- #1
Husserliana97
- 42
- 5
Lately I've been trying my hand at consistent/deconsistent histories.
To measure my understanding, I'm trying to apply formalism to concrete cases - in this case, the sacrosanct double-slit experiment. I would therefore have liked to submit this little sketch to you.
There are two main stories, defined by projectors as a function of time
(The projector being here a Hermitian operator acting on the Hilbert space, in bijection with a subspace which itself represents a ‘property’ of the system - in other words, the specific value, at a given instant, of a quantity of this system).
So here we have:
$${H}_\text{A} = P_\text{ga}(t_1) P_x(t_2)$$ for the passage through the left-hand slit;
And $${H}_\text{B} = P_\text{dr}(t_1) P_x(t_2)$$ for the right-hand slot.
These two ‘histories’, or sequences of temporally ordered propositions, are said to be (very) coarse-grained, in that only two moments are specified for the location of the particle; these locations are not tracked with arbitrarily high precision, but only as a function of the width of the slits; and finally, the intermediate positions are not specified, any more than the other magnitudes. Such a story therefore covers a large number of possible trajectories (there are, so to speak, an infinite number of ways for the particle to pass through the left-hand slit); it also covers an infinite number of finer-grained trajectories.. This can be formalised using the path integral: for our two stories, the probability amplitude corresponds to
$$\Psi_\text{A}(x) = \int_{C_\text{ga}} e^{\frac{i}{\hbar} S[C_\text{ga}]} \mathcal{D}C_\text{ga}, \quad$$
and
$$\Psi_\text{B}(x) = \int_{C_\text{dr}} e^{\frac{i}{\hbar} S[C_\text{dr}]} \mathcal{D}C_\text{dr}$$
With ##S[C] = \int_{t_1}^{t_2} L(x(t), \dot{x}(t), t) \, dt## the classical action for a given path (=a fine-grained trajectory), and ##L(x(t), \dot{x}(t), t)## the Lagrangian of the system.
As with Feynman, there is interference between the amplitudes associated with each of the particular paths of the integral; in other words, between all the ‘fine grained histories’. But what about between the two coarse grained histories in our case?
Without a measuring device placed at the height of the slits, we will inevitably observe interference patterns. It is therefore the case that the (amplitudes of the) two coarse-grained histories interfere with each other within the configuration space.
##\Psi_\text{ga}(x)## and $\Psi_\text{dr}(x)$ interfere with each other. The probability observed at ##x## on the screen is given by $$P(x) = |\Psi_\text{ga}(x) + \Psi_\text{dr}(x)|^2$$
or:
$$P(x) = |\Psi_\text{A}(x)|^2 + |\Psi_\text{B}(x)|^2 + 2\text{Re}\(\Psi_\text{A}^*(x) \Psi_\text{B}(x)$$
The cross term ##2\text{Re}\(\Psi_\text{A}^*(x) \Psi_\text{B}(x))## reflects the quantum interference between the two stories.
These coarse-grained histories are therefore rigorously inconsistent, just like the fine-grained histories that each covers. The presence of the crossed terms attests to the impossibility of assigning them a classical probability (violation of the additivity property; and for good reason, what we are summing up are the amplitudes, not the probabilities!) And we know that the whole point of interpreting consistent histories is to determine a family of sequences that can be evaluated probabilistically (and therefore, that can be observed while conforming to our bivalent and ‘Boolean’ logic).
Another way of writing this conflict with probabilities is to introduce the ‘Decoherence Functional’, which quantifies the degree of interference between two amplitudes/histories.
$$D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}] = \int_{C_\text{A}} \int_{C_\text{B}} \rho[C_\text{A}, C_\text{B}] e^{\frac{i}{\hbar}(S[C_\text{A}] - S[C_\text{B}])} \mathcal{D}C_\text{A} \mathcal{D}C_\text{B}$$
With ##\rho[C_\text{ga}, C_\text{dr}]## which encodes the correlations between the trajectories ##( C_\text{ga}## and ##( C_\text{dr}##; and the exponential phase term ##e^{\frac{i}{\hbar}(S[C_\text{ga}] - S[C_\text{dr}])}## which introduces oscillations due to the difference in action between the two histories.
