# Continuous quantum measurements and path integrals Lastly, we show that such framework to calculate probability amplitude can be explicitly implemented using Feynman Path Integral The impacts of this framework are fundamental as it is the basis for deriving the formulations that are mathematically equivalent to the laws in traditional quantum mechanics. Wave function is found to be a mathematical tool representing the summation of relational probability amplitude.

Thus, the notion of wave function collapse during measurement is just a consequence of changes of relational properties. On the other hand, when there is change in the entanglement measure, quantum measurement theory is obtained. Although the formulation presented here is mathematically equivalent to the traditional quantum mechanics, the theory presented here provides new understanding on the origin of quantum probability.

It shows that an essence of quantum mechanics is a new set of rules to calculate the measurement probability from an interaction process. The most important outcome of this paper is that quantum mechanics can be constructed with the relational properties among quantum systems as the most fundamental building blocks.

The paper is organized as following. We first clarify the definitions of key terminologies. In the Result Section, we introduce the postulates and frameworks to calculate quantum probability. In the Discussion and Conclusion Section, we provide a comparison between this works and the original RQM theory, discuss the limitations, and summarize the conclusions.

An explicit calculation of the relational probability amplitude using Feynman Path Integral formulation is presented in the Method Section. To avoid potential confusion, it is useful to define several key terms before proceeding. A Quantum System , denoted by symbol S , is an object under study and follows the postulates that will be introduced in next section. An Apparatus , denoted as A , can refer to the measuring devices, the environment that S is interacting with, or the system from which S is created.

It is another quantum system that can interact with S , can acquire or encode information from S. We will strictly follow the assumptions that all systems are quantum systems, including any apparatus. Depending on the selection of observer, the boundary between a system and an apparatus can change. For example, in a measurement setup, the measuring system is an apparatus A , the measured system is S.

In an ideal measurement to measure an observable of S , the apparatus is designed in such a way that at the end of the measurement, the pointer state of A has a distinguishable, one to one correlation with the eigenvalue of the observable of S. The definition of Observer is associated with an apparatus. An observer, denoted as O , is an intelligent entity who can operate and read the pointer variable of the apparatus. This can be a human being, or an artificial intelligent computer.

The distinction between an observer and an apparatus is that an apparatus directly interacts with S , while an observer does not. Whether or not this observer is a quantum system is irrelevant in our formulation. However, there is a restriction that is imposed by the principle of locality. An observer is defined to be physically local to the apparatus he associates with.

This prevents the situation that O can instantaneously read the pointer variable of the apparatus that is space-like separated from O. Receiving the information from A at a speed faster than the speed of light is prohibited. This locality requirement is crucial in resolving the EPR argument 13 , An observer cannot be associated with two or more apparatuses in the same time if these apparatuses are space-like separated.

Given the hypothesis that a quantum system should be described relative to another system, the first question to ask is which another system the description is relative to? A quantum system, at any given time, is either being measured by an apparatus, or interacting with its environment, or is in an isolated environment. It is intuitive to select a reference system that has been previously interacting with the quantum system. A brief review of the traditional quantum measurement theory is helpful since it brings important insights on the meaning of a quantum state.

In the traditional quantum measurement theory proposed by von Neumann 20 , both the quantum system and the measuring apparatus follow the same quantum mechanics laws.

Von Neumann further distinguished two separated measurement stages, Process 1 and Process 2. Mathematically, an ideal measurement process is expressed as. In Process 2, referring to the first arrow in Eq. However, as a combined system they are isolated from interaction with any other system. Process 2 establishes a one to one correlation between the eigenstate of observable of S and the pointer state of A.

After Process 2, there are many possible outcomes to choose from. In the next step which is called Process 1, referring to the second arrow in Eq. An observer knows the outcome of the measurement via the pointer variable of the apparatus. Both systems encode information each other, allowing an observer to infer measurement results of S by reading pointer variable of A.

This observation is also applicable in the case that a quantum system is prepared in a particular state. The term preparation refers to the situation that S is measured by an apparatus, or is prepared with a particular lab setup for instance, a spin half particle passes through a Stern-Gerlach Apparatus , such that its state is explicitly known to an observer. The measuring system, and the environment that S interacts with, are collectively termed as the apparatus A.

