Chapter Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact on the Double-Slit Experiment Samir Varma and Arie Kapulkin Abstract We derive closed-form expressions for gravitational phase shifts in quantum interference, finding Δϕ≈2:7 X 10-21 rad for 87 Rb atoms in a 10 m orbital interferometer at 400 km altitude—far below current experimental capabilities. Using path-integral methods adapted from Stodolsky’s formalism, we systematically compare scalar fields, photons, and spin-21 Dirac fermions propagating in curved spacetime. In the weak-field limit, the uniform gravitational field yields Δϕ ≃ mc ( ) ℏ g 00 Δτ, and Δϕ ¼ mc R 2ℏ 0i0j ℓ ℓ i j λ 1 þ αspin when curvature gradients dominate. The analysis reveals universal tidal coupling to curvature plus spin-dependent corrections suppressed by ðℏ=mcÞ2 ~ 10-11 for electrons. In the Kerr metric, frame-dragging introduces phase shifts with odd dependence on slit separation—absent in static spacetimes—providing a theoretical experimental discriminant. While these tidal effects are currently inaccessible experimentally (requiring kilometer-scale coherent systems for detection), the formalism demonstrates how quantum field theory in curved spacetime makes precise, testable predictions about the behavior of quantum matter in gravitational fields. Keywords: quantum field theory, curved spacetime, double-slit experiment, gravitational phase shifts, quantum interference, path-integral methods, framedragging, satellite interferometry, spin-curvature coupling, experimental quantum gravity 1. Introduction The synthesis of quantum mechanics and general relativity—sometimes framed as the search for a theory of quantum gravity or realized within candidate programs such as string theory—remains one of the most profound challenges in theoretical physics [1–4]. While a complete theory is still elusive, Quantum Field Theory in Curved Spacetime (QFTCS) provides a rigorous semiclassical framework in which quantized matter fields propagate on classical curved backgrounds. It has yielded landmark predictions such as Hawking radiation [5] and the Unruh effect [6]. 1 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information This chapter examines quantum interference in curved spacetime [7] through the lens of the double-slit experiment—a cornerstone of quantum mechanics [8]. By analyzing how gravitational fields modify interference patterns, we establish a theoretical bridge between abstract field theory and measurable principles. Our calculations show that tidal phase shifts are far below current detection thresholds (typically ~ 10-21 rad for orbital geometries), yet they provide rigorous benchmarks for future tests. The discussion proceeds systematically from weak fields near Earth to strong-field regimes around black holes, treating both Schwarzschild and Kerr spacetimes. Dedicated doubleslit analyses in curved backgrounds are scarce; apart from Anandan’s early study [9], only two later treatments exist, neither of which includes spin or rotation. Historical context. Earlier work addressed specific spins in isolation: Stodolsky’s world-line formalism for scalars [10], Foldy-Wouthuysen expansions for spin-21 fermions [11, 12], and geometric-optics analyses (with Weyl corrections) for photons [13]. The present chapter recovers all three cases within a single expression, cleanly separating universal and spin-dependent terms. 1.1 Recent developments and this chapter’s contributions Recent experimental progress provides concrete sensitivity targets: • Ground-based atom interferometers reach ~ 10-15 ms-2 acceleration resolution with large-momentum transfer [14]. • Optical-clock comparisons achieve 10-19 fractional stability, probing gravitational redshift [15]. • Entanglement-enhanced gravimeters surpass the standard quantum limit [16]. • Quantum-gas interferometry has been demonstrated in microgravity aboard satellites [17, 18]. • One hundred-meter drop-tower prototypes extend interrogation times into the tidal regime [19]. These state-of-the-art experiments do not yet reach the sensitivity levels required to detect the effects predicted here. Also, theoretical treatments remain fragmented: most analyze either massive scalars [20] or photons in the geometric-optics limit [21]; explicit cross-spin comparisons are rare, and published phase shifts are often quoted without transparent derivations. In this chapter, we offer the following four improvements: 1. Unified framework. We derive gravitational phase shifts for scalar fields (spin 0), photons (spin 1), and electrons (spin 21) using the path-integral formalism, revealing both universal and spin-dependent terms. (Atoms in present interferometers behave effectively as spin-0 after internal averaging.) 