# 1. What needs explaining In the double-slit experiment, single particles produce an interference pattern. When a which-way detector is introduced, interference visibility decreases or vanishes, even if the detector does not macroscopically fire. This effect is often explained using abstract notions such as observation, information, or wavefunction collapse. These explanations obscure the underlying physical mechanism. Here we show that no such concepts are required. The phenomenon follows directly from standard quantum mechanics once detector coupling and partial interference are treated correctly. --- # 2. Interference as Hilbert-space overlap Let the particle state be a coherent superposition of two paths: \[ |\psi\rangle = |\psi_L\rangle + |\psi_R\rangle \] The probability density on the detection screen is: \[ |\psi|^2 = |\psi_L|^2 + |\psi_R|^2 + 2\,\Re\!\left(\psi_L^*\psi_R\right) \] The interference term depends on the **overlap** between the two path amplitudes. Interference does not require the states to be identical, only that their inner product be nonzero. Interference strength is therefore **continuous**, not binary. --- # 3. Phase shifts do not destroy interference A relative phase shift, \[ |\psi\rangle = |\psi_L\rangle + e^{i\phi}|\psi_R\rangle, \] does not reduce interference visibility. It only shifts fringe positions. Thus, phase rotation alone cannot explain the disappearance of interference in which-way experiments. --- # 4. What a detector physically does A detector is a physical system that interacts with the particle. Crucially, it becomes **correlated** with the particle’s path. After interaction, the joint state is: \[ |\Psi\rangle = |\psi_L\rangle|D_L\rangle + |\psi_R\rangle|D_R\rangle \] Here, \(|D_L\rangle\) and \(|D_R\rangle\) are detector states correlated with each path. No assumption of macroscopic triggering or classical thresholds is required. --- # 5. Partial interference and visibility The observable probability distribution is obtained by tracing over detector degrees of freedom: \[ P(x) = |\psi_L|^2 + |\psi_R|^2 + 2\,\Re\!\left(\psi_L^*\psi_R\langle D_L|D_R\rangle\right) \] Define the visibility factor: \[ V = |\langle D_L|D_R\rangle| \in [0,1] \] - \(V = 1\): detector states identical → full interference - \(0 < V < 1\): partial distinguishability → partial interference - \(V = 0\): detector states orthogonal → no interference This continuous behavior matches experimental results. --- # 6. Which-way information without energy exchange The loss of interference does not require net energy transfer. Which-way information can be acquired through entanglement alone, as demonstrated in quantum nondemolition and cavity-based experiments. Therefore, interference loss is not governed by energy minimization, but by state distinguishability. --- # 7. No collapse, no observer, no paradox The disappearance of interference is not caused by: - observation, - consciousness, - knowledge, - or wavefunction collapse. It is caused by **entanglement with uncontrolled degrees of freedom** and the resulting suppression of off-diagonal terms in the reduced density matrix. The process is unitary and reversible in principle. --- # 8. Interpretation The double-slit which-way experiment demonstrates a single principle: > Interference exists exactly to the degree that alternative paths remain > indistinguishable in Hilbert space. This statement is quantitative, testable, and free of metaphysical assumptions. --- # 9. Conclusion The which-way problem contains no mystery. Interference disappears because: 1. Detectors couple to the particle. 2. Coupling correlates path and detector states. 3. Path distinguishability suppresses interference continuously. The phenomenon is fully explained within standard quantum mechanics, without appeal to observation, collapse, or interpretation-dependent language. --- # References - B.-G. Englert, *Fringe Visibility and Which-Way Information*, Phys. Rev. Lett. 77, 2154 (1996). - W. H. Zurek, *Decoherence and the Transition from Quantum to Classical*, Rev. Mod. Phys. 75, 715 (2003). - M. O. Scully, B.-G. Englert, H. Walther, *Quantum Optical Tests of Complementarity*, Nature 351, 111 (1991).