Because of the correlation established between S and A during the state preparation process, it is natural to describe the state of S in reference to A. After the state preparation, suppose the interaction Hamiltonian between S and A vanishes, S starts its unitary time evolution. During time evolution, S can still be described in reference to the original apparatus A.

After the interaction finishes, S enters unitary time evolution again. This process can be repeated continuously. The key insight of quantum measurement is that it is a question-and-answer bidirectional process. The measuring system interacts or, disturbs the measured system.

The interaction in turn alters the state of the measuring system. As a result, a correlation is established, allowing the measurement result for S to be inferred from the pointer variable of A. The notion of information in ref. Information exchange between the observed system and the observing apparatus implies change of correlation between the two systems. Correlation is relational and observer-dependent. There are many ways to mathematically define correlation, one of them is introduced in the Result Section.

However, in this paper, we use the notion of information in a more general sense. It can be understood as data that represents values attributed to parameters or properties, or knowledge that describes understanding of physical systems or abstract concepts. Correlation is one type of information. From the examination on the measurement process and the interaction history of a quantum system, we consider a quantum state encodes the information relative to the measuring system or the environment that the system previously interacted with.

In this sense, the quantum state of S is described relative to A. The idea that a quantum state encodes information from previous interactions is also proposed in ref. The information encoded in the quantum state is the complete knowledge an observer can say about S , as it determines the possible outcomes of the next measurement. However, in this paper we consider observer-dependent relational properties more basic. With the clarifications of the key terminologies, we can proceed to introduce the postulates and start the reformulation of quantum mechanics.

Suppose there is no quantum mechanics formulation yet and the goal is to construct a quantum theory that describes a quantum system S in the context of measurement by an apparatus A. We start the reconstruction process by asking a basic question - what are the possible outcomes if one performs a measurement on S using apparatus A? From experimental observations, the measurement yields multiple possible outcomes randomly.

Each potential outcome is obtained with a certain probability. We call each measurement with a distinct outcome a quantum event. They reflect the experimental observations that there can be many distinct measurement outcomes when a variable of S , q , is measured.

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## Path Integrals and Quantum Anomalies - Oxford Scholarship

Here finite number of measurement outcomes is assumed. It is always possible to extend the notation to infinite number of events. With such a representation, the next step is to develop a mathematically framework to calculate the probabilities of possible events. This prediction is carried out before a measurement is actually performed. It is subtle to assign a probability of an outcome from a quantum measurement process. As mentioned earlier, a measurement is an inferring process that depends on the physical interaction between S and A.

The interaction process consists A probing or, disturbing S , and S in the same time altering A. In other words, it is a bidirectional process. Accordingly, p ij is called an interactional probability. This process is true for measurement in either classical or quantum mechanics. The difference is that in classical mechanics, the measurement can be setup such that there is only one measurement outcome deterministically. This means there is a one-to-one correlation between the macroscopic state of measured object and the pointer variable in the measuring device. The probability of this correlation always equals to one.

On the other hand, in quantum mechanics, measurement of a variable q of the quantum system S yields multiple possible results. This is a two-way process, or, a questioning and answering pair in terms of quantum logic approach We expect the framework to calculate p ij should model this bidirectional process. This implies p ij should be derived as a product of two numbers, with each number associated with one direction. Here we assume process of each direction is independent from each other See Note 2 in Supplemental Information. The requirements for the interactional probability p ij can be summarized as following:.

To satisfy requirement 2, we rewrite these two quantities as matrix elements, i. Equation 2 becomes. Thus, requirement 2 is satisfied. We should not consider these variables themselves as probability quantities in the classical sense. They can be complex numbers See Note 3 in Supplemental Information. The direction from S to A is significant here and explicitly called out in the superscript. In this notation, index i is reserved for S while index j is reserved for A. Given Eq. Equation 3 then becomes. Equation 4 can be intuitively understood as this: viewed from A or viewed from S , the probabilistic quantities have the same magnitude, but different in phase.