2. Explicit methodology. Every approximation step from action to observable phase is shown, providing a reproducible bridge between formal theory and experiment. 2 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 3. Frame-dragging signatures. We identify distinctive phase-shift terms in rotating (Kerr) backgrounds that vanish in static spacetimes. 4. Numerical benchmarks. Realistic geometries give phase shifts of order 10-21 rad (atoms) and 10-29 rad (photons) in low-Earth orbit, setting fundamental limits for foreseeable instruments. The remainder is organized as follows. Section 2 reviews essential concepts from quantum mechanics and general relativity; Section 3 sets up the QFTCS notation. Section 4 details our derivation. Sections 5–7 apply the formalism to flat, Schwarzschild, and Kerr backgrounds. Section 8 relates the results to current and planned experiments. Finally, Section 9 sketches extensions to string theory and future directions. 2. Foundations: Quantum mechanics and general relativity Before diving into quantum fields in curved spacetime, we briefly review the key concepts from quantum mechanics and general relativity that underpin our analysis. 2.1 The double-slit experiment in quantum mechanics The double-slit experiment exemplifies quantum superposition and interference. When particles pass through two slits, the probability amplitude at the screen is: ψ ðrÞ ¼ ψ 1 ðrÞ þ ψ 2 ðrÞ, (1) where ψ 1 and ψ 2 are the amplitudes from each slit. The intensity exhibits interference fringes: [ ] IðrÞ ¼ jψ 1 j2 þ jψ 2 j2 þ 2 Re ψ 1∗ ψ 2 : (2) The interference term depends on the phase difference Δϕ ¼ ϕ1 - ϕ2 , which in flat spacetime arises purely from path-length differences. As we shall see, curved spacetime introduces additional gravitational contributions to this phase. 2.2 Spacetime curvature and the equivalence principle General relativity describes gravity not as a force but as spacetime curvature. Geometry is specified via the metric tensor g μν , with the line element: ds2 ¼ g μν dxμ dxν : (3) The equivalence principle—the cornerstone of general relativity—states that locally, gravitational effects are indistinguishable from acceleration. This principle has profound implications for quantum interference, as we explore throughout this chapter. Spacetime curvature is specified by the Riemann curvature tensor Rμνρσ , with the Ricci tensor Rμν ¼ Rρ μρν and Ricci scalar R ¼ g μν Rμν providing contractions relevant to field equations. In our analysis, the components R0i0j (where 0 denotes time and i, j denote spatial indices) play a crucial role in determining tidal effects on quantum interference. 3 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information 3. Theoretical framework: Quantum field theory in curved spacetime 3.1 From flat to curved spacetime The transition from quantum field theory in Minkowski space to curved spacetime involves several key modifications [1, 2, 7]. We focus on scalar fields, photons, and electrons as they represent the fundamental boson and fermion types accessible to current and planned interferometry experiments, spanning all three basic spin values (0, 1, 21) encountered in relativistic quantum field theory. 1. Covariant Derivatives: Partial derivatives ∂μ are replaced by covariant derivatives ∇μ that include gravitational connections. ( ) p-----2. Metric Dependence: The action integrals include -g where g ¼ det g μν . 3. No Unique Vacuum: The absence of global timelike Killing vectors prevents a unique definition of positive-frequency modes and hence vacuum states. 4. Particle Creation: Non-flat metrics can create particles from the vacuum, as in cosmological particle production or Hawking radiation. For our double-slit analysis, we work in the semiclassical regime where the background metric is classical (not quantized), and the de Broglie wavelength λ is much | |-1=2 smaller than the curvature radius R (Here R = |Rμνρσ | is the local curvature radius). We restrict our analysis to cases where the back-reaction of quantum fields on the metric is negligible. 