It appears that D ≠ 0; our two coarse-grained histories, our two integrals, remain coherent, their phases preserved, and their associated amplitudes combined in the probability finally observed - which will therefore inevitably include cross terms.
So it appears that consistency cannot be achieved in the presence of interference: it will be a question of making the latter negligible for the cause.
If we then introduce a gas at the slits, we can say goodbye to the interference pattern. And in fact then, the functional is written: $$D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}]=\int_{C_\text{ga}}\int_{C_\text{dr}}\rho[C_\text{ga},C_\text{dr}] e^{\frac{i}{\hbar}(S[C_\text{ga}] - S[C_\text{dr}])} e^{-\Gamma[C_\text{ga}, C_\text{dr}]} \mathcal{D}C_\text{ga} \mathcal{D}C_\text{dr}.$$
Here, the decoherence suppression factor
##e^{-\Gamma[C_\text{ga}, C_\text{dr}]}## accounts for the gas-particle interaction and the loss of coherence between the paths.
If, in fact, the gas is introduced, the functional ##D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}]## becomes approximately zero because of the suppression factor ##e^{-\Gamma[C_\text{left}, C_\text{right}]}## This means that our two coarse histories can no longer interfere; and that they will, consequently, allow themselves to be assigned probabilities
(The probability of observing the particle at position ##x## then becomes ##P(x) = P_\text{A}(x) + P_\text{B}(x)##
where: $$P_\text{A}(x) = |\Psi_\text{A}(x)|^2, \quad P_\text{B}(x) = |\Psi_\text{B}(x)|^2$$
So that's what I think I can say about the interpretation of decoherent histories in path integrals. Have I made mistakes? Too many approximations?
To measure my understanding, I'm trying to apply formalism to concrete cases - in this case, the sacrosanct double-slit experiment. I would therefore have liked to submit this little sketch to you.
There are two main stories, defined by projectors as a function of time
(The projector being here a Hermitian operator acting on the Hilbert space, in bijection with a subspace which itself represents a ‘property’ of the system - in other words, the specific value, at a given instant, of a quantity of this system).
So here we have:
$${H}_\text{A} = P_\text{ga}(t_1) P_x(t_2)$$ for the passage through the left-hand slit;
And $${H}_\text{B} = P_\text{dr}(t_1) P_x(t_2)$$ for the right-hand slot.
These two ‘histories’, or sequences of temporally ordered propositions, are said to be (very) coarse-grained, in that only two moments are specified for the location of the particle; these locations are not tracked with arbitrarily high precision, but only as a function of the width of the slits; and finally, the intermediate positions are not specified, any more than the other magnitudes. Such a story therefore covers a large number of possible trajectories (there are, so to speak, an infinite number of ways for the particle to pass through the left-hand slit); it also covers an infinite number of finer-grained trajectories.. This can be formalised using the path integral: for our two stories, the probability amplitude corresponds to
$$\Psi_\text{A}(x) = \int_{C_\text{ga}} e^{\frac{i}{\hbar} S[C_\text{ga}]} \mathcal{D}C_\text{ga}, \quad$$
and
$$\Psi_\text{B}(x) = \int_{C_\text{dr}} e^{\frac{i}{\hbar} S[C_\text{dr}]} \mathcal{D}C_\text{dr}$$
With ##S[C] = \int_{t_1}^{t_2} L(x(t), \dot{x}(t), t) \, dt## the classical action for a given path (=a fine-grained trajectory), and ##L(x(t), \dot{x}(t), t)## the Lagrangian of the system.