Physically it ensures there is no preferred choice of S and A in defining the relational variables See Note 4 in Supplemental Information. Given the relation in Eq. The relational matrix R SA gives the complete description of S. It provides a framework to derive the probability of future measurement outcome. We summarize the ideas presented in this section with the following two postulates. Probability of a measurement outcome is calculated by modeling the potential interaction process , i.

There are two important notes. Physical interaction between S and A may cause change of S p , which is the phase of the probability amplitude. So far, we have not yet introduced the notion of quantum state for S. The next step is to derive the properties of S based on R SA. This can be achieved by examining how the probability of measuring S with a particular outcome of variable q is calculated.

We will take a move on mathematical notation before proceeding further. It is more convenient to introduce a Hilbert space for the quantum system S. But there is a limitation for such specification if we wish to calculate the probability of measuring S with a particular outcome of variable q. We do not know that which event will occur to the quantum system A since it is completely probabilistic. The only way an observer can determine which event occurs is to perform actual measurement, or to infer from another system.

Correspondingly, we generalize Eq. The second step utilizes Eq. The probability for a measurement outcome can be calculated by identifying the appropriate alternatives and summing up their weights. The indeterminacy on which event will occur to a quantum system influences the way possible measurement configurations can be arranged. Consequently, it influences how the applicable configurations are counted and then how the probability is calculated See Note 5 in Supplemental Information.

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Such generalized framework of calculating probability is stated by extending Postulate 2. Probability of a measurement outcome is calculated by modeling the potential interaction process. The probability is proportional to the sum of weights from all applicable measurement configurations , where the weight is defined as the product of two relational probability amplitudes corresponding to the configuration. With this framework, the remaining task to calculate the probability is to correctly count the applicable alternatives of measurement configuration.

This task depends on the expected measurement outcome. Some typical cases are analyzed next. To see why this quantity can be considered a probability number, we note that Eq. It can be rewritten as. A notation move is made in the above equation by omitting the superscript in R SA , with the convention that R refers to the relational matrix from S to A. The definition of the wave function naturally emerges out from Eq. In summary,. Adding these terms together, the probability is.

Equation 12 captures the characteristics of superposition. This is a generalization of Eq. Equations 8 and 13 are introduced on the condition that there is no correlation between quantum system S and A. If there is correlation between them, the summation in Eq. But first, from the relational matrix R , how to determine whether there is a correlation between S and A?

Correlation between two quantum system means one can infer the information on one system from information on the other system. The relational variable R ij itself does not quantify an inference relation between S and A. We need to define a different parameter that can quantify the quantum correlation between S and A. The capability of inferring information of a quantum state of one system from information of the other system is defined as entanglement. Since S and A both are quantum systems, they form a bipartite quantum system. Entanglement between two composite system is quantified by an entanglement measure E.

There are many forms of entanglement measure 22 , 23 , the simplest one is the von Neumann entropy. Given the relational matrix R , the von Neumann entropy is defined as following. A change in H R implies there is change of entanglement between S and A. Unless explicitly pointed out, we only consider the situation that S is described by a single relational matrix R. The definition of H R enables us to distinguish different quantum dynamics.

Given a quantum system S and its referencing apparatus A , there are two types of the dynamics between them. In the first type of dynamics, there is no physical interaction and no change in the entanglement measure between S and A. S is not necessarily isolated in the sense that it can still be entangled with A , but the entanglement measure remains unchanged.

This type of dynamics is defined as time evolution. In the second type of dynamics, there is a physical interaction and correlation information exchange between S and A , i. This type of dynamics is defined as quantum operation. Quantum measurement is a special type of quantum operation with a particular outcome. Whether the entanglement measure changes distinguishes a dynamic as either a time evolution or a quantum operation. This is summarized in the following postulate.

In a time evolution process , the entanglement measure of relational matrix is unchanged , while in a quantum operation process , there is change in the entanglement measure of relational matrix. The following theorem allows us to detect whether relational matrix R is entangled. The theorem will be used extensively later. The proof is left to the Method Section. When there is entanglement between S and A , A and S can infer information from each other. The way probability is calculated in Eqs 8 and 12 must be modified because the summation in Eq. More This book provides an introduction to the path integral formulation of quantum field theory and its applications to the analyses of symmetry breaking by the quantization procedure.