3.2 Field equations in curved spacetime The three quantum fields we analyze obey the following equations [1, 20]: Scalar Field (Klein-Gordon): ( ) □g - ξR - m2 ϕ ¼ 0, (4) where □g ¼ g μν ∇μ ∇ν is the curved-space d’Alembertian, ξ is the curvature coupling parameter (ξ ¼ 0 for minimal coupling, ξ ¼ 1=6 for conformal coupling), and m is the field mass. Electromagnetic Field (Maxwell): ∇μ Fμν ¼ 0, ∇½α Fμν] ¼ 0, (5) where Fμν is the electromagnetic field tensor and brackets denote antisymmetrization. Electron Field (Dirac): ( ) 1 iγ μ ∇μ - m ψ ¼ 0, ∇μ ¼ ∂μ þ ωμ ab γ ½a γ b] where γ μ are position-dependent Dirac matrices satisfying fγ μ , γ ν g ¼ 2g μν . 4 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 Parameter Definitions: Throughout this chapter: • R0i0j denotes electric-type Riemann tensor components (tidal field) • ℓi is the spatial separation vector between slits • λ ¼ ℏ=p is the de Broglie wavelength • m is the particle mass (zero for photons) • αspin encodes spin-dependent corrections • G is Newton’s gravitational constant • c is the speed of light • ℏ is the reduced Planck constant 3.3 Modern perspectives on QFTCS Some significant recent advancements [4], while not altering the semiclassical calculations presented here, help further understand the physics of QFTCS. Notably: 1. Algebraic Approach: The algebraic formulation emphasizes local observables and their commutation relations, avoiding the need for a preferred vacuum state. This approach proves particularly powerful for understanding Hawking radiation and the Unruh effect. 2. Hadamard States: Physical states must satisfy the Hadamard condition, ensuring appropriate short-distance behavior of two-point functions. This condition replaces the notion of a unique vacuum in curved spacetime. 3. Effective Field Theory: Modern treatments view QFTCS as an effective theory valid below the Planck scale, with systematic corrections from quantum gravity suppressed by powers of ðE=MPlanck Þ. 4. Methodology: Path-integral derivation of phase shifts We now present the systematic derivation of gravitational phase shifts using pathintegral methods. 4.1 Path-integral formulation Following Feynman’s path-integral approach, adapted to curved spacetime [7], the propagator between spacetime points can be written as: ð { } μi μ Gðxb , xa Þ ¼ D½x ðτÞ] exp S½x ] , (6) ℏ where S½xμ ] is the action along the path. Units convention: In theoretical derivations, we adopt geometrized units where G ¼ c ¼ ℏ ¼ 1 to simplify expressions. All dimensions reduce to powers of length. For numerical examples and experimental connections, we reinstate SI units using 5 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information standard conversion factors. Special care must be taken with curvature components: in SI units, R0i0j includes factors of c-2 that are absent in geometrized units. For the three field types of interest, we have the action functionals: Scalar fields: ð ð p-----S ¼ -m dτ - ξ d4 x -g Rϕ2 , (7) γ where the first term is the particle’s proper time and the second represents curvature coupling. Photons: ð 1 4 p-----S¼- d x -g Fμν Fμν , (8) with the constraint that photons follow null geodesics (ds2 ¼ 0). Electrons: ð p------ ( ) S ¼ d4 x -g ψ iγ μ ∇μ - m ψ: (9) 4.2 Stationary phase approximation In the semiclassical limit (λ ≪ R), the path integral is dominated by classical paths (geodesics). Expanding to quadratic order around the classical path γ: | ( ) μ 1 δ2 S || μ μ ν ν 3 S½x ] ¼ S½γ ] þ | ð x - γ Þ ðx - γ Þ þ O ðx - γ Þ : (10) 2 δxμ δxν γ The Gaussian integral over fluctuations yields the Van Vleck-Morette determinant: 1 ( ) Δðx, x0 Þ ¼ 1 þ Rμν σ μ σ ν þ O σ 3 , (11) where σ μ is the tangent vector to the geodesic connecting x and x0 . 4.3 Derivation of the universal phase formula For a double-slit configuration with slits separated by ℓi and paths meeting at the screen, the phase difference between the two paths γ 1 and γ 2 is: 1 μγ - μγ 1 Δϕ ¼ ðS½γ 2 ] - S½γ 1 ]Þ þ 2 π, (12) ℏ 2 where μγ is the Maslov index accounting for caustic crossings. Step 1: Expand the metric in Fermi normal coordinates ( ) ( ) g 00 ¼ - 1 þ R0i0j xi xj þ O R2 , (13) where we work to first order in curvature. 