As with Feynman, there is interference between the amplitudes associated with each of the particular paths of the integral; in other words, between all the ‘fine grained histories’. But what about between the two coarse grained histories in our case?
Without a measuring device placed at the height of the slits, we will inevitably observe interference patterns. It is therefore the case that the (amplitudes of the) two coarse-grained histories interfere with each other within the configuration space.
##\Psi_\text{ga}(x)## and $\Psi_\text{dr}(x)$ interfere with each other. The probability observed at ##x## on the screen is given by $$P(x) = |\Psi_\text{ga}(x) + \Psi_\text{dr}(x)|^2$$
or:
$$P(x) = |\Psi_\text{A}(x)|^2 + |\Psi_\text{B}(x)|^2 + 2\text{Re}\(\Psi_\text{A}^*(x) \Psi_\text{B}(x)$$
The cross term ##2\text{Re}\(\Psi_\text{A}^*(x) \Psi_\text{B}(x))## reflects the quantum interference between the two stories.
These coarse-grained histories are therefore rigorously inconsistent, just like the fine-grained histories that each covers. The presence of the crossed terms attests to the impossibility of assigning them a classical probability (violation of the additivity property; and for good reason, what we are summing up are the amplitudes, not the probabilities!) And we know that the whole point of interpreting consistent histories is to determine a family of sequences that can be evaluated probabilistically (and therefore, that can be observed while conforming to our bivalent and ‘Boolean’ logic).
Another way of writing this conflict with probabilities is to introduce the ‘Decoherence Functional’, which quantifies the degree of interference between two amplitudes/histories.
$$D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}] = \int_{C_\text{A}} \int_{C_\text{B}} \rho[C_\text{A}, C_\text{B}] e^{\frac{i}{\hbar}(S[C_\text{A}] - S[C_\text{B}])} \mathcal{D}C_\text{A} \mathcal{D}C_\text{B}$$
With ##\rho[C_\text{ga}, C_\text{dr}]## which encodes the correlations between the trajectories ##( C_\text{ga}## and ##( C_\text{dr}##; and the exponential phase term ##e^{\frac{i}{\hbar}(S[C_\text{ga}] - S[C_\text{dr}])}## which introduces oscillations due to the difference in action between the two histories.
It appears that D ≠ 0; our two coarse-grained histories, our two integrals, remain coherent, their phases preserved, and their associated amplitudes combined in the probability finally observed - which will therefore inevitably include cross terms.
So it appears that consistency cannot be achieved in the presence of interference: it will be a question of making the latter negligible for the cause.
If we then introduce a gas at the slits, we can say goodbye to the interference pattern. And in fact then, the functional is written: $$D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}]=\int_{C_\text{ga}}\int_{C_\text{dr}}\rho[C_\text{ga},C_\text{dr}] e^{\frac{i}{\hbar}(S[C_\text{ga}] - S[C_\text{dr}])} e^{-\Gamma[C_\text{ga}, C_\text{dr}]} \mathcal{D}C_\text{ga} \mathcal{D}C_\text{dr}.$$
Here, the decoherence suppression factor
##e^{-\Gamma[C_\text{ga}, C_\text{dr}]}## accounts for the gas-particle interaction and the loss of coherence between the paths.
If, in fact, the gas is introduced, the functional ##D[\mathcal{H}_\text{A}, \mathcal{H}_\text{B}]## becomes approximately zero because of the suppression factor ##e^{-\Gamma[C_\text{left}, C_\text{right}]}## This means that our two coarse histories can no longer interfere; and that they will, consequently, allow themselves to be assigned probabilities
(The probability of observing the particle at position ##x## then becomes ##P(x) = P_\text{A}(x) + P_\text{B}(x)##
where: $$P_\text{A}(x) = |\Psi_\text{A}(x)|^2, \quad P_\text{B}(x) = |\Psi_\text{B}(x)|^2$$
So that's what I think I can say about the interpretation of decoherent histories in path integrals. Have I made mistakes? Too many approximations?