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Powered by: Safari Books Online. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes elaborate curlicues , curves in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns to all these amplitudes equal weight but varying phase , or argument of the complex number. Contributions from paths wildly different from the classical trajectory may be suppressed by interference see below. Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is at most quadratic in the momentum.

The path integral formulation of quantum field theory represents the transition amplitude corresponding to the classical correlation function as a weighted sum of all possible histories of the system from the initial to the final state. A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude. One common approach to deriving the path integral formula is to divide the time interval into small pieces.

Once this is done, the Trotter product formula tells us that the noncommutativity of the kinetic and potential energy operators can be ignored. For a particle in a smooth potential, the path integral is approximated by zigzag paths, which in one dimension is a product of ordinary integrals.

For the motion of the particle from position x a at time t a to x b at time t b , the time sequence. Actually L is the classical Lagrangian of the one-dimensional system considered,. Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.

In terms of the wave function in the position representation, the path integral formula reads as follows:. The path integral representation gives the quantum amplitude to go from point x to point y as an integral over all paths. To do this, it is convenient to start without the factor i in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude or Kernel reads:. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:.

The Fourier transform of the Gaussian G is another Gaussian of reciprocal variance:. The Fourier transform gives K , and it is a Gaussian again with reciprocal variance:. The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined.

The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process. The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path.

The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the central limit theorem , which can be interpreted as the first historical evaluation of a statistical path integral.

The probability interpretation gives a natural normalization choice. The path integral should be defined so that. For oscillatory path integrals, ones with an i in the numerator, the time slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. This is closely related to Wick rotation. Then the same convolution argument as before gives the propagation kernel:.

This means that any superposition of K s will also obey the same equation, by linearity. The classical trajectory can be written as. Using the infinite-product representation of the sinc function ,. One may write this propagator in terms of energy eigenstates as. Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator,. Only after replacing the time t by another path-dependent pseudo-time parameter. This is easiest to see by taking a path-integral over infinitesimally separated times. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial.

This separation of the kinetic and potential energy terms in the exponent is essentially the Trotter product formula. The exponential of the action is. The second term is the free particle propagator, corresponding to i times a diffusion process. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case.

An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics. Now x t at each separate time is a separate integration variable. The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:.

This can be shown using the method of stationary phase applied to the propagator. The formulation of the path integral does not make it clear at first sight that the quantities x and p do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present. To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by i :. The quantity x t is fluctuating, and the derivative is defined as the limit of a discrete difference. This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.

But in this case, the difference between the two is not Then f t is a rapidly fluctuating statistical quantity, whose average value is 1, i. The fluctuations of such a quantity can be described by a statistical Lagrangian. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1".

In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation nonholonomic mapping explained here. Sometimes e. This measure cannot be expressed as a functional multiplying the D x measure because they belong to entirely different classes. It is very common in path integrals to perform a Wick rotation from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.

This change is known as a Wick rotation. If we repeat the derivation of the path-integral formula in this setting, we obtain . Note the sign change between this and the normal action, where the potential energy term is negative. The term Euclidean is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.

The Wiener measure, constructed by Norbert Wiener gives a rigorous foundation to Einstein's mathematical model of Brownian motion.

We then have a rigorous version of the Feynman path integral, known as the Feynman—Kac formula : . Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory.

The path integral is just the generalization of the integral above to all quantum mechanical problems—. The connection with statistical mechanics follows. Strictly speaking, though, this is the partition function for a statistical field theory. Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation.

In the canonical formulation, one sees that the unitary evolution operator of a state is given by.

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If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in imaginary time iT is given by. Note, however, that the Euclidean path integral is actually in the form of a classical statistical mechanics model. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation.

The results of a calculation are covariant, but the symmetry is not apparent in intermediate stages.

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If naive field-theory calculations did not produce infinite answers in the continuum limit, this would not have been such a big problem — it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry.

This makes it difficult to extract the physical predictions, which require a careful limiting procedure. The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It extends the Heisenberg-type operator algebra to operator product rules , which are new relations difficult to see in the old formalism. Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. Continuous quantum measurements and path integrals Continuous quantum measurements and path integrals Continuous quantum measurements and path integrals Continuous quantum measurements and path integrals Continuous quantum measurements and path integrals

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