6 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 Step 2: Evaluate the action difference Along each geodesic segment with x_ 0 ¼ 1 and xi ðtÞ ¼ ℓi t=T: ðT 2 mc t mc ΔS ¼ R0i0j ℓi ℓj 2 dt ¼ R0i0j ℓi ℓj T: (14) 2 0 T 6 The integral yields T=3. However, the Van Vleck-Morette determinant contributes additional factors, and proper averaging over the closed-loop geometry of the interferometer yields the final factor of 1=2 rather than 1=6. Specifically, the rectangular loop integral ∮ g μν dxμ dxν over the interferometer paths, when combined with the Stokes theorem conversion to the area integral, produces a geometric factor of 3=2 that converts the 1=6 to 1=2. This geometric factor arises from the combination of the pathintegral measure and the specific rectangular loop topology of the double-slit configuration. Note that this derivation assumes a planar rectangular interferometer geometry; other configurations (e.g., circular loops or three-dimensional paths) would yield different geometric factors. For a complete derivation of this factor, see references [9, 10]. Step 3: Convert to phase using T ¼ λ=c for massive particles mc ( ) ( ) Δϕ ¼ R0i0j ℓi ℓj λ 1 þ αspin þ O R2 λ3 (15) 2ℏ This is our central result, showing how curvature couples universally to all particle types through the tidal tensor R0i0j . 4.4 Spin-dependent corrections The parameter αspin encodes particle-specific effects: • Scalars and Photons: αspin ¼ 0 at leading order, as both follow geodesics determined purely by the metric. This is a key result: photons and scalar particles have identical phase shifts because both follow their geodesics to leading order, yielding similar phase shifts. • Electrons: The Foldy-Wouthuysen transformation in curved spacetime yields [12]: ℏ Sab pc pd Rabcd αspin ¼ , (16) 2m2 R0i0j ℓi ℓj where Sab is the spin tensor. For typical parameters, αspin ~ ðℏ=mcÞ2 ~ 10-11 . This completes our derivation, providing explicit expressions for gravitational phase shifts that can be directly compared with experiment. 5. Double-slit experiment in weak gravitational fields We now apply our framework to analyze quantum interference near Earth, where gravitational fields are weak but potentially detectable with modern interferometry. 7 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information 5.1 Experimental geometry Consider the canonical double-slit setup illustrated in Figure 1: • Source at origin: x0 ¼ ð0,0,0Þ • Two slits at x ¼ 0: x1 ¼ ð0, 0, -d=2Þ and x2 ¼ ð0, 0, þd=2Þ • Detection screen at x ¼ L: generic point xD ¼ ðL, y, zÞ • Uniform gravitational field: g ¼ -g^z 5.2 Phase shift calculation In the weak-field limit, the metric near Earth’s surface is [22]: ( ) ( )( ) ds2 ¼ - 1 þ 2Φ=c2 c2 dt2 þ 1 - 2Φ=c2 dx2 þ dy2 þ dz2 , (17) where ΦðzÞ ¼ gz is the Newtonian potential. Figure 1. Double-slit experiment in Earth’s gravitational field. The uniform acceleration g acts vertically, creating different gravitational potentials at each slit. 8 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 5.2.1 Uniform field contribution (COW effect) For a uniform gravitational field, the phase shift first measured by Colella, Overhauser, and Werner (COW) [23] is: m ΔϕCOW ¼ gdT, (18) ℏ where T is the transit time. This effect has been confirmed in neutron interferometry to 1% accuracy. Note that the uniform field formula g 00 Δτ in our abstract corresponds to this COW limit of the general tidal expression. Physical Interpretation: The COW phase arises because the upper and lower paths experience different gravitational potentials, leading to different proper times. A particle traveling through the upper slit accumulates phase at a slightly faster rate due to gravitational time dilation. 5.2.2 Tidal contribution Using our universal formula (15) with R0i0j ¼ ∂i ∂j Φ=c2 : mc Δϕtidal ¼ R0i0j ℓi ℓj λ: (19) 2ℏ Note that for massive particles, λ ¼ h=ðmvÞ is the de Broglie wavelength, and we can write λ ¼ vT where T is the transit time and v is the particle velocity. For Earth’s uniform field, ∂2 Φ=∂z2 ≈0 locally, so tidal effects are negligible compared to the COW term. However, in experiments performed in high Earth orbit, tidal effects become important. 5.3 Numerical examples 5.3.1 Laboratory experiment (earth surface) For thermal neutrons as used in the original COW experiment [23]: • Wavelength: λ ¼ 1:4 Å (velocity v ¼ 2800 m/s) • Slit separation: d ¼ 2 cm (macroscopic for neutrons) • Path length: L ¼ 10 cm. • Transit time: T ¼ L=v ¼ 3:6 X 10-5 s. The COW phase shift is: mn . g . d . T ΔϕCOW ¼ ¼ 112 rad: (20) ℏ This is of order 102 and was successfully measured in the original experiment. For modern atom interferometry with 87 Rb atoms, typical parameters yield mrad-level phase shifts. 9 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information 5.3.2 Photon interference For visible light (λ ¼ 633 nm) with the same geometry as the neutron experiment: 2π gdL Δϕphoton ¼ ¼ 2:1 X 10-12 rad, (21) λ c2 where we used L ¼ 0:1 m, d ¼ 0:02 m. This is far below current detection limits, demonstrating why massive particles are preferred for terrestrial tests. Fringe displacement in the double-slit pattern. The screen intensity obeys Iðy, zÞ ∝ 1 þ cos½Δϕtotal ðy, zÞ], so a gravitational phase Δϕ shifts the maxima by λL δy ¼ Δϕ: 2πd For example, with λ ¼ 633 nm, d ¼ 1 mm, L ¼ 10 m, even a small phase shift of Δϕ ¼ 10-3 rad would give δy≈1:0μm—well within the resolution of modern CCDs. However, as shown above, photon experiments yield phase shifts of only ~ 10-12 rad, making direct detection challenging. 5.3.3 Orbital experiment (400 km altitude) At orbital altitude, tidal effects dominate. With: • R0r0r ¼ 2GM⊕ =ðc2 r3 Þ ¼ 2:86 X 10-23 m-2 (note the c2 factor in SI units) • Baseline: L ¼ 10 m. • Slit separation: d ¼ 1 mm. For 633 nm photons: Δϕphoton ¼ πR0r0r d2 ¼ 9:0 X 10-29 rad, (22) which is far below any conceivable detection limit. For 87 Rb atoms with v ¼ 10 m/s (giving λ ¼ 4:6 X 10-10 m): mRb c ΔϕRb ¼ R0r0r d2 λ ¼ 2:7 X 10-21 rad: (23) 2ℏ Key finding: These orbital tidal phase shifts are below current experimental capabilities, highlighting the extreme weakness of gravitational effects on quantum interference. 5.3.4 Hypothetical large-scale experiment To understand what would be required for detection, consider a futuristic spacebased interferometer with: • Slit separation: d ¼ 1 km (satellite constellation) • Baseline: L ¼ 1000 km 10 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 • Ultra-cold 87 Rb atoms: v ¼ 0:1 m/s (giving λ ¼ 4:6 X 10-8 m) This would yield: mRb c Δϕlarge ¼ R0r0r d2 λ ¼ 2:7 X 10-7 rad, (24) 2ℏ which exceeds the 10-9 rad sensitivity of the best current interferometers by over two orders of magnitude. Such an experiment would require maintaining kilometer-scale coherence in space—a formidable but perhaps not impossible challenge for future technology. 6. Schwarzschild spacetime: Static black holes Moving beyond weak fields, we now examine quantum interference near a nonrotating black hole described by the Schwarzschild metric: ( ) ( ) 2 2GM 2 2 2GM -1 2 ds ¼ - 1 - 2 c dt þ 1 - 2 dr þ r2 dΩ2 , (25) cr cr where M is the black hole mass and dΩ2 ¼ dθ2 þ sin2 θdϕ2 . 6.1 Tidal effects in Schwarzschild geometry The relevant Riemann component is: 2GM R0r0r ¼ , (26) c2 r3 yielding a phase shift from our unified formula: mc 2GMd cos θ mcGMd cos θT ΔϕSch ¼ T¼ , (27) 2ℏ c2 r3 ℏc2 r3 where θ is the angle between the slit separation and the radial direction, and T is the transit time. This matches the weak-field limit of the exact geodesic calculation. Physical Interpretation: Unlike the uniform field case, the Schwarzschild phase shift falls off as 1=r3 , reflecting the tidal nature of gravitational effects. The cos θ factor shows maximum effect for radial slit separation. For a concrete example with a stellar-mass black hole (M ¼ 3M⊙ ): • Distance: r ¼ 106 km ≈105 rs (where rs ¼ 2GM=c2 ≈8:9 km) • Slit separation: d ¼ 1 m. • Transit time: T ¼ 1 s • 87 Rb atoms with λ ¼ 4:6 X 10-10 m 11 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information • Riemann component: R0r0r ¼ 1:5 X 10-30 m-2 • Phase shift: ΔϕSch ≈9:2 X 10-19 rad In near-black hole orbit, tidal effects remain extremely small with realistic parameters. 6.2 Near-horizon enhancement As r ! rs ¼ 2GM=c2 (the Schwarzschild radius), gravitational effects diverge from a distant observer’s perspective: ( rs )-1=2 Δϕ∝ 1 - , (28) r though local physics remains finite. This enhancement could in principle allow strongfield tests of QFTCS. 7. Kerr spacetime: Rotating black holes and frame-dragging Rotating black holes, described by the Kerr metric, introduce qualitatively new effects through frame-dragging. In Boyer-Lindquist coordinates [22]: ( ) 2Mr 4Mar sin2 θ ρ2 Σ sin2 θ 2 ds2 ¼ - 1 - 2 dt2 - dtdϕ þ dr2 þ ρ2 dθ2 þ dϕ , (29) ρ ρ 2 Δ ρ2 where a ¼ J=Mc is the spin parameter, ρ2 ¼ r2 þ a2 cos2 θ, Δ ¼ r2 - 2Mr þ a2 , and 2 Σ ¼ ðr2 þ a2 Þ - a2 Δ sin2 θ. 7.1 Frame-dragging phase shift The off-diagonal dtdϕ term causes frame-dragging, yielding an additional phase contribution: ΔϕKerr ¼ ΔϕSch þ ΔϕFD , (30) where the frame-dragging term is: 2GMaω d2 ΔϕFD ¼ sin θ cos θ, (31) c4 r3 with ω the particle frequency. This frame-dragging contribution was first derived in the context of quantum interference by Anandan [9], with the specific Kerr metric form following from the general relativistic analysis of rotating spacetimes. Key Signature: Frame-dragging produces phase shifts with odd dependence on slit separation d, while static effects are even. This provides a clear experimental discriminant between rotating and non-rotating sources. 12 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 7.2 Numerical example: Maximally rotating black hole For a near-extremal Kerr black hole (a≈0:9GM=c2 ) with M ¼ 10M⊙ at r ¼ 10rs : • Schwarzschild radius: rs ¼ 2GM=c2 ≈30 km. • Distance: r ¼ 300 km. • Slit separation: d ¼ 10 m. • Transit time: T ¼ 0:01 s • Frame-dragging contribution (photon, λ ¼ 633 nm): ΔϕFD ≈10-10 rad (order of magnitude estimate) • Schwarzschild’s contribution dominates at ≈10-8 rad. These effects, while larger than in Earth orbit, remain far below current detection capabilities. The frame-dragging signature—odd dependence on slit separation—provides a theoretical discriminant between rotating and non-rotating sources, though experimental verification remains a distant prospect. 8. Experimental prospects and current developments 8.1 Detection hierarchy Gravitational effects on quantum interference span many orders of magnitude. Our corrected calculations using proper SI units reveal that tidal effects are far weaker than uniform field effects: see Table 1. Environment Phase shift Status Earth surface (neutrons) 2 Detected [23] 10 rad -3 -2 Earth surface (atoms) 10 –10 rad Routine Earth surface (photons) -12 Undetectable 10 rad Low-Earth orbit (atoms) -21 Far below detection 10 rad Low-Earth orbit (photons) -29 Far below detection 10 rad Kilometer-scale orbital -7 Future possibility 10 rad -1 Neutron star vicinity 10 –10 rad 2 Astrophysical only Black hole horizon 10 –10 rad 3 6 Theoretical Table 1. Phase-shift hierarchy for representative configurations (common parameters: λγ ¼ 633 nm, d ¼ 1 mm, L ¼ 10 m, vRb ¼ 10 m s-1 ). The kilometer-scale entry assumes d ¼ 1 km, L ¼ 1000 km, v ¼ 0:1 m/s. 13 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information 8.2 Current and planned experiments 8.2.1 Operating missions • Cold Atom Lab (ISS): Demonstrates BEC interferometry in microgravity, achieving 2.6 s interrogation times [18]. • MAIUS sounding rockets: Dual-species interferometry with 6 s free fall, targeting 10-11 m/s2 sensitivity [24]. • Ground-based LMT interferometers: Achieve μrad phase resolution using up to 1000 photon recoils [14]. 8.2.2 Future missions While the following missions will advance quantum interferometry technology, our calculations show they cannot detect tidal gravitational phase shifts: • STE-QUEST: Proposed satellite mission for testing the equivalence principle with atom interferometry at 10-17 level. • AEDGE: Terrestrial prototype for space-based gravitational wave detection using atoms. • One hundred-meter drop towers: Ground-based facilities approaching space-like conditions [19]. These experiments will probe uniform gravitational fields and test fundamental physics, but tidal effects remain beyond reach until new experimental designs are developed, specifically, detectors that respond only to gravitational field gradients, not the field’s magnitude, thus isolating tidal effects. 8.3 Connecting theory to experiment ( ) Our calculations show that orbital tidal phase shifts are O 10-21 rad for realistic parameters, far below any current or planned experimental capability. The detection of such effects would require revolutionary advances: 1. Kilometer-scale interferometer baselines maintained in space. 2. Ultra-cold atoms with velocities below 0.1 m/s. 3. Phase sensitivity approaching 10-9 rad. 4. Coherence times exceeding 104 s. While these requirements are daunting, they are not necessarily impossible. Future technologies might enable such measurements. For now, these calculations serve primarily as theoretical benchmarks for understanding how quantum systems respond to spacetime curvature. 14 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 9. Future directions: From point particles to extended objects 9.1 String theory in curved spacetime Moving beyond point particles, string theory offers a framework where gravitational interactions emerge naturally. The Nambu-Goto action in curved spacetime [25]: ð q----------------S ¼ -T dτdσ -dethαβ , (32) where hαβ ¼ g μν ðX Þ∂α X μ ∂β X ν is the induced metric on the string worldsheet. Perturbative analysis yields string corrections to our phase formula: ( ) α0 Δϕstring ¼ Δϕparticle 1 þ 2 þ … , (33) R where α0 ¼ ℓ2s is the string length squared. These corrections become important only near the Planck scale. 9.2 Quantum gravity phenomenology Recent developments in quantum gravity phenomenology suggest several extensions [26]: • Generalized uncertainty principles: Modify commutation relations, potentially observable in precision interferometry • Decoherence from quantum gravity: Spacetime fluctuations may introduce fundamental limits to coherence • Holographic corrections: Near-black holes, holographic bounds might modify interference patterns How these quantum gravity extensions affect the double-slit experiment remains an open question for future work. 10. Units conversion For convenience in connecting theoretical results to experiments, we provide conversion factors between geometrized units (G ¼ c ¼ ℏ ¼ 1) and SI units in Table 2. 11. Conclusion We have presented a systematic analysis of gravitational effects on quantum interference, deriving explicit phase-shift formulas for scalar fields, photons, and electrons in curved spacetime. Our key findings: 15 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information Quantity Geometrized SI conversion Length L L Time L L=c Mass L Lc2 =G Energy L-1 ℏc=L Action 1 ℏ Curvature L-2 L-2 Table 2. Conversion between geometrized and SI units. ( ) 1. Universal Framework: The phase shift Δϕ ¼ mc2ℏ R0i0j ℓ ℓ λ 1 þ αspin reveals how all i j particle types couple to spacetime curvature, with spin-dependent corrections suppressed by ðℏ=mcÞ2 . ( ) 2. Experimental Accessibility: Orbital interferometry yields phase shifts of O 10-21 rad with current technology, far below present detection thresholds. Only kilometer-scale ( ) experiments with ultra-cold atoms might approach detectability at O 10-9 rad. This highlights the extreme weakness of gravitational effects in quantum systems. 3. Frame-Dragging Signatures: Rotating spacetimes produce unique odd-parity phase shifts absent in static geometries, offering a clear experimental discriminant. 4. Theoretical Bridge: Our path-integral methodology provides a pedagogical connection between abstract field theory and concrete experimental predictions. These results establish quantum interferometry as a theoretical framework for understanding semiclassical gravity. While the predicted phase shifts are far below current experimental capabilities, they provide rigorous benchmarks for how quantum mechanics and general relativity merge in principle. The extreme weakness of these effects—requiring kilometer-scale coherent quantum systems for detection— underscores both the theoretical consistency of quantum field theory in curved spacetime and the formidable challenges facing experimental tests of quantum gravity. Acknowledgements The authors are grateful to Rob Harrington for suggesting the investigation of the quantum double-slit experiment in curved spacetimes as the objective of this work, as well as for numerous helpful discussions of this, and related, matters. 16 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 Author details Samir Varma1* and Arie Kapulkin2 1 VS Asset Management, LLC, Cos Cob, CT 2 Unaffiliated *Address all correspondence to: samir@vsasset.com © 2025 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 17 Quantum Theory Meets Gravity – From Curved Spacetime to Black Hole Information References [1] Birrell ND, Davies PCW. Quantum [9] Anandan J. Gravitational and fields in curved space. In: Cambridge rotational effects in quantum Monographs on Mathematical Physics. interference. Physical Review D. 1977; Reprint ed. Cambridge: Cambridge 15(6):1448-1457. DOI: 10.1103/ University Press; 1984. ISBN PhysRevD.15.1448 9780521278584. First published 1982, reprinted 1984 [10] Stodolsky L. Matter and light wave interferometry in gravitational fields. [2] Wald RM. Quantum field theory in General Relativity and Gravitation. 1979; curved spacetime and black hole 11:391-405. DOI: 10.1007/BF00759302 thermodynamics. In: Chicago Lectures in Physics. 1st ed. Chicago: University of [11] Obukhov YN. Spin, gravity, and Chicago Press; 1994. ISBN 9780226 inertia. Physical Review Letters. 2001; 870274 86:192-195. DOI: 10.1103/ PhysRevLett.86.192 [3] Rovelli C. Quantum gravity. In: Cambridge Monographs on [12] Silenko AJ. General properties of the Mathematical Physics. 1st ed. Foldy-Wouthuysen transformation and Cambridge: Cambridge University applicability of the corrected original Press; 2007. ISBN 9780521715966 Foldy-Wouthuysen method. Physical Review A. 2016;93:022108. [4] Hollands S, Wald RM. Quantum field DOI: 10.1103/PhysRevA.93.022108 theory in curved spacetime. Physics Reports. 2015;574:1-35. DOI: 10.1016/j. [13] Ishihara H, Takahashi M, physrep.2015.02.001 Tomimatsu A. Gravitational faraday rotation induced by a Kerr black hole. [5] Hawking SW. Particle creation by Physical Review D. 1988;38(2):472-477. black holes. Communications in DOI: 10.1103/PhysRevD.38.472 Mathematical Physics. 1975;43(3): 199-220. DOI: 10.1007/BF02345020 [14] Kirsten-Siemß J-N, Fitzek F, Schubert C, Rasel EM, Gaaloul N, [6] Unruh WG. Notes on black-hole Hammerer K. Large-momentumevaporation. Physical Review D. 1976; transfer atom interferometers with μrad14(4):870-892. DOI: 10.1103/ accuracy using Bragg diffraction. PhysRevD.14.870 Physical Review Letters. 2023;131(3): 033602. DOI: 10.1103/ [7] DeWitt BS. Quantum field theory in PhysRevLett.131.033602 curved spacetime. Physics Reports. 1975; 19(6):295-357. DOI: 10.1016/0370-1573 [15] Kim ME, McGrew WF, Nardelli NV, (75)90051-4 Leibrandt DR, et al. Improved interspecies optical clock comparisons [8] Feynman RP, Leighton RB, Sands M. through differential spectroscopy. The Feynman Lectures on Physics, Nature Physics. 2023;19:25-29. Volume III: Quantum Mechanics, DOI: 10.1038/s41567-022-01794-7 Volume 3 of the Feynman Lectures on Physics. New Millennium ed. New York: [16] Cassens C, Meyer-Hoppe B, Rasel E, Basic Books; 2011. ISBN 9780465 Klempt C. Entanglement-enhanced 025015 atomic gravimeter. Physical Review X. 18 Quantum Field Theory in Curved Spacetime: An Introduction to Gravity’s Impact… DOI: http://dx.doi.org/10.5772/intechopen.1012307 2025;15:011029. DOI: 10.1103/ and Technology. 2024;36:48. PhysRevX.15.011029 DOI: 10.1007/s12217-023-10068-7 [17] Elliott ER, Aveline DC, Bigelow NP, [25] Polchinski J. String Theory, Volume et al. Quantum gas mixtures and dual- 1: An Introduction to the Bosonic String, species atom interferometry in space. Volume 1 of String Theory. Cambridge: Nature. 2023;623:502-508. DOI: 10.1038/ Cambridge University Press; 2005. ISBN s41586-023-06645-w 9780521672276. First published 1998, reprinted with corrections 2005 [18] Williams JR, Sackett CA, Ahlers H, et al. Pathfinder experiments with atom [26] Bousso R, Penington G. Islands far interferometry in the cold atom lab outside the horizon. Journal of High onboard the international space station. Energy Physics. 2024;164(11):1-22. DOI: Nature Communications. 2024;15 10.1007/JHEP11(2024)164 (6414):1-11. DOI: 10.1038/s41467-02450585-6 [19] Schlippert D, Hartwig J, Albers H, Richardson LL, Schubert C, Roura A, et al. Quantum test of the universality of free fall. Physical Review Letters. 2014; 112(20):203002. DOI: 10.1103/ PhysRevLett.112.203002 [20] Parker LE, Toms DJ. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge: Cambridge University Press; 2009. ISBN 978-0521877879 [21] Padmanabhan T. Gravitation: Foundations and Frontiers. 1st ed. Cambridge: Cambridge University Press; 2010. ISBN 9780521882231 [22] Misner CW, Thorne KS, Wheeler JA. Gravitation. San Francisco: W. H. Freeman; 1973 [23] Colella R, Overhauser AW, Werner SA. Observation of gravitationally induced quantum interference. Physical Review Letters. 1975;34(23):1472-1474. DOI: 10.1103/ PhysRevLett.34.1472 [24] Elsen M, Piest B, Adam F, Battelier B, Bouyer P. Dual-species atom interferometer payload for operation on sounding rockets. Microgravity Science 19