probes—such as triple-path interferometry or Ramsey-3 schemes—could isolate the predicted E∼k2/3E \sim k^{2/3}E∼k2/3 dispersion and the visible/hidden phase channels. This formulation provides a unified description in which quantum behavior, the time operator, and curvature are facets of a single cyclic geometry. It offers a mathematically closed, physically interpretable, and experimentally testable framework that extends the Schrödinger equation into the C3C_{3}C3 algebraic domain, opening a possible path toward higher-order phase systems C4,C5,C6C_{4}, C_{5}, C_{6}C4,C5,C6 and the full geometric hierarchy of multicarrier quantum dynamics.
La fonction d'onde du photon en principe et en pratique Pendant ces trois ans, nous nous sommes intéressés à quelques sujets choisis en optique et en électrodynamique quantiques. Le fil rouge de nos interrogations est la fonction d’onde du photon. Les expériences d’optique et d’électrodynamique quantique peuvent-elles être décrites de manière simple, dans l’espace des positions, à l’aide d’une fonction d’onde décrivant le ou les photon(s) impliqués dans l’expérience ? Ce n’est pas entièrement évident :la description usuelle des photons se fait dans l’espace réciproque des vecteurs d’onde. Mais ces expériences gagnent à être décrites par la mécanique ondulatoire en représentation position, comme cela est fait dans les manuels de mécanique quantique pour des situations impliquant des particules massives. De surcroît, une expérience récente[1] a conduit à l’observation de trajectoires de photons uniques à travers un interféromètre à deux fentes d’Young.Pour essayer de décrire formellement ces trajectoires, il est naturel de formuler une mécanique ondulatoire pour les photons. Nous avons donc examiné en détail la construction formelle de la fonction d’onde du photon, un objet qui est resté peu étudié jusqu’aux années 1990. Nous avons également étudié les propriétés de la fonction d’onde du photon en présence de sources, et considéré pour ce faire divers systèmes quantiques ouverts (en interaction). Nous avons vu qu’il existe, en principe, une infinité de possibilités pour le choix de la fonction d’onde du photon.Nous avons mis en évidence un certain nombre de critères sur la base desquels il apparaît que seuls trois choix parmi tous ceux possibles sont intéressants, l’un d’entre eux ramenant à un objet introduit par Glauber [2] pour étudier la détection de la lumière et les corrélations du champ électromagnétique. Nous avons également vu qu’en l’absence de sources l’équation quantique de propagation des photons est formellement identique aux équations de Maxwell.À bas nombre de photons, le formalisme de la fonction d’onde peut se révéler très pratique. Nous avons adapté l’approche aux systèmes en interaction, en nous intéressant dans un premier temps à l’électrodynamique quantique1en cavité [3], en particulier aux expériences réalisées par le groupe de Serge Haroche [4]. Nous avons proposé un modèle simple pour la description des photons dans les cavités d’électrodynamique. À l’aide de ce modèle, et de la fonction d’onde du photon, nous avons étudié la propagation des photons s’échappant de la cavité. Nous avons également construit l’équation maîtresse de Lindblad sans introduire de sauts quantiques non unitaires (voir également [5]). Nous nous sommes enfin intéressés à la question de l’évolution spatiotemporelle d’un photon émis lors d’une désexcitation d’un électron atomique. Après avoir étudié soigneusement la dynamique de la désexcitation de l’électron, notamment aux temps très courts [6, 7], nous nous sommes attachés à décrire, aussi rigoureusement que possible, le champ électromagnétique émis. Celui-ci, de manière surprenante, n’évolue pas causalement. Si cela n’est pas entièrement inattendu au vu du théorème de Hegerfeldt, qui stipule [8] que la causalité est exclue pour les systèmes décrits par un Hamiltonien dont le spectre est borné inférieurement, nous avons identifié [9] deux autres sources de non-causalité, l’une, prédite qualitativement par Shirokov [10], et l’autre, entièrement nouvelle à notre connaissance, et dont la compréhension reste